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2-EPT probability density function
In probability theory, a 2-EPT probability density function is a class of probability density functions on the real line. The class contains the density functions of all distributions that have characteristic functions that are strictly proper rational functions (i.e., the degree of the numerator is strictly less than the degree of the denominator).
2-EPT Density Function
Parameters
$({\textbf {A}}_{N},{\textbf {b}}_{N},{\textbf {c}}_{N},{\textbf {A}}_{P},{\textbf {b}}_{P},{\textbf {c}}_{P})$
${\mathfrak {Re}}(\sigma ({\textbf {A}}_{P}))<0$
${\mathfrak {Re}}(\sigma ({\textbf {A}}_{N}))>0$
Support $x\in (-\infty ;+\infty )\!$ ;+\infty )\!}
PDF $f(x)=\left\{{\begin{matrix}{\textbf {c}}_{N}e^{{\textbf {A}}_{N}x}{\textbf {b}}_{N}&{\text{if }}x<0\\[8pt]{\textbf {c}}_{P}e^{{\textbf {A}}_{P}x}{\textbf {b}}_{P}&{\text{if }}x\geq 0\end{matrix}}\right.$
CDF $F(x)=\left\{{\begin{matrix}{\textbf {c}}_{N}{\textbf {A}}_{N}^{-1}e^{{\textbf {A}}_{N}x}{\textbf {b}}_{N}&{\text{if }}x<0\\[8pt]1+{\textbf {c}}_{P}{\textbf {A}}_{P}^{-1}e^{{\textbf {A}}_{P}x}{\textbf {b}}_{P}&{\text{if }}x\geq 0\end{matrix}}\right.$
Mean $-{\textbf {c}}_{N}(-{\textbf {A}}_{N})^{-2}{\textbf {b}}_{N}+{\textbf {c}}_{P}(-{\textbf {A}}_{P})^{-2}{\textbf {b}}_{P}$
CF $-{\textbf {c}}_{N}(Iiu-{\textbf {A}}_{N})^{-1}{\textbf {b}}_{N}+{\textbf {c}}_{P}(Iiu-{\textbf {A}}_{P})^{-1}{\textbf {b}}_{P}$
Definition
A 2-EPT probability density function is a probability density function on $\mathbb {R} $ with a strictly proper rational characteristic function. On either $[0,+\infty )$ or $(-\infty ,0)$ these probability density functions are exponential-polynomial-trigonometric (EPT) functions.
Any EPT density function on $(-\infty ,0)$ can be represented as
$f(x)={\textbf {c}}_{N}e^{{\textbf {A}}_{N}x}{\textbf {b}}_{N},$
where e represents a matrix exponential, $({\textbf {A}}_{N},{\textbf {A}}_{P})$ are square matrices, $({\textbf {b}}_{N},{\textbf {b}}_{P})$ are column vectors and $({\textbf {c}}_{N},{\textbf {c}}_{P})$ are row vectors. Similarly the EPT density function on $[0,-\infty )$ is expressed as
$f(x)={\textbf {c}}_{P}e^{{\textbf {A}}_{P}x}{\textbf {b}}_{P}.$
The parameterization $({\textbf {A}}_{N},{\textbf {b}}_{N},{\textbf {c}}_{N},{\textbf {A}}_{P},{\textbf {b}}_{P},{\textbf {c}}_{P})$ is the minimal realization[1] of the 2-EPT function.
The general class of probability measures on $\mathbb {R} $ with (proper) rational characteristic functions are densities corresponding to mixtures of the pointmass at zero ("delta distribution") and 2-EPT densities. Unlike phase-type and matrix geometric[2] distributions, the 2-EPT probability density functions are defined on the whole real line. It has been shown that the class of 2-EPT densities is closed under many operations and using minimal realizations these calculations have been illustrated for the two-sided framework in Sexton and Hanzon.[3] The most involved operation is the convolution of 2-EPT densities using state space techniques. Much of the work centers on the ability to decompose the rational characteristic function into the sum of two rational functions with poles located in either the open left or open right half plane. The variance-gamma distribution density has been shown to be a 2-EPT density under a parameter restriction and the variance gamma process[4] can be implemented to demonstrate the benefits of adopting such an approach for financial modelling purposes.
It can be shown using Parseval's theorem and an isometry that approximating the discrete time rational transform is equivalent to approximating the 2-EPT density itself in the L-2 Norm sense. The rational approximation software RARL2 is used to approximate the discrete time rational characteristic function of the density.[5]
Applications
Examples of applications include option pricing, computing the Greeks and risk management calculations. Fitting 2-EPT density functions to empirical data has also been considered.[6]
Notes
1. Kailath, T. (1980) Linear Systems, Prentice Hall, 1980
2. Neuts, M. "Probability Distributions of Phase Type", Liber Amicorum Prof. Emeritus H. Florin pages 173-206, Department of Mathematics, University of Louvain, Belgium 1975
3. Sexton, C. and Hanzon,B.,"State Space Calculations for two-sided EPT Densities with Financial Modelling Applications", www.2-ept.com
4. Madan, D., Carr, P., Chang, E. (1998) "The Variance Gamma Process and Option Pricing", European Finance Review 2: 79–105
5. Olivi, M. (2010) "Parametrization of Rational Lossless Matrices with Applications to Linear System Theory", HDR Thesis
6. Sexton, C., Olivi, M., Hanzon, B, "Rational Approximation of Transfer Functions for Non-Negative EPT Densities", Draft paper Archived 2020-12-01 at the Wayback Machine
External links
• 2 - Exponential-Polynomial-Trigonometric (2-EPT) Probability Density Functions Archived 2020-07-08 at the Wayback Machine Website for background and Matlab implementations
Probability distributions (list)
Discrete
univariate
with finite
support
• Benford
• Bernoulli
• beta-binomial
• binomial
• categorical
• hypergeometric
• negative
• Poisson binomial
• Rademacher
• soliton
• discrete uniform
• Zipf
• Zipf–Mandelbrot
with infinite
support
• beta negative binomial
• Borel
• Conway–Maxwell–Poisson
• discrete phase-type
• Delaporte
• extended negative binomial
• Flory–Schulz
• Gauss–Kuzmin
• geometric
• logarithmic
• mixed Poisson
• negative binomial
• Panjer
• parabolic fractal
• Poisson
• Skellam
• Yule–Simon
• zeta
Continuous
univariate
supported on a
bounded interval
• arcsine
• ARGUS
• Balding–Nichols
• Bates
• beta
• beta rectangular
• continuous Bernoulli
• Irwin–Hall
• Kumaraswamy
• logit-normal
• noncentral beta
• PERT
• raised cosine
• reciprocal
• triangular
• U-quadratic
• uniform
• Wigner semicircle
supported on a
semi-infinite
interval
• Benini
• Benktander 1st kind
• Benktander 2nd kind
• beta prime
• Burr
• chi
• chi-squared
• noncentral
• inverse
• scaled
• Dagum
• Davis
• Erlang
• hyper
• exponential
• hyperexponential
• hypoexponential
• logarithmic
• F
• noncentral
• folded normal
• Fréchet
• gamma
• generalized
• inverse
• gamma/Gompertz
• Gompertz
• shifted
• half-logistic
• half-normal
• Hotelling's T-squared
• inverse Gaussian
• generalized
• Kolmogorov
• Lévy
• log-Cauchy
• log-Laplace
• log-logistic
• log-normal
• log-t
• Lomax
• matrix-exponential
• Maxwell–Boltzmann
• Maxwell–Jüttner
• Mittag-Leffler
• Nakagami
• Pareto
• phase-type
• Poly-Weibull
• Rayleigh
• relativistic Breit–Wigner
• Rice
• truncated normal
• type-2 Gumbel
• Weibull
• discrete
• Wilks's lambda
supported
on the whole
real line
• Cauchy
• exponential power
• Fisher's z
• Kaniadakis κ-Gaussian
• Gaussian q
• generalized normal
• generalized hyperbolic
• geometric stable
• Gumbel
• Holtsmark
• hyperbolic secant
• Johnson's SU
• Landau
• Laplace
• asymmetric
• logistic
• noncentral t
• normal (Gaussian)
• normal-inverse Gaussian
• skew normal
• slash
• stable
• Student's t
• Tracy–Widom
• variance-gamma
• Voigt
with support
whose type varies
• generalized chi-squared
• generalized extreme value
• generalized Pareto
• Marchenko–Pastur
• Kaniadakis κ-exponential
• Kaniadakis κ-Gamma
• Kaniadakis κ-Weibull
• Kaniadakis κ-Logistic
• Kaniadakis κ-Erlang
• q-exponential
• q-Gaussian
• q-Weibull
• shifted log-logistic
• Tukey lambda
Mixed
univariate
continuous-
discrete
• Rectified Gaussian
Multivariate
(joint)
• Discrete:
• Ewens
• multinomial
• Dirichlet
• negative
• Continuous:
• Dirichlet
• generalized
• multivariate Laplace
• multivariate normal
• multivariate stable
• multivariate t
• normal-gamma
• inverse
• Matrix-valued:
• LKJ
• matrix normal
• matrix t
• matrix gamma
• inverse
• Wishart
• normal
• inverse
• normal-inverse
• complex
Directional
Univariate (circular) directional
Circular uniform
univariate von Mises
wrapped normal
wrapped Cauchy
wrapped exponential
wrapped asymmetric Laplace
wrapped Lévy
Bivariate (spherical)
Kent
Bivariate (toroidal)
bivariate von Mises
Multivariate
von Mises–Fisher
Bingham
Degenerate
and singular
Degenerate
Dirac delta function
Singular
Cantor
Families
• Circular
• compound Poisson
• elliptical
• exponential
• natural exponential
• location–scale
• maximum entropy
• mixture
• Pearson
• Tweedie
• wrapped
• Category
• Commons
| Wikipedia |
2-EXPTIME
In computational complexity theory, the complexity class 2-EXPTIME (sometimes called 2-EXP) is the set of all decision problems solvable by a deterministic Turing machine in O(22p(n)) time, where p(n) is a polynomial function of n.
In terms of DTIME,
${\mathsf {2{\mbox{-}}EXPTIME}}=\bigcup _{k\in \mathbb {N} }{\mathsf {DTIME}}\left(2^{2^{n^{k}}}\right).$
We know
P ⊆ NP ⊆ PSPACE ⊆ EXPTIME ⊆ NEXPTIME ⊆ EXPSPACE ⊆ 2-EXPTIME ⊆ ELEMENTARY.
2-EXPTIME can also be reformulated as the space class AEXPSPACE, the problems that can be solved by an alternating Turing machine in exponential space. This is one way to see that EXPSPACE ⊆ 2-EXPTIME, since an alternating Turing machine is at least as powerful as a deterministic Turing machine.[1]
2-EXPTIME is one class in a hierarchy of complexity classes with increasingly higher time bounds. The class 3-EXPTIME is defined similarly to 2-EXPTIME but with a triply exponential time bound $2^{2^{2^{n^{k}}}}$. This can be generalized to higher and higher time bounds.
Examples
Examples of algorithms that require at least 2-EXPTIME include:
• Each decision procedure for Presburger arithmetic provably requires at least doubly exponential time[2]
• Computing a Gröbner basis over a field. In the worst case, a Gröbner basis may have a number of elements which is doubly exponential in the number of variables. On the other hand, the worst-case complexity of Gröbner basis algorithms is doubly exponential in the number of variables as well as in the entry size.[3]
• Finding a complete set of associative-commutative unifiers[4]
• Satisfying CTL+ (which is, in fact, 2-EXPTIME-complete)[5]
• Quantifier elimination on real closed fields takes doubly exponential time (see Cylindrical algebraic decomposition).
• Calculating the complement of a regular expression[6]
2-EXPTIME-complete problems
Generalizations of many fully observable games are EXPTIME-complete. These games can be viewed as particular instances of a class of transition systems defined in terms of a set of state variables and actions/events that change the values of the state variables, together with the question of whether a winning strategy exists. A generalization of this class of fully observable problems to partially observable problems lifts the complexity from EXPTIME-complete to 2-EXPTIME-complete.[7]
See also
• Double exponential function
References
1. Christos Papadimitriou, Computational Complexity (1994), ISBN 978-0-201-53082-7. Section 20.1, corollary 3, page 495.
2. Fischer, M. J., and Michael O. Rabin, 1974, ""Super-Exponential Complexity of Presburger Arithmetic. Archived 2006-09-15 at the Wayback Machine" Proceedings of the SIAM-AMS Symposium in Applied Mathematics Vol. 7: 27–41
3. Dubé, Thomas W. (August 1990). "The Structure of Polynomial Ideals and Gröbner Bases". SIAM Journal on Computing. 19 (4): 750–773. doi:10.1137/0219053.
4. Kapur, Deepak; Narendran, Paliath (1992), "Double-exponential complexity of computing a complete set of AC-unifiers", [1992] Proceedings of the Seventh Annual IEEE Symposium on Logic in Computer Science, pp. 11–21, doi:10.1109/LICS.1992.185515, ISBN 0-8186-2735-2, S2CID 206437926.
5. Johannsen, Jan; Lange, Martin (2003), "CTL+ is complete for double exponential time", in Baeten, Jos C. M.; Lenstra, Jan Karel; Parrow, Joachim; Woeginger, Gerhard J. (eds.), Proceedings of the 30th International Colloquium on Automata, Languages and Programming (ICALP 2003) (PDF), Lecture Notes in Computer Science, vol. 2719, Springer-Verlag, pp. 767–775, doi:10.1007/3-540-45061-0_60, ISBN 978-3-540-40493-4, archived from the original (PDF) on 2007-09-30, retrieved 2006-12-22.
6. Gruber, Hermann; Holzer, Markus (2008). "Finite Automata, Digraph Connectivity, and Regular Expression Size" (PDF). Proceedings of the 35th International Colloquium on Automata, Languages and Programming (ICALP 2008). Vol. 5126. pp. 39–50. doi:10.1007/978-3-540-70583-3_4.
7. Jussi Rintanen (2004). "Complexity of Planning with Partial Observability" (PDF). Proceedings of International Conference on Automated Planning and Scheduling. AAAI Press: 345–354.
Important complexity classes
Considered feasible
• DLOGTIME
• AC0
• ACC0
• TC0
• L
• SL
• RL
• NL
• NL-complete
• NC
• SC
• CC
• P
• P-complete
• ZPP
• RP
• BPP
• BQP
• APX
• FP
Suspected infeasible
• UP
• NP
• NP-complete
• NP-hard
• co-NP
• co-NP-complete
• AM
• QMA
• PH
• ⊕P
• PP
• #P
• #P-complete
• IP
• PSPACE
• PSPACE-complete
Considered infeasible
• EXPTIME
• NEXPTIME
• EXPSPACE
• 2-EXPTIME
• ELEMENTARY
• PR
• R
• RE
• ALL
Class hierarchies
• Polynomial hierarchy
• Exponential hierarchy
• Grzegorczyk hierarchy
• Arithmetical hierarchy
• Boolean hierarchy
Families of classes
• DTIME
• NTIME
• DSPACE
• NSPACE
• Probabilistically checkable proof
• Interactive proof system
List of complexity classes
| Wikipedia |
Strict 2-category
In category theory, a strict 2-category is a category with "morphisms between morphisms", that is, where each hom-set itself carries the structure of a category. It can be formally defined as a category enriched over Cat (the category of categories and functors, with the monoidal structure given by product of categories).
For the notion of a weak 2-category, see bicategory.
The concept of 2-category was first introduced by Charles Ehresmann in his work on enriched categories in 1965.[1] The more general concept of bicategory (or weak 2-category), where composition of morphisms is associative only up to a 2-isomorphism, was introduced in 1968 by Jean Bénabou.[2]
Definition
A 2-category C consists of:
• A class of 0-cells (or objects) A, B, ....
• For all objects A and B, a category $\mathbf {C} (A,B)$. The objects $f,g:A\to B$ of this category are called 1-cells and its morphisms $\alpha :f\Rightarrow g$ are called 2-cells; the composition in this category is usually written $\circ $ or $\circ _{1}$ and called vertical composition or composition along a 1-cell.
• For any object A there is a functor from the terminal category (with one object and one arrow) to $\mathbf {C} (A,A)$ that picks out the identity 1-cell idA on A and its identity 2-cell ididA. In practice these two are often denoted simply by A.
• For all objects A, B and C, there is a functor $\circ _{0}\colon \mathbf {C} (B,C)\times \mathbf {C} (A,B)\to \mathbf {C} (A,C)$, called horizontal composition or composition along a 0-cell, which is associative and admits the identity 1 and 2-cells of idA as identities. Here, associativity for $\circ _{0}$ means that horizontally composing $\mathbf {C} (C,D)\times \mathbf {C} (B,C)\times \mathbf {C} (A,B)$ twice to $\mathbf {C} (A,D)$ is independent of which of the two $\mathbf {C} (C,D)\times \mathbf {C} (B,C)$ and $\mathbf {C} (B,C)\times \mathbf {C} (A,B)$ are composed first. The composition symbol $\circ _{0}$ is often omitted, the horizontal composite of 2-cells $\alpha \colon f\Rightarrow g\colon A\to B$ and $\beta \colon f'\Rightarrow g'\colon B\to C$ being written simply as $\beta \alpha \colon f'f\Rightarrow g'g\colon A\to C$.
The 0-cells, 1-cells, and 2-cells terminology is replaced by 0-morphisms, 1-morphisms, and 2-morphisms in some sources[3] (see also Higher category theory).
The notion of 2-category differs from the more general notion of a bicategory in that composition of 1-cells (horizontal composition) is required to be strictly associative, whereas in a bicategory it needs only be associative up to a 2-isomorphism. The axioms of a 2-category are consequences of their definition as Cat-enriched categories:
• Vertical composition is associative and unital, the units being the identity 2-cells idf.
• Horizontal composition is also (strictly) associative and unital, the units being the identity 2-cells ididA on the identity 1-cells idA.
• The interchange law holds; i.e. it is true that for composable 2-cells $\alpha ,\beta ,\gamma ,\delta $
$(\alpha \circ _{0}\beta )\circ _{1}(\gamma \circ _{0}\delta )=(\alpha \circ _{1}\gamma )\circ _{0}(\beta \circ _{1}\delta )$
The interchange law follows from the fact that $\circ _{0}$ is a functor between hom categories. It can be drawn as a pasting diagram as follows:
= = $\circ _{0}$
$\circ _{1}$
Here the left-hand diagram denotes the vertical composition of horizontal composites, the right-hand diagram denotes the horizontal composition of vertical composites, and the diagram in the centre is the customary representation of both. The 2-cell are drawn with double arrows ⇒, the 1-cell with single arrows →, and the 0-cell with points.
Examples
The category Ord (of preordered sets) is a 2-category since preordered sets can easily be interpreted as categories.
Category of small categories
The archetypal 2-category is the category of small categories, with natural transformations serving as 2-morphisms; typically 2-morphisms are given by Greek letters (such as $\alpha $ above) for this reason.
The objects (0-cells) are all small categories, and for all objects A and B the category $\mathbf {C} (A,B)$ is a functor category. In this context, vertical composition is[4] the composition of natural transformations.
Doctrines
In mathematics, a doctrine is simply a 2-category which is heuristically regarded as a system of theories. For example, algebraic theories, as invented by William Lawvere, is an example of a doctrine, as are multi-sorted theories, operads, categories, and toposes.
The objects of the 2-category are called theories, the 1-morphisms $f\colon A\rightarrow B$ are called models of the A in B, and the 2-morphisms are called morphisms between models.
The distinction between a 2-category and a doctrine is really only heuristic: one does not typically consider a 2-category to be populated by theories as objects and models as morphisms. It is this vocabulary that makes the theory of doctrines worth while.
For example, the 2-category Cat of categories, functors, and natural transformations is a doctrine. One sees immediately that all presheaf categories are categories of models.
As another example, one may take the subcategory of Cat consisting only of categories with finite products as objects and product-preserving functors as 1-morphisms. This is the doctrine of multi-sorted algebraic theories. If one only wanted 1-sorted algebraic theories, one would restrict the objects to only those categories that are generated under products by a single object.
Doctrines were discovered by Jonathan Mock Beck.
See also
• n-category
• 2-category at the nLab
References
1. Charles Ehresmann, Catégories et structures, Dunod, Paris 1965.
2. Jean Bénabou, Introduction to bicategories, in Reports of the Midwest Category Seminar, Springer, Berlin, 1967, pp. 1--77.
3. "2-category in nLab". ncatlab.org. Retrieved 2023-02-20.
4. "vertical composition in nLab". ncatlab.org. Retrieved 2023-02-20.
Footnotes
• Generalised algebraic models, by Claudia Centazzo.
External links
• Media related to Strict 2-category at Wikimedia Commons
Category theory
Key concepts
Key concepts
• Category
• Adjoint functors
• CCC
• Commutative diagram
• Concrete category
• End
• Exponential
• Functor
• Kan extension
• Morphism
• Natural transformation
• Universal property
Universal constructions
Limits
• Terminal objects
• Products
• Equalizers
• Kernels
• Pullbacks
• Inverse limit
Colimits
• Initial objects
• Coproducts
• Coequalizers
• Cokernels and quotients
• Pushout
• Direct limit
Algebraic categories
• Sets
• Relations
• Magmas
• Groups
• Abelian groups
• Rings (Fields)
• Modules (Vector spaces)
Constructions on categories
• Free category
• Functor category
• Kleisli category
• Opposite category
• Quotient category
• Product category
• Comma category
• Subcategory
Higher category theory
Key concepts
• Categorification
• Enriched category
• Higher-dimensional algebra
• Homotopy hypothesis
• Model category
• Simplex category
• String diagram
• Topos
n-categories
Weak n-categories
• Bicategory (pseudofunctor)
• Tricategory
• Tetracategory
• Kan complex
• ∞-groupoid
• ∞-topos
Strict n-categories
• 2-category (2-functor)
• 3-category
Categorified concepts
• 2-group
• 2-ring
• En-ring
• (Traced)(Symmetric) monoidal category
• n-group
• n-monoid
• Category
• Outline
• Glossary
| Wikipedia |
2-functor
In mathematics, specifically, in category theory, a 2-functor is a morphism between 2-categories.[1] They may be defined formally using enrichment by saying that a 2-category is exactly a Cat-enriched category and a 2-functor is a Cat-functor.[2]
Explicitly, if C and D are 2-categories then a 2-functor $F\colon C\to D$ consists of
• a function $F\colon {\text{Ob}}C\to {\text{Ob}}D$, and
• for each pair of objects $c,c'\in {\text{Ob}}C$, a functor $F_{c,c'}\colon {\text{Hom}}_{C}(c,c')\to {\text{Hom}}_{D}(Fc,Fc')$
such that each $F_{c,c}$ strictly preserves identity objects and they commute with horizontal composition in C and D.
See [3] for more details and for lax versions.
References
1. Kelly, G.M.; Street, R. (1974). "Review of the elements of 2-categories". Category Seminar. 420: 75--103.
2. G. M. Kelly. Basic concepts of enriched category theory. Reprints in Theory and Applications of Categories, (10), 2005.
3. 2-functor at the nLab
Category theory
Key concepts
Key concepts
• Category
• Adjoint functors
• CCC
• Commutative diagram
• Concrete category
• End
• Exponential
• Functor
• Kan extension
• Morphism
• Natural transformation
• Universal property
Universal constructions
Limits
• Terminal objects
• Products
• Equalizers
• Kernels
• Pullbacks
• Inverse limit
Colimits
• Initial objects
• Coproducts
• Coequalizers
• Cokernels and quotients
• Pushout
• Direct limit
Algebraic categories
• Sets
• Relations
• Magmas
• Groups
• Abelian groups
• Rings (Fields)
• Modules (Vector spaces)
Constructions on categories
• Free category
• Functor category
• Kleisli category
• Opposite category
• Quotient category
• Product category
• Comma category
• Subcategory
Higher category theory
Key concepts
• Categorification
• Enriched category
• Higher-dimensional algebra
• Homotopy hypothesis
• Model category
• Simplex category
• String diagram
• Topos
n-categories
Weak n-categories
• Bicategory (pseudofunctor)
• Tricategory
• Tetracategory
• Kan complex
• ∞-groupoid
• ∞-topos
Strict n-categories
• 2-category (2-functor)
• 3-category
Categorified concepts
• 2-group
• 2-ring
• En-ring
• (Traced)(Symmetric) monoidal category
• n-group
• n-monoid
• Category
• Outline
• Glossary
| Wikipedia |
2-group
In mathematics, a 2-group, or 2-dimensional higher group, is a certain combination of group and groupoid. The 2-groups are part of a larger hierarchy of n-groups. In some of the literature, 2-groups are also called gr-categories or groupal groupoids.
This article is about 2-dimensional higher groups. For p-groups with p = 2, see p-group.
Definition
A 2-group is a monoidal category G in which every morphism is invertible and every object has a weak inverse. (Here, a weak inverse of an object x is an object y such that xy and yx are both isomorphic to the unit object.)
Strict 2-groups
Much of the literature focuses on strict 2-groups. A strict 2-group is a strict monoidal category in which every morphism is invertible and every object has a strict inverse (so that xy and yx are actually equal to the unit object).
A strict 2-group is a group object in a category of categories; as such, they are also called groupal categories. Conversely, a strict 2-group is a category object in the category of groups; as such, they are also called categorical groups. They can also be identified with crossed modules, and are most often studied in that form. Thus, 2-groups in general can be seen as a weakening of crossed modules.
Every 2-group is equivalent to a strict 2-group, although this can't be done coherently: it doesn't extend to 2-group homomorphisms.
Properties
Weak inverses can always be assigned coherently: one can define a functor on any 2-group G that assigns a weak inverse to each object and makes that object an adjoint equivalence in the monoidal category G.
Given a bicategory B and an object x of B, there is an automorphism 2-group of x in B, written AutB(x). The objects are the automorphisms of x, with multiplication given by composition, and the morphisms are the invertible 2-morphisms between these. If B is a 2-groupoid (so all objects and morphisms are weakly invertible) and x is its only object, then AutB(x) is the only data left in B. Thus, 2-groups may be identified with one-object 2-groupoids, much as groups may be identified with one-object groupoids and monoidal categories may be identified with one-object bicategories.
If G is a strict 2-group, then the objects of G form a group, called the underlying group of G and written G0. This will not work for arbitrary 2-groups; however, if one identifies isomorphic objects, then the equivalence classes form a group, called the fundamental group of G and written π1(G). (Note that even for a strict 2-group, the fundamental group will only be a quotient group of the underlying group.)
As a monoidal category, any 2-group G has a unit object IG. The automorphism group of IG is an abelian group by the Eckmann–Hilton argument, written Aut(IG) or π2(G).
The fundamental group of G acts on either side of π2(G), and the associator of G (as a monoidal category) defines an element of the cohomology group H3(π1(G),π2(G)). In fact, 2-groups are classified in this way: given a group π1, an abelian group π2, a group action of π1 on π2, and an element of H3(π1,π2), there is a unique (up to equivalence) 2-group G with π1(G) isomorphic to π1, π2(G) isomorphic to π2, and the other data corresponding.
The element of H3(π1,π2) associated to a 2-group is sometimes called its Sinh invariant, as it was developed by Grothendieck's student Hoàng Xuân Sính.
Fundamental 2-group
Given a topological space X and a point x in that space, there is a fundamental 2-group of X at x, written Π2(X,x). As a monoidal category, the objects are loops at x, with multiplication given by concatenation, and the morphisms are basepoint-preserving homotopies between loops, with these morphisms identified if they are themselves homotopic.
Conversely, given any 2-group G, one can find a unique (up to weak homotopy equivalence) pointed connected space (X,x) whose fundamental 2-group is G and whose homotopy groups πn are trivial for n > 2. In this way, 2-groups classify pointed connected weak homotopy 2-types. This is a generalisation of the construction of Eilenberg–Mac Lane spaces.
If X is a topological space with basepoint x, then the fundamental group of X at x is the same as the fundamental group of the fundamental 2-group of X at x; that is,
$\pi _{1}(X,x)=\pi _{1}(\Pi _{2}(X,x)).\!$
This fact is the origin of the term "fundamental" in both of its 2-group instances.
Similarly,
$\pi _{2}(X,x)=\pi _{2}(\Pi _{2}(X,x)).\!$
Thus, both the first and second homotopy groups of a space are contained within its fundamental 2-group. As this 2-group also defines an action of π1(X,x) on π2(X,x) and an element of the cohomology group H3(π1(X,x),π2(X,x)), this is precisely the data needed to form the Postnikov tower of X if X is a pointed connected homotopy 2-type.
See also
• N-group (category theory)
• Abelian 2-group
References
• Baez, John C.; Lauda, Aaron D. (2004), "Higher-dimensional algebra V: 2-groups" (PDF), Theory and Applications of Categories, 12: 423–491, arXiv:math.QA/0307200
• Baez, John C.; Stevenson, Danny (2009), "The classifying space of a topological 2-group", in Baas, Nils; Friedlander, Eric; Jahren, Bjørn; Østvær, Paul Arne (eds.), Algebraic Topology. The Abel Symposium 2007, Springer, Berlin, pp. 1–31, arXiv:0801.3843
• Brown, Ronald; Higgins, Philip J. (July 1991), "The classifying space of a crossed complex", Mathematical Proceedings of the Cambridge Philosophical Society, 110 (1): 95–120, Bibcode:1991MPCPS.110...95B, doi:10.1017/S0305004100070158
• Brown, Ronald; Higgins, Philip J.; Sivera, Rafael (August 2011), Nonabelian algebraic topology: filtered spaces, crossed complexes, cubical homotopy groupoids, EMS Tracts in Mathematics, vol. 15, arXiv:math/0407275, doi:10.4171/083, ISBN 978-3-03719-083-8, MR 2841564, Zbl 1237.55001
• Pfeiffer, Hendryk (2007), "2-Groups, trialgebras and their Hopf categories of representations", Advances in Mathematics, 212 (1): 62–108, arXiv:math/0411468, doi:10.1016/j.aim.2006.09.014
• Hoàng, Xuân Sính (1975), "Gr-catégories", Thesis, archived from the original on 2015-07-21
• Cegarra, Antonio Martínez; Heredia, Benjamín A.; Remedios, Josué (2012), "Double groupoids and homotopy 2-types", Applied Categorical Structures, 20 (4): 323–378, arXiv:1003.3820, doi:10.1007/s10485-010-9240-1
External links
• 2-group at the nLab
• 2008 Workshop on Categorical Groups at the Centre de Recerca Matemàtica
Category theory
Key concepts
Key concepts
• Category
• Adjoint functors
• CCC
• Commutative diagram
• Concrete category
• End
• Exponential
• Functor
• Kan extension
• Morphism
• Natural transformation
• Universal property
Universal constructions
Limits
• Terminal objects
• Products
• Equalizers
• Kernels
• Pullbacks
• Inverse limit
Colimits
• Initial objects
• Coproducts
• Coequalizers
• Cokernels and quotients
• Pushout
• Direct limit
Algebraic categories
• Sets
• Relations
• Magmas
• Groups
• Abelian groups
• Rings (Fields)
• Modules (Vector spaces)
Constructions on categories
• Free category
• Functor category
• Kleisli category
• Opposite category
• Quotient category
• Product category
• Comma category
• Subcategory
Higher category theory
Key concepts
• Categorification
• Enriched category
• Higher-dimensional algebra
• Homotopy hypothesis
• Model category
• Simplex category
• String diagram
• Topos
n-categories
Weak n-categories
• Bicategory (pseudofunctor)
• Tricategory
• Tetracategory
• Kan complex
• ∞-groupoid
• ∞-topos
Strict n-categories
• 2-category (2-functor)
• 3-category
Categorified concepts
• 2-group
• 2-ring
• En-ring
• (Traced)(Symmetric) monoidal category
• n-group
• n-monoid
• Category
• Outline
• Glossary
| Wikipedia |
2-opt
In optimization, 2-opt is a simple local search algorithm for solving the traveling salesman problem. The 2-opt algorithm was first proposed by Croes in 1958,[1] although the basic move had already been suggested by Flood.[2] The main idea behind it is to take a route that crosses over itself and reorder it so that it does not. A complete 2-opt local search will compare every possible valid combination of the swapping mechanism. This technique can be applied to the traveling salesman problem as well as many related problems. These include the vehicle routing problem (VRP) as well as the capacitated VRP, which require minor modification of the algorithm.
Pseudocode
Visually, one swap looks like:
- A B - - A - B -
× ==>
- C D - - C - D -
In pseudocode, the mechanism by which the 2-opt swap manipulates a given route is as follows. Here v1 and v2 are the first vertices of the edges you wish to swap when traversing through the route:
procedure 2optSwap(route, v1, v2) {
1. take route[0] to route[v1] and add them in order to new_route
2. take route[v1+1] to route[v2] and add them in reverse order to new_route
3. take route[v2+1] to route[start] and add them in order to new_route
return new_route;
}
Here is an example of the above with arbitrary input:
• Example route: A → B → E → D → C → F → G → H → A
• Example parameters: v1=1, v2=4 (assuming starting index is 0)
• Contents of new_route by step:
1. (A → B)
2. A → B → (C → D → E)
3. A → B → C → D → E → (F → G → H → A)
This is the complete 2-opt swap making use of the above mechanism:
repeat until no improvement is made {
best_distance = calculateTotalDistance(existing_route)
start_again:
for (i = 0; i <= number of nodes eligible to be swapped - 1; i++) {
for (j = i + 1; j <= number of nodes eligible to be swapped; j++) {
new_route = 2optSwap(existing_route, i, j)
new_distance = calculateTotalDistance(new_route)
if (new_distance < best_distance) {
existing_route = new_route
best_distance = new_distance
goto start_again
}
}
}
}
Note: If you start/end at a particular node or depot, then you must remove this from the search as an eligible candidate for swapping, as reversing the order will cause an invalid path.
For example, with depot at A:
A → B → C → D → A
Swapping using node[0] and node[2] would yield
C → B → A → D → A
which is not valid (does not leave from A, the depot).
Efficient Implementation
Building the new route and calculating the distance of the new route can be a very expensive operation, usually $O(n)$ where n is the number of vertices in the route. This can be skipped in the symmetrical case (where the distance between two nodes is the same in each opposite direction) by performing an $O(1)$ operation. Since a 2-opt operation involves removing 2 edges and adding 2 different edges we can subtract and add the distances of only those edges.
lengthDelta = - dist(route[v1], route[v1+1]) - dist(route[v2], route[v2+1]) + dist(route[v1+1], route[v2+1]) + dist(route[v1], route[v2])
If lengthDelta is negative that would mean that the new distance after the swap would be smaller. Once it is known that lengthDelta is negative, then we perform a 2-opt swap. This saves us a lot of computation.
Also using squared distances there helps reduce the computation by skipping a square root function call. Since we only care about comparing two distances and not the exact distance, this will help speed things up. It's not much, but it helps with large datasets that have millions of vertices
C++ code
#include <algorithm>
#include <random>
#include <stdio.h>
#include <vector>
using namespace std;
class Point {
public:
int x, y;
Point(int x, int y) {
this->x = x;
this->y = y;
}
Point() {
this->x = 0;
this->y = 0;
}
// Distance between two points squared
inline int dist2(const Point &other) const {
return (x - other.x) * (x - other.x) + (y - other.y) * (y - other.y);
}
};
// Calculate the distance of the whole path (Squared Distances between points)
int pathLengthSq(vector<Point> &path) {
int length = 0;
for (int i = 0; i < path.size(); i++) {
length += path[i].dist2(path[(i + 1) % path.size()]);
}
return length;
}
// Perform a 2-opt swap
void do2Opt(vector<Point> &path, int i, int j) {
reverse(begin(path) + i + 1, begin(path) + j + 1);
}
// Print the path.
void printPath(string pathName, vector<Point> &path) {
printf("%s = [", pathName.c_str());
for (int i = 0; i < path.size(); i++) {
if (i % 10 == 0) {
printf("\n ");
}
if (i < path.size() - 1) {
printf("[ %3d, %3d], ", path[i].x, path[i].y);
}
else {
printf("[ %3d, %3d]", path[i].x, path[i].y);
}
}
printf("\n];\n");
}
// Create a path of length n with random points between 0 and 1000
vector<Point> createRandomPath(int n) {
vector<Point> path;
for (int i = 0; i < n; i++) {
path.push_back(Point(rand() % 1000, rand() % 1000));
}
return path;
}
int main() {
vector<Point> path = createRandomPath(100);
printPath("path1", path);
int curLength = pathLengthSq(path);
int n = path.size();
bool foundImprovement = true;
while (foundImprovement) {
foundImprovement = false;
for (int i = 0; i <= n - 2; i++) {
for (int j = i + 1; j <= n - 1; j++) {
int lengthDelta = -path[i].dist2(path[(i + 1) % n]) - path[j].dist2(path[(j + 1) % n]) + path[i].dist2(path[j]) + path[(i + 1) % n].dist2(path[(j + 1) % n]);
// If the length of the path is reduced, do a 2-opt swap
if (lengthDelta < 0) {
do2Opt(path, i, j);
curLength += lengthDelta;
foundImprovement = true;
}
}
}
}
printPath("path2", path);
return 0;
}
Output
path1 = [
[ 743, 933], [ 529, 262], [ 508, 700], [ 256, 752], [ 119, 256], [ 351, 711], [ 705, 843], [ 393, 108], [ 366, 330], [ 932, 169],
[ 847, 917], [ 868, 972], [ 223, 980], [ 592, 549], [ 169, 164], [ 427, 551], [ 624, 190], [ 920, 635], [ 310, 944], [ 484, 862],
[ 301, 363], [ 236, 710], [ 431, 876], [ 397, 929], [ 491, 675], [ 344, 190], [ 425, 134], [ 30, 629], [ 126, 727], [ 334, 743],
[ 760, 104], [ 620, 749], [ 932, 256], [ 613, 572], [ 509, 490], [ 405, 119], [ 49, 695], [ 719, 327], [ 824, 497], [ 649, 596],
[ 184, 356], [ 245, 93], [ 306, 7], [ 754, 509], [ 665, 352], [ 738, 783], [ 690, 801], [ 337, 330], [ 656, 195], [ 11, 963],
[ 42, 427], [ 968, 106], [ 1, 212], [ 480, 510], [ 571, 658], [ 814, 331], [ 564, 847], [ 625, 197], [ 931, 438], [ 487, 18],
[ 187, 151], [ 179, 913], [ 750, 995], [ 913, 750], [ 134, 562], [ 547, 273], [ 830, 521], [ 557, 140], [ 726, 678], [ 597, 503],
[ 893, 408], [ 238, 988], [ 93, 85], [ 720, 188], [ 746, 211], [ 710, 387], [ 887, 209], [ 103, 668], [ 900, 473], [ 105, 674],
[ 952, 183], [ 787, 370], [ 410, 302], [ 110, 905], [ 996, 400], [ 585, 142], [ 47, 860], [ 731, 924], [ 386, 158], [ 400, 219],
[ 55, 415], [ 874, 682], [ 6, 61], [ 268, 602], [ 470, 365], [ 723, 518], [ 106, 89], [ 130, 319], [ 732, 655], [ 974, 993]
];
path2 = [
[ 743, 933], [ 750, 995], [ 847, 917], [ 868, 972], [ 974, 993], [ 913, 750], [ 920, 635], [ 874, 682], [ 726, 678], [ 732, 655],
[ 830, 521], [ 900, 473], [ 893, 408], [ 931, 438], [ 996, 400], [ 932, 256], [ 952, 183], [ 968, 106], [ 932, 169], [ 887, 209],
[ 760, 104], [ 746, 211], [ 720, 188], [ 656, 195], [ 625, 197], [ 624, 190], [ 585, 142], [ 557, 140], [ 487, 18], [ 306, 7],
[ 245, 93], [ 187, 151], [ 169, 164], [ 106, 89], [ 93, 85], [ 6, 61], [ 1, 212], [ 119, 256], [ 130, 319], [ 184, 356],
[ 301, 363], [ 337, 330], [ 366, 330], [ 410, 302], [ 344, 190], [ 393, 108], [ 405, 119], [ 425, 134], [ 386, 158], [ 400, 219],
[ 529, 262], [ 547, 273], [ 470, 365], [ 509, 490], [ 597, 503], [ 710, 387], [ 665, 352], [ 719, 327], [ 814, 331], [ 787, 370],
[ 824, 497], [ 754, 509], [ 723, 518], [ 649, 596], [ 571, 658], [ 613, 572], [ 592, 549], [ 480, 510], [ 427, 551], [ 268, 602],
[ 134, 562], [ 55, 415], [ 42, 427], [ 30, 629], [ 49, 695], [ 103, 668], [ 105, 674], [ 126, 727], [ 47, 860], [ 11, 963],
[ 110, 905], [ 179, 913], [ 223, 980], [ 238, 988], [ 310, 944], [ 256, 752], [ 236, 710], [ 334, 743], [ 351, 711], [ 491, 675],
[ 508, 700], [ 431, 876], [ 397, 929], [ 484, 862], [ 564, 847], [ 620, 749], [ 690, 801], [ 738, 783], [ 705, 843], [ 731, 924]
];
Visualization
See also
• 3-opt
• local search (optimization)
• Lin–Kernighan heuristic
References
1. G. A. Croes, A method for solving traveling salesman problems. Operations Res. 6 (1958), pp., 791-812.
2. M. M. Flood, The traveling-salesman problem. Operations Res. 4 (1956), pp., 61-75.
• G. A. CROES (1958). A method for solving traveling salesman problems. Operations Res. 6 (1958), pp., 791-812.
• M. M. FLOOD (1956). The traveling-salesman problem. Operations Res. 4 (1956), pp., 61-75.
External links
• The Traveling Salesman Problem: A Case Study in Local Optimization
• Improving Solutions: 2-opt Exchanges
| Wikipedia |
2-valued morphism
In mathematics, a 2-valued morphism[1] is a homomorphism that sends a Boolean algebra B onto the two-element Boolean algebra 2 = {0,1}. It is essentially the same thing as an ultrafilter on B, and, in a different way, also the same things as a maximal ideal of B. 2-valued morphisms have also been proposed as a tool for unifying the language of physics.[2]
2-valued morphisms, ultrafilters and maximal ideals
Suppose B is a Boolean algebra.
• If s : B → 2 is a 2-valued morphism, then the set of elements of B that are sent to 1 is an ultrafilter on B, and the set of elements of B that are sent to 0 is a maximal ideal of B.
• If U is an ultrafilter on B, then the complement of U is a maximal ideal of B, and there is exactly one 2-valued morphism s : B → 2 that sends the ultrafilter to 1 and the maximal ideal to 0.
• If M is a maximal ideal of B, then the complement of M is an ultrafilter on B, and there is exactly one 2-valued morphism s : B → 2 that sends the ultrafilter to 1 and the maximal ideal to 0.
Physics
If the elements of B are viewed as "propositions about some object", then a 2-valued morphism on B can be interpreted as representing a particular "state of that object", namely the one where the propositions of B which are mapped to 1 are true, and the propositions mapped to 0 are false. Since the morphism conserves the Boolean operators (negation, conjunction, etc.), the set of true propositions will not be inconsistent but will correspond to a particular maximal conjunction of propositions, denoting the (atomic) state. (The true propositions form an ultrafilter, the false propositions form a maximal ideal, as mentioned above.)
The transition between two states s1 and s2 of B, represented by 2-valued morphisms, can then be represented by an automorphism f from B to B, such that s2 o f = s1.
The possible states of different objects defined in this way can be conceived as representing potential events. The set of events can then be structured in the same way as invariance of causal structure, or local-to-global causal connections or even formal properties of global causal connections.
The morphisms between (non-trivial) objects could be viewed as representing causal connections leading from one event to another one. For example, the morphism f above leads form event s1 to event s2. The sequences or "paths" of morphisms for which there is no inverse morphism, could then be interpreted as defining horismotic or chronological precedence relations. These relations would then determine a temporal order, a topology, and possibly a metric.
According to,[2] "A minimal realization of such a relationally determined space-time structure can be found". In this model there are, however, no explicit distinctions. This is equivalent to a model where each object is characterized by only one distinction: (presence, absence) or (existence, non-existence) of an event. In this manner, "the 'arrows' or the 'structural language' can then be interpreted as morphisms which conserve this unique distinction".[2]
If more than one distinction is considered, however, the model becomes much more complex, and the interpretation of distinction states as events, or morphisms as processes, is much less straightforward.
References
1. Fleischer, Isidore (1993), "A Boolean formalization of predicate calculus", Algebras and orders (Montreal, PQ, 1991), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 389, Kluwer Acad. Publ., Dordrecht, pp. 193–198, MR 1233791.
2. Heylighen, Francis (1990). A Structural Language for the Foundations of Physics. Brussels: International Journal of General Systems 18, p. 93-112.
External links
• "Representation and Change - A metarepresentational framework for the foundations of physical and cognitive science"
| Wikipedia |
Square root of 5
The square root of 5 is the positive real number that, when multiplied by itself, gives the prime number 5. It is more precisely called the principal square root of 5, to distinguish it from the negative number with the same property. This number appears in the fractional expression for the golden ratio. It can be denoted in surd form as:
${\sqrt {5}}.\,$
Square root of 5
RationalityIrrational
Representations
Decimal2.23606797749978969...
Algebraic form${\sqrt {5}}$
Continued fraction$2+{\cfrac {1}{4+{\cfrac {1}{4+{\cfrac {1}{4+{\cfrac {1}{4+\ddots }}}}}}}}$
Binary10.0011110001101110...
Hexadecimal2.3C6EF372FE94F82C...
It is an irrational algebraic number.[1] The first sixty significant digits of its decimal expansion are:
2.23606797749978969640917366873127623544061835961152572427089... (sequence A002163 in the OEIS).
which can be rounded down to 2.236 to within 99.99% accuracy. The approximation 161/72 (≈ 2.23611) for the square root of five can be used. Despite having a denominator of only 72, it differs from the correct value by less than 1/10,000 (approx. 4.3×10−5). As of January 2022, its numerical value in decimal has been computed to at least 2,250,000,000,000 digits.[2]
Rational approximations
The square root of 5 can be expressed as the continued fraction
$[2;4,4,4,4,4,\ldots ]=2+{\cfrac {1}{4+{\cfrac {1}{4+{\cfrac {1}{4+{\cfrac {1}{4+{{} \atop \displaystyle \ddots }}}}}}}}}.$ (sequence A040002 in the OEIS)
The successive partial evaluations of the continued fraction, which are called its convergents, approach ${\sqrt {5}}$:
${\frac {2}{1}},{\frac {9}{4}},{\frac {38}{17}},{\frac {161}{72}},{\frac {682}{305}},{\frac {2889}{1292}},{\frac {12238}{5473}},{\frac {51841}{23184}},\dots $
Their numerators are 2, 9, 38, 161, … (sequence A001077 in the OEIS), and their denominators are 1, 4, 17, 72, … (sequence A001076 in the OEIS).
Each of these is a best rational approximation of ${\sqrt {5}}$; in other words, it is closer to ${\sqrt {5}}$ than any rational with a smaller denominator.
The convergents, expressed as x/y, satisfy alternately the Pell's equations[3]
$x^{2}-5y^{2}=-1\quad {\text{and}}\quad x^{2}-5y^{2}=1$
When ${\sqrt {5}}$ is approximated with the Babylonian method, starting with x0 = 2 and using xn+1 = 1/2(xn + 5/xn), the nth approximant xn is equal to the 2nth convergent of the continued fraction:
$x_{0}=2.0,\quad x_{1}={\frac {9}{4}}=2.25,\quad x_{2}={\frac {161}{72}}=2.23611\dots ,\quad x_{3}={\frac {51841}{23184}}=2.2360679779\ldots ,\quad x_{4}={\frac {5374978561}{2403763488}}=2.23606797749979\ldots $
The Babylonian method is equivalent to Newton's method for root finding applied to the polynomial $x^{2}-5$. The Newton's method update, $x_{n+1}=x_{n}-f(x_{n})/f'(x_{n})$, is equal to $(x_{n}+5/x_{n})/2$ when $f(x)=x^{2}-5$. The method therefore converges quadratically.
Relation to the golden ratio and Fibonacci numbers
The golden ratio φ is the arithmetic mean of 1 and ${\sqrt {5}}$.[4] The algebraic relationship between ${\sqrt {5}}$, the golden ratio and the conjugate of the golden ratio (Φ = –1/φ = 1 − φ) is expressed in the following formulae:
${\begin{aligned}{\sqrt {5}}&=\varphi -\Phi =2\varphi -1=1-2\Phi \\[5pt]\varphi &={\frac {1+{\sqrt {5}}}{2}}\\[5pt]\Phi &={\frac {1-{\sqrt {5}}}{2}}.\end{aligned}}$
(See the section below for their geometrical interpretation as decompositions of a ${\sqrt {5}}$ rectangle.)
${\sqrt {5}}$ then naturally figures in the closed form expression for the Fibonacci numbers, a formula which is usually written in terms of the golden ratio:
$F(n)={\frac {\varphi ^{n}-(1-\varphi )^{n}}{\sqrt {5}}}.$
The quotient of ${\sqrt {5}}$ and φ (or the product of ${\sqrt {5}}$ and Φ), and its reciprocal, provide an interesting pattern of continued fractions and are related to the ratios between the Fibonacci numbers and the Lucas numbers:[5]
${\begin{aligned}{\frac {\sqrt {5}}{\varphi }}=\Phi \cdot {\sqrt {5}}={\frac {5-{\sqrt {5}}}{2}}&=1.3819660112501051518\dots \\&=[1;2,1,1,1,1,1,1,1,\ldots ]\\[5pt]{\frac {\varphi }{\sqrt {5}}}={\frac {1}{\Phi \cdot {\sqrt {5}}}}={\frac {5+{\sqrt {5}}}{10}}&=0.72360679774997896964\ldots \\&=[0;1,2,1,1,1,1,1,1,\ldots ].\end{aligned}}$
The series of convergents to these values feature the series of Fibonacci numbers and the series of Lucas numbers as numerators and denominators, and vice versa, respectively:
${\begin{aligned}&{1,{\frac {3}{2}},{\frac {4}{3}},{\frac {7}{5}},{\frac {11}{8}},{\frac {18}{13}},{\frac {29}{21}},{\frac {47}{34}},{\frac {76}{55}},{\frac {123}{89}}},\ldots \ldots [1;2,1,1,1,1,1,1,1,\ldots ]\\[8pt]&{1,{\frac {2}{3}},{\frac {3}{4}},{\frac {5}{7}},{\frac {8}{11}},{\frac {13}{18}},{\frac {21}{29}},{\frac {34}{47}},{\frac {55}{76}},{\frac {89}{123}}},\dots \dots [0;1,2,1,1,1,1,1,1,\dots ].\end{aligned}}$
In fact, the limit of the quotient of the $n^{th}$ Lucas number $L_{n}$ and the $n^{th}$ Fibonacci number $F_{n}$ is directly equal to the square root of $5$:
$\lim _{n\to \infty }{\frac {L_{n}}{F_{n}}}={\sqrt {5}}.$
Geometry
Geometrically, ${\sqrt {5}}$ corresponds to the diagonal of a rectangle whose sides are of length 1 and 2, as is evident from the Pythagorean theorem. Such a rectangle can be obtained by halving a square, or by placing two equal squares side by side. This can be used to subdivide a square grid into a tilted square grid with five times as many squares, forming the basis for a subdivision surface.[6] Together with the algebraic relationship between ${\sqrt {5}}$ and φ, this forms the basis for the geometrical construction of a golden rectangle from a square, and for the construction of a regular pentagon given its side (since the side-to-diagonal ratio in a regular pentagon is φ).
Since two adjacent faces of a cube would unfold into a 1:2 rectangle, the ratio between the length of the cube's edge and the shortest distance from one of its vertices to the opposite one, when traversing the cube surface, is ${\sqrt {5}}$. By contrast, the shortest distance when traversing through the inside of the cube corresponds to the length of the cube diagonal, which is the square root of three times the edge.
A rectangle with side proportions 1:${\sqrt {5}}$ is called a root-five rectangle and is part of the series of root rectangles, a subset of dynamic rectangles, which are based on ${\sqrt {1}}$ (= 1), ${\sqrt {2}}$, ${\sqrt {3}}$, ${\sqrt {4}}$ (= 2), ${\sqrt {5}}$... and successively constructed using the diagonal of the previous root rectangle, starting from a square.[7] A root-5 rectangle is particularly notable in that it can be split into a square and two equal golden rectangles (of dimensions Φ × 1), or into two golden rectangles of different sizes (of dimensions Φ × 1 and 1 × φ).[8] It can also be decomposed as the union of two equal golden rectangles (of dimensions 1 × φ) whose intersection forms a square. All this is can be seen as the geometric interpretation of the algebraic relationships between ${\sqrt {5}}$, φ and Φ mentioned above. The root-5 rectangle can be constructed from a 1:2 rectangle (the root-4 rectangle), or directly from a square in a manner similar to the one for the golden rectangle shown in the illustration, but extending the arc of length ${\sqrt {5}}/2$ to both sides.
Trigonometry
Like ${\sqrt {2}}$ and ${\sqrt {3}}$, the square root of 5 appears extensively in the formulae for exact trigonometric constants, including in the sines and cosines of every angle whose measure in degrees is divisible by 3 but not by 15.[9] The simplest of these are
${\begin{aligned}\sin {\frac {\pi }{10}}=\sin 18^{\circ }&={\tfrac {1}{4}}({\sqrt {5}}-1)={\frac {1}{{\sqrt {5}}+1}},\\[5pt]\sin {\frac {\pi }{5}}=\sin 36^{\circ }&={\tfrac {1}{4}}{\sqrt {2(5-{\sqrt {5}})}},\\[5pt]\sin {\frac {3\pi }{10}}=\sin 54^{\circ }&={\tfrac {1}{4}}({\sqrt {5}}+1)={\frac {1}{{\sqrt {5}}-1}},\\[5pt]\sin {\frac {2\pi }{5}}=\sin 72^{\circ }&={\tfrac {1}{4}}{\sqrt {2(5+{\sqrt {5}})}}\,.\end{aligned}}$
As such the computation of its value is important for generating trigonometric tables. Since ${\sqrt {5}}$ is geometrically linked to half-square rectangles and to pentagons, it also appears frequently in formulae for the geometric properties of figures derived from them, such as in the formula for the volume of a dodecahedron.
Diophantine approximations
Hurwitz's theorem in Diophantine approximations states that every irrational number x can be approximated by infinitely many rational numbers m/n in lowest terms in such a way that
$\left|x-{\frac {m}{n}}\right|<{\frac {1}{{\sqrt {5}}\,n^{2}}}$
and that ${\sqrt {5}}$ is best possible, in the sense that for any larger constant than ${\sqrt {5}}$, there are some irrational numbers x for which only finitely many such approximations exist.[10]
Closely related to this is the theorem[11] that of any three consecutive convergents pi/qi, pi+1/qi+1, pi+2/qi+2, of a number α, at least one of the three inequalities holds:
$\left|\alpha -{p_{i} \over q_{i}}\right|<{1 \over {\sqrt {5}}q_{i}^{2}},\qquad \left|\alpha -{p_{i+1} \over q_{i+1}}\right|<{1 \over {\sqrt {5}}q_{i+1}^{2}},\qquad \left|\alpha -{p_{i+2} \over q_{i+2}}\right|<{1 \over {\sqrt {5}}q_{i+2}^{2}}.$
And the ${\sqrt {5}}$ in the denominator is the best bound possible since the convergents of the golden ratio make the difference on the left-hand side arbitrarily close to the value on the right-hand side. In particular, one cannot obtain a tighter bound by considering sequences of four or more consecutive convergents.[11]
Algebra
The ring $\mathbb {Z} [{\sqrt {-5}}]$ contains numbers of the form $a+b{\sqrt {-5}}$, where a and b are integers and ${\sqrt {-5}}$ is the imaginary number $i{\sqrt {5}}$. This ring is a frequently cited example of an integral domain that is not a unique factorization domain.[12] The number 6 has two inequivalent factorizations within this ring:
$6=2\cdot 3=(1-{\sqrt {-5}})(1+{\sqrt {-5}}).\,$
The field $\mathbb {Q} [{\sqrt {-5}}],$ like any other quadratic field, is an abelian extension of the rational numbers. The Kronecker–Weber theorem therefore guarantees that the square root of five can be written as a rational linear combination of roots of unity:
${\sqrt {5}}=e^{{\frac {2\pi }{5}}i}-e^{{\frac {4\pi }{5}}i}-e^{{\frac {6\pi }{5}}i}+e^{{\frac {8\pi }{5}}i}.\,$
Identities of Ramanujan
The square root of 5 appears in various identities discovered by Srinivasa Ramanujan involving continued fractions.[13][14]
For example, this case of the Rogers–Ramanujan continued fraction:
${\cfrac {1}{1+{\cfrac {e^{-2\pi }}{1+{\cfrac {e^{-4\pi }}{1+{\cfrac {e^{-6\pi }}{1+{{} \atop \displaystyle \ddots }}}}}}}}}=\left({\sqrt {\frac {5+{\sqrt {5}}}{2}}}-{\frac {{\sqrt {5}}+1}{2}}\right)e^{\frac {2\pi }{5}}=e^{\frac {2\pi }{5}}\left({\sqrt {\varphi {\sqrt {5}}}}-\varphi \right).$
${\cfrac {1}{1+{\cfrac {e^{-2\pi {\sqrt {5}}}}{1+{\cfrac {e^{-4\pi {\sqrt {5}}}}{1+{\cfrac {e^{-6\pi {\sqrt {5}}}}{1+{{} \atop \displaystyle \ddots }}}}}}}}}=\left({{\sqrt {5}} \over 1+{\sqrt[{5}]{5^{\frac {3}{4}}(\varphi -1)^{\frac {5}{2}}-1}}}-\varphi \right)e^{\frac {2\pi }{\sqrt {5}}}.$
$4\int _{0}^{\infty }{\frac {xe^{-x{\sqrt {5}}}}{\cosh x}}\,dx={\cfrac {1}{1+{\cfrac {1^{2}}{1+{\cfrac {1^{2}}{1+{\cfrac {2^{2}}{1+{\cfrac {2^{2}}{1+{\cfrac {3^{2}}{1+{\cfrac {3^{2}}{1+{{} \atop \displaystyle \ddots }}}}}}}}}}}}}}}.$
See also
• Golden ratio
• Square root
• Square root of 2
• Square root of 3
• Square root of 6
• Square root of 7
References
1. Dauben, Joseph W. (June 1983) Scientific American Georg Cantor and the origins of transfinite set theory. Volume 248; Page 122.
2. Yee, Alexander. "Records Set by y-cruncher".
3. Conrad, Keith. "Pell's Equation II" (PDF). uconn.edu. Retrieved 17 March 2022.
4. Browne, Malcolm W. (July 30, 1985) New York Times Puzzling Crystals Plunge Scientists into Uncertainty. Section: C; Page 1. (Note: this is a widely cited article).
5. Richard K. Guy: "The Strong Law of Small Numbers". American Mathematical Monthly, vol. 95, 1988, pp. 675–712
6. Ivrissimtzis, Ioannis P.; Dodgson, Neil A.; Sabin, Malcolm (2005), "${\sqrt {5}}$-subdivision", in Dodgson, Neil A.; Floater, Michael S.; Sabin, Malcolm A. (eds.), Advances in multiresolution for geometric modelling: Papers from the workshop (MINGLE 2003) held in Cambridge, September 9–11, 2003, Mathematics and Visualization, Berlin: Springer, pp. 285–299, doi:10.1007/3-540-26808-1_16, MR 2112357
7. Kimberly Elam (2001), Geometry of Design: Studies in Proportion and Composition, New York: Princeton Architectural Press, ISBN 1-56898-249-6
8. Jay Hambidge (1967), The Elements of Dynamic Symmetry, Courier Dover Publications, ISBN 0-486-21776-0
9. Julian D. A. Wiseman, "Sin and cos in surds"
10. LeVeque, William Judson (1956), Topics in number theory, Addison-Wesley Publishing Co., Inc., Reading, Mass., MR 0080682
11. Khinchin, Aleksandr Yakovlevich (1964), Continued Fractions, University of Chicago Press, Chicago and London
12. Chapman, Scott T.; Gotti, Felix; Gotti, Marly (2019), "How do elements really factor in $\mathbb {Z} [{\sqrt {-5}}]$?", in Badawi, Ayman; Coykendall, Jim (eds.), Advances in Commutative Algebra: Dedicated to David F. Anderson, Trends in Mathematics, Singapore: Birkhäuser/Springer, pp. 171–195, arXiv:1711.10842, doi:10.1007/978-981-13-7028-1_9, MR 3991169, S2CID 119142526, Most undergraduate level abstract algebra texts use $\mathbb {Z} [{\sqrt {-5}}]$ as an example of an integral domain which is not a unique factorization domain
13. Ramanathan, K. G. (1984), "On the Rogers-Ramanujan continued fraction", Proceedings of the Indian Academy of Sciences, Section A, 93 (2): 67–77, doi:10.1007/BF02840651, ISSN 0253-4142, MR 0813071, S2CID 121808904
14. Eric W. Weisstein, Ramanujan Continued Fractions at MathWorld
Algebraic numbers
• Algebraic integer
• Chebyshev nodes
• Constructible number
• Conway's constant
• Cyclotomic field
• Eisenstein integer
• Gaussian integer
• Golden ratio (φ)
• Perron number
• Pisot–Vijayaraghavan number
• Quadratic irrational number
• Rational number
• Root of unity
• Salem number
• Silver ratio (δS)
• Square root of 2
• Square root of 3
• Square root of 5
• Square root of 6
• Square root of 7
• Doubling the cube
• Twelfth root of two
Mathematics portal
Irrational numbers
• Chaitin's (Ω)
• Liouville
• Prime (ρ)
• Omega
• Cahen
• Logarithm of 2
• Gauss's (G)
• Twelfth root of 2
• Apéry's (ζ(3))
• Plastic (ρ)
• Square root of 2
• Supergolden ratio (ψ)
• Erdős–Borwein (E)
• Golden ratio (φ)
• Square root of 3
• Square root of pi (√π)
• Square root of 5
• Silver ratio (δS)
• Square root of 6
• Square root of 7
• Euler's (e)
• Pi (π)
• Schizophrenic
• Transcendental
• Trigonometric
| Wikipedia |
2.5D (visual perception)
2.5D is an effect in visual perception. It is the construction of an apparently three-dimensional environment from 2D retinal projections.[1][2][3] While the result is technically 2D, it allows for the illusion of depth. It is easier for the eye to discern the distance between two items than the depth of a single object in the view field.[4] Computers can use 2.5D to make images of human faces look lifelike.[5]
Perception of the physical environment is limited because of visual and cognitive issues. The visual problem is the lack of objects in three-dimensional space to be imaged with the same projection, while the cognitive problem is that the perception of an object depends on the observer.[2] David Marr found that 2.5D has visual projection constraints that exist because "parts of images are always (deformed) discontinuities in luminance".[2] Therefore, in reality, the observer does not see all of the surroundings but constructs a viewer-centred three-dimensional view.
Blur perception
A primary aspect of the human visual system is blur perception. Blur perception plays a key role in focusing on near or far objects. Retinal focus patterns are critical in blur perception as these patterns are composed of distal and proximal retinal defocus. Depending on the object's distance and motion from the observer, these patterns contain a balance and an imbalance of focus in both directions.[6]
Human blur perceptions involve blur detection and blur discrimination. Blur goes across the central and peripheral retina. The model has a changing nature and a model of blur perception is in dioptric space while in near viewing. The model can have suggestions according to depth perception and accommodating control.[6]
Digital synthesis
The 2.5D range data is obtained by a range imaging system, and the 2D colour image is taken by a regular camera. These two data sets are processed individually and then combined. Human face output can be lifelike and be manipulated by computer graphics tools. In facial recognition, this tool can provide complete facial details.[7] Three different approaches are used in colour edge detection:
• Analyze each colour independently and then combine them;
• Analyze the 'luminance channel' and use the chrominance channels to aid other decisions;
• Treat the colour image as a vector field, and use derivatives of the vector field as the colour gradient.
2.5D (visual perception) offers an automatic approach to making human face models. It analyzes a range data set and a color perception image. The sources are analyzed separately to identify the anatomical sites of features, craft the geometry of the face and produce a volumetric facial model.[8] The two methods of feature localization are a deformable template and chromatic edge detection.[9]
The range imaging system contains benefits such as having problems become avoided through contact measurement. This would be easier to keep and is much safer and other advantages also include how it is needless to calibrate when measuring an object of similarity, and enabling the machine to be appropriate for facial range data measurement.[5]
2.5D datasets can be conveniently represented in a framework of boxels (axis-aligned, non-overlapping boxes). They can be used to directly represent objects in the scene or as bounding volumes. Leonidas J. Guibas and Yuan Yao's work showed that axis-aligned disjoint rectangles can be ordered into four orders so that any ray meets them in one of the four orders. This is applicable to boxels and has shown that four different partitionings of the boxels into ordered sequences of disjoint sets exist. These are called antichains and enable boxels in one antichain to occlude boxels in subsequent antichains. The expected runtime for the antichain partitioning is O(n log n), where n is the number of boxels. This partitioning can be used for the efficient implementation of virtual drive-throughs and ray tracing.[10]
A person's perception of a visual representation involves three successive stages
• The 2D representation component yields an approximate description.
• The 2.5D representation component adds visuospatial properties to the object's surface.
• The 3D representation component adds depth and volume.[11]
Applications
Uses for a human face model include medicine, identification, computer animation, and intelligent coding.[12]
References
1. MacEachren, Alan M. (2008). "GVIS Facilitating Visual Thinking". How maps work : representation, visualization, and design. Guilford Press. pp. 355–458. ISBN 978-1-57230-040-8. OCLC 698536855.
2. Watt, R.J. and B.J. Rogers. "Human Vision and Cognitive Science." In Cognitive Psychology Research Directions in Cognitive Science: European Perspectives Vol. 1, edited by Alan Baddeley and Niels Ole Bernsen, 10–12. East Sussex: Lawrence Erlbaum Associates, 1989.
3. Wood, Jo; Kirschenbauer, Sabine; Dollner, Jurgen; Lopes, Adriano; Bodum, Lars (2005). "Using 3D in Visualization". Exploring geovisualization. International Cartographic Association/Elsevier. ISBN 0-08-044531-4. OCLC 988646788.
4. Read, JCA; Phillipson, GP; Serrano-Pedraza, I; Milner, AD; Parker, AJ (2010). "Stereoscopic Vision on the Absence of the Lateral Occipital Cortex". PLOS ONE. 5 (9): e12608. Bibcode:2010PLoSO...512608R. doi:10.1371/journal.pone.0012608. PMC 2935377. PMID 20830303.
5. Kang, C.-Y.; Chen, Y.-S.; Hsu, W.-H. (1993). "Mapping a lifelike 2.5 D human face via an automatic approach". Proceedings of IEEE Conference on Computer Vision and Pattern Recognition. IEEE Comput. Soc. Press. pp. 611–612. doi:10.1109/cvpr.1993.341061. ISBN 0-8186-3880-X. S2CID 10957251.
6. Ciuffreda, Kenneth J.; Wang, Bin; Vasudevan, Balamurali (April 2007). "Conceptual model of human blur perception". Vision Research. 47 (9): 1245–1252. doi:10.1016/j.visres.2006.12.001. PMID 17223154. S2CID 10320448.
7. Chii-Yuan, Kang (January 1, 1994). "Automatic approach to mapping a lifelike 2.5D human face". Image and Vision Computing. 12 (1): 5–14. doi:10.1016/0262-8856(94)90051-5.
8. Chii-Yuan, Kang; Yung-Sheng, Chen; Wen-Hsing, Hsu (1994). "Automatic approach to mapping a lifelike 2.5D human face". Image and Vision Computing. 12: 5–14. doi:10.1016/0262-8856(94)90051-5.
9. Automatic identification of human faces by the 3-D shape of surfaces – using vertices of B spline surface Syst. & Computers in Japan, v.Vol 22 (No 7), p. 96, 1991, Abe T et al.
10. Goldschmidt, Nir; Gordon, Dan (November 2008). "The BOXEL framework for 2.5D data with applications to virtual drivethroughs and ray tracing". Computational Geometry. 41 (3): 167–187. doi:10.1016/j.comgeo.2007.09.003. ISSN 0925-7721.
11. Bouaziz, Serge; Magnan, Annie (January 2007). "Contribution of the visual perception and graphic production systems to the copying of complex geometrical drawings: A developmental study". Cognitive Development. 22 (1): 5–15. doi:10.1016/j.cogdev.2006.10.002. ISSN 0885-2014.
12. Kang, C. Y.; Chen, Y. S.; Hsu, W. H. (1994). "Automatic approach to mapping a lifelike 2.5d human face ". Image and Vision Computing. 12 (1): 5–14. doi:10.1016/0262-8856(94)90051-5.
| Wikipedia |
25 great circles of the spherical octahedron
In geometry, the 25 great circles of the spherical octahedron is an arrangement of 25 great circles in octahedral symmetry.[1] It was first identified by Buckminster Fuller and is used in construction of geodesic domes.
Construction
The 25 great circles can be seen in 3 sets: 12, 9, and 4, each representing edges of a polyhedron projected onto a sphere. Nine great circles represent the edges of a disdyakis dodecahedron, the dual of a truncated cuboctahedron. Four more great circles represent the edges of a cuboctahedron, and the last twelve great circles connect edge-centers of the octahedron to centers of other triangles.
See also
• 31 great circles of the spherical icosahedron
References
1. "Fig. 450.11B".
• Edward Popko, Divided Spheres: Geodesics and the Orderly Subdivision of the Sphere, 2012, pp 21–22.
• Vector Equilibrium and its Transformation Pathways
| Wikipedia |
2D geometric model
A 2D geometric model is a geometric model of an object as a two-dimensional figure, usually on the Euclidean or Cartesian plane.
Even though all material objects are three-dimensional, a 2D geometric model is often adequate for certain flat objects, such as paper cut-outs and machine parts made of sheet metal. Other examples include circles used as a model of thunderstorms, which can be considered flat when viewed from above.[1]
2D geometric models are also convenient for describing certain types of artificial images, such as technical diagrams, logos, the glyphs of a font, etc. They are an essential tool of 2D computer graphics and often used as components of 3D geometric models, e.g. to describe the decals to be applied to a car model. Modern architecture practice "digital rendering" which is a technique used to form a perception of a 2-D geometric model as of a 3-D geometric model designed through descriptive geometry and computerized equipment.[2]
2D geometric modeling techniques
• simple geometric shapes
• boundary representation
• Boolean operations on polygons
See also
• 2D geometric primitive
• Computational geometry
• Digital image
References
1. Nissen, Silas Boye; Haerter, Jan O. (September 24, 2021). "Circling in on Convective Self-Aggregation". Journal of Geophysical Research: Atmospheres. 126. arXiv:1911.12849. doi:10.1029/2021JD035331.
2. Dresp, Birgitta; Silvestri, Chiara; Motro, René (2007). "Which geometric model for the curvature of 2-D shape contours?". Spatial Vision. 20 (3): 219–64. doi:10.1163/156856807780421165. PMID 17524256. S2CID 35702710.
| Wikipedia |
4,294,967,295
The number 4,294,967,295 is a whole number equal to 232 − 1. It is a perfect totient number, meaning it is equal to the sum of its iterated totients.[1][2] It follows 4,294,967,294 and precedes 4,294,967,296. It has a factorization of $3\cdot 5\cdot 17\cdot 257\cdot 65537$.
4294967295
• List of numbers
• Integers
← 100 101 102 103 104 105 106 107 108 109
Cardinalfour billion two hundred ninety-four million nine hundred sixty-seven thousand two hundred ninety-five
Ordinal4294967295th
(four billion two hundred ninety-four million nine hundred sixty-seven thousand two hundred ninety-fifth)
Factorization3 × 5 × 17 × 257 × 65537
Greek numeral${\stackrel {\mu \beta \theta \upsilon \mathrm {\koppa} \digamma }{\mathrm {M} }}$͵ζσϟε´
Roman numeralN/A
Binary111111111111111111111111111111112
Ternary1020020222012211112103
Senary15501040155036
Octal377777777778
Duodecimal9BA46159312
HexadecimalFFFFFFFF16
In computing, 4,294,967,295 is the highest unsigned (that is, not negative) 32-bit integer, which makes it the highest possible number a 32-bit system can store in memory.
In geometry
Since the prime factors of 232 − 1 are exactly the five known Fermat primes, this number is the largest known odd value n for which a regular n-sided polygon is constructible using compass and straightedge.[3][4] Equivalently, it is the largest known odd number n for which the angle $2\pi /n$ can be constructed, or for which $\cos(2\pi /n)$ can be expressed in terms of square roots.
Not only is 4,294,967,295 the largest known odd number of sides of a constructible polygon, but since constructibility is related to factorization, the list of odd numbers n for which an n-sided polygon is constructible begins with the list of factors of 4,294,967,295. If there are no more Fermat primes, then the two lists are identical. Namely (assuming 65537 is the largest Fermat prime), an odd-sided polygon is constructible if and only if it has 3, 5, 15, 17, 51, 85, 255, 257, 771, 1285, 3855, 4369, 13107, 21845, 65535, 65537, 196611, 327685, 983055, 1114129, 3342387, 5570645, 16711935, 16843009, 50529027, 84215045, 252645135, 286331153, 858993459, 1431655765, or 4294967295 sides.[4] If there are more numbers in this list, they must be at least 2233+1 (approximately 102585827973), because every intervening Fermat number is known to be composite.[5]
In computing
The number 4,294,967,295, equivalent to the hexadecimal value FFFF,FFFF16, is the maximum value for a 32-bit unsigned integer in computing.[6] It is therefore the maximum value for a variable declared as an unsigned integer (usually indicated by the unsigned codeword) in many programming languages running on modern computers. The presence of the value may reflect an error, overflow condition, or missing value.
This value is also the largest memory address for CPUs using a 32-bit address bus.[7] Being an odd value, its appearance may reflect an erroneous (misaligned) memory address. Such a value may also be used as a sentinel value to initialize newly allocated memory for debugging purposes.
Internet Protocol version 4 (IPv4) uses a 32-bit addresses which limits the address space to 4294967296 (232) unique addresses.
In 2004, 800 aircraft over Los Angeles were put in danger when the LA Air Route Traffic Control Center lost radio contact with all of the aircraft for about three hours, delaying 400 flights and cancelling 600, due to a computer design that kept time by starting at 4,294,967.295 seconds and counting down to zero, or 49 days, 17 hours, 2 minutes and 47.295 seconds. Some people were aware that the system needed to be restarted at least every 30 days, but the root problem was the choice of such a small number.[8]
On May 4, 2021, Nasdaq temporarily suspended price feeds for Berkshire Hathaway Class A shares (Nasdaq: BRK.A), which reached $421,000. Nasdaq stores stock prices as 32-bit unsigned integers in increments of ten-thousandths of a dollar, so the maximum price that could be represented was $429,496.7295.[9]
See also
• 2,147,483,647 – Mersenne primePages displaying wikidata descriptions as a fallback
• Power of two – Two raised to an integer power
• Equilateral triangle – Shape with three equal sides
• Pentagon – Shape with five sides
• Heptadecagon (17-sides)
• 257-gon – polygon with 257 sidesPages displaying wikidata descriptions as a fallback
• 65537-gon – Shape with 65537 sides
References
1. Loomis, Paul; Plytage, Michael; Polhill, John (2008). "Summing up the Euler φ Function". College Mathematics Journal. 39 (1): 34–42. doi:10.1080/07468342.2008.11922272. JSTOR 27646564. S2CID 44013467.
2. Iannucci, Douglas E.; Deng, Moujie; Cohen, Graeme L. (2003). "On perfect totient numbers" (PDF). Journal of Integer Sequences. 6 (4): 03.4.5. Bibcode:2003JIntS...6...45I. MR 2051959.
3. Lines, Malcolm E (1986). A Number for your Thoughts: Facts and Speculations About Numbers from Euclid to the latest Computers... (2 ed.). Taylor & Francis. p. 17. ISBN 9780852744956.
4. Sloane, N. J. A. (ed.). "Sequence A004729 (Divisors of 2^32 - 1 (for a(1) to a(31), the 31 regular polygons with an odd number of sides constructible with ruler and compass))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
5. "Fermat Number". Wolfram MathWorld.
6. Simpson, Alan (2005). "58: Editing the Windows Registry". Alan Simpson's Windows XP bible (2nd ed.). Indianapolis, Indiana: J. Wiley. p. 999. ISBN 9780764588969.
7. Spector, Lincoln (19 November 2012). "Why can't 32-bit Windows access 4GB of RAM?". PC World. IDG Consumer & SMB. Archived from the original on 7 March 2016.
8. Parker, Matt. "Chapter One: Losing Track of Time". Humble Pi: A Comedy of Maths Errors. Penguin Random House UK.
9. Osipovich, Alexander (4 May 2021). "Berkshire Hathaway's Stock Price Is Too Much for Computers". The Wall Street Journal. Retrieved 6 May 2021.
| Wikipedia |
Mersenne prime
In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form Mn = 2n − 1 for some integer n. They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17th century. If n is a composite number then so is 2n − 1. Therefore, an equivalent definition of the Mersenne primes is that they are the prime numbers of the form Mp = 2p − 1 for some prime p.
Mersenne prime
Named afterMarin Mersenne
No. of known terms51
Conjectured no. of termsInfinite
Subsequence ofMersenne numbers
First terms3, 7, 31, 127, 8191
Largest known term282,589,933 − 1 (December 7, 2018)
OEIS index
• A000668
• Mersenne primes (of form 2^p − 1 where p is a prime)
The exponents n which give Mersenne primes are 2, 3, 5, 7, 13, 17, 19, 31, ... (sequence A000043 in the OEIS) and the resulting Mersenne primes are 3, 7, 31, 127, 8191, 131071, 524287, 2147483647, ... (sequence A000668 in the OEIS).
Numbers of the form Mn = 2n − 1 without the primality requirement may be called Mersenne numbers. Sometimes, however, Mersenne numbers are defined to have the additional requirement that n be prime. The smallest composite Mersenne number with prime exponent n is 211 − 1 = 2047 = 23 × 89.
Mersenne primes were studied in antiquity because of their close connection to perfect numbers: the Euclid–Euler theorem asserts a one-to-one correspondence between even perfect numbers and Mersenne primes. Many of the largest known primes are Mersenne primes because Mersenne numbers are easier to check for primality.
As of 2023, 51 Mersenne primes are known. The largest known prime number, 282,589,933 − 1, is a Mersenne prime.[1] Since 1997, all newly found Mersenne primes have been discovered by the Great Internet Mersenne Prime Search, a distributed computing project. In December 2020, a major milestone in the project was passed after all exponents below 100 million were checked at least once.[2]
About Mersenne primes
Unsolved problem in mathematics:
Are there infinitely many Mersenne primes?
(more unsolved problems in mathematics)
Many fundamental questions about Mersenne primes remain unresolved. It is not even known whether the set of Mersenne primes is finite or infinite. The Lenstra–Pomerance–Wagstaff conjecture asserts that there are infinitely many Mersenne primes and predicts their order of growth. It is also not known whether infinitely many Mersenne numbers with prime exponents are composite, although this would follow from widely believed conjectures about prime numbers, for example, the infinitude of Sophie Germain primes congruent to 3 (mod 4). For these primes p, 2p + 1 (which is also prime) will divide Mp, for example, 23 | M11, 47 | M23, 167 | M83, 263 | M131, 359 | M179, 383 | M191, 479 | M239, and 503 | M251 (sequence A002515 in the OEIS). Since for these primes p, 2p + 1 is congruent to 7 mod 8, so 2 is a quadratic residue mod 2p + 1, and the multiplicative order of 2 mod 2p + 1 must divide $ {\frac {(2p+1)-1}{2}}=p$. Since p is a prime, it must be p or 1. However, it cannot be 1 since $\Phi _{1}(2)=1$ and 1 has no prime factors, so it must be p. Hence, 2p + 1 divides $\Phi _{p}(2)=2^{p}-1$ and $2^{p}-1=M_{p}$ cannot be prime.
The first four Mersenne primes are M2 = 3, M3 = 7, M5 = 31 and M7 = 127 and because the first Mersenne prime starts at M2, all Mersenne primes are congruent to 3 (mod 4). Other than M0 = 0 and M1 = 1, all other Mersenne numbers are also congruent to 3 (mod 4). Consequently, in the prime factorization of a Mersenne number ( ≥ M2 ) there must be at least one prime factor congruent to 3 (mod 4).
A basic theorem about Mersenne numbers states that if Mp is prime, then the exponent p must also be prime. This follows from the identity
${\begin{aligned}2^{ab}-1&=(2^{a}-1)\cdot \left(1+2^{a}+2^{2a}+2^{3a}+\cdots +2^{(b-1)a}\right)\\&=(2^{b}-1)\cdot \left(1+2^{b}+2^{2b}+2^{3b}+\cdots +2^{(a-1)b}\right).\end{aligned}}$
This rules out primality for Mersenne numbers with a composite exponent, such as M4 = 24 − 1 = 15 = 3 × 5 = (22 − 1) × (1 + 22).
Though the above examples might suggest that Mp is prime for all primes p, this is not the case, and the smallest counterexample is the Mersenne number
M11 = 211 − 1 = 2047 = 23 × 89.
The evidence at hand suggests that a randomly selected Mersenne number is much more likely to be prime than an arbitrary randomly selected odd integer of similar size.[3] Nonetheless, prime values of Mp appear to grow increasingly sparse as p increases. For example, eight of the first 11 primes p give rise to a Mersenne prime Mp (the correct terms on Mersenne's original list), while Mp is prime for only 43 of the first two million prime numbers (up to 32,452,843).
The current lack of any simple test to determine whether a given Mersenne number is prime makes the search for Mersenne primes a difficult task, since Mersenne numbers grow very rapidly. The Lucas–Lehmer primality test (LLT) is an efficient primality test that greatly aids this task, making it much easier to test the primality of Mersenne numbers than that of most other numbers of the same size. The search for the largest known prime has somewhat of a cult following. Consequently, a large amount of computer power has been expended searching for new Mersenne primes, much of which is now done using distributed computing.
Arithmetic modulo a Mersenne number is particularly efficient on a binary computer, making them popular choices when a prime modulus is desired, such as the Park–Miller random number generator. To find a primitive polynomial of Mersenne number order requires knowing the factorization of that number, so Mersenne primes allow one to find primitive polynomials of very high order. Such primitive trinomials are used in pseudorandom number generators with very large periods such as the Mersenne twister, generalized shift register and Lagged Fibonacci generators.
Perfect numbers
Main article: Euclid–Euler theorem
Mersenne primes Mp are closely connected to perfect numbers. In the 4th century BC, Euclid proved that if 2p − 1 is prime, then 2p − 1(2p − 1) is a perfect number. In the 18th century, Leonhard Euler proved that, conversely, all even perfect numbers have this form.[4] This is known as the Euclid–Euler theorem. It is unknown whether there are any odd perfect numbers.
History
2 3 5 7 11 13 17 19
23 29 31 37 41 43 47 53
59 61 67 71 73 79 83 89
97 101 103 107 109 113 127 131
137 139 149 151 157 163 167 173
179 181 191 193 197 199 211 223
227 229 233 239 241 251 257 263
269 271 277 281 283 293 307 311
The first 64 prime exponents with those corresponding to Mersenne primes shaded in cyan and in bold, and those thought to do so by Mersenne in red and bold
Mersenne primes take their name from the 17th-century French scholar Marin Mersenne, who compiled what was supposed to be a list of Mersenne primes with exponents up to 257. The exponents listed by Mersenne in 1644 were as follows:
2, 3, 5, 7, 13, 17, 19, 31, 67, 127, 257.
His list replicated the known primes of his time with exponents up to 19. His next entry, 31, was correct, but the list then became largely incorrect, as Mersenne mistakenly included M67 and M257 (which are composite) and omitted M61, M89, and M107 (which are prime). Mersenne gave little indication of how he came up with his list.[5]
Édouard Lucas proved in 1876 that M127 is indeed prime, as Mersenne claimed. This was the largest known prime number for 75 years until 1951, when Ferrier found a larger prime, $(2^{148}+1)/17$, using a desk calculating machine.[6]: page 22 M61 was determined to be prime in 1883 by Ivan Mikheevich Pervushin, though Mersenne claimed it was composite, and for this reason it is sometimes called Pervushin's number. This was the second-largest known prime number, and it remained so until 1911. Lucas had shown another error in Mersenne's list in 1876. Without finding a factor, Lucas demonstrated that M67 is actually composite. No factor was found until a famous talk by Frank Nelson Cole in 1903.[7] Without speaking a word, he went to a blackboard and raised 2 to the 67th power, then subtracted one, resulting in the number 147,573,952,589,676,412,927. On the other side of the board, he multiplied 193,707,721 × 761,838,257,287 and got the same number, then returned to his seat (to applause) without speaking.[8] He later said that the result had taken him "three years of Sundays" to find.[9] A correct list of all Mersenne primes in this number range was completed and rigorously verified only about three centuries after Mersenne published his list.
Searching for Mersenne primes
Fast algorithms for finding Mersenne primes are available, and as of June 2023, the six largest known prime numbers are Mersenne primes.
The first four Mersenne primes M2 = 3, M3 = 7, M5 = 31 and M7 = 127 were known in antiquity. The fifth, M13 = 8191, was discovered anonymously before 1461; the next two (M17 and M19) were found by Pietro Cataldi in 1588. After nearly two centuries, M31 was verified to be prime by Leonhard Euler in 1772. The next (in historical, not numerical order) was M127, found by Édouard Lucas in 1876, then M61 by Ivan Mikheevich Pervushin in 1883. Two more (M89 and M107) were found early in the 20th century, by R. E. Powers in 1911 and 1914, respectively.
The most efficient method presently known for testing the primality of Mersenne numbers is the Lucas–Lehmer primality test. Specifically, it can be shown that for prime p > 2, Mp = 2p − 1 is prime if and only if Mp divides Sp − 2, where S0 = 4 and Sk = (Sk − 1)2 − 2 for k > 0.
During the era of manual calculation, all the exponents up to and including 257 were tested with the Lucas–Lehmer test and found to be composite. A notable contribution was made by retired Yale physics professor Horace Scudder Uhler, who did the calculations for exponents 157, 167, 193, 199, 227, and 229.[10] Unfortunately for those investigators, the interval they were testing contains the largest known relative gap between Mersenne primes: the next Mersenne prime exponent, 521, would turn out to be more than four times as large as the previous record of 127.
The search for Mersenne primes was revolutionized by the introduction of the electronic digital computer. Alan Turing searched for them on the Manchester Mark 1 in 1949,[11] but the first successful identification of a Mersenne prime, M521, by this means was achieved at 10:00 pm on January 30, 1952, using the U.S. National Bureau of Standards Western Automatic Computer (SWAC) at the Institute for Numerical Analysis at the University of California, Los Angeles, under the direction of D. H. Lehmer, with a computer search program written and run by Prof. R. M. Robinson. It was the first Mersenne prime to be identified in thirty-eight years; the next one, M607, was found by the computer a little less than two hours later. Three more — M1279, M2203, and M2281 — were found by the same program in the next several months. M4,423 was the first prime discovered with more than 1000 digits, M44,497 was the first with more than 10,000, and M6,972,593 was the first with more than a million. In general, the number of digits in the decimal representation of Mn equals ⌊n × log102⌋ + 1, where ⌊x⌋ denotes the floor function (or equivalently ⌊log10Mn⌋ + 1).
In September 2008, mathematicians at UCLA participating in the Great Internet Mersenne Prime Search (GIMPS) won part of a $100,000 prize from the Electronic Frontier Foundation for their discovery of a very nearly 13-million-digit Mersenne prime. The prize, finally confirmed in October 2009, is for the first known prime with at least 10 million digits. The prime was found on a Dell OptiPlex 745 on August 23, 2008. This was the eighth Mersenne prime discovered at UCLA.[12]
On April 12, 2009, a GIMPS server log reported that a 47th Mersenne prime had possibly been found. The find was first noticed on June 4, 2009, and verified a week later. The prime is 242,643,801 − 1. Although it is chronologically the 47th Mersenne prime to be discovered, it is smaller than the largest known at the time, which was the 45th to be discovered.
On January 25, 2013, Curtis Cooper, a mathematician at the University of Central Missouri, discovered a 48th Mersenne prime, 257,885,161 − 1 (a number with 17,425,170 digits), as a result of a search executed by a GIMPS server network.[13]
On January 19, 2016, Cooper published his discovery of a 49th Mersenne prime, 274,207,281 − 1 (a number with 22,338,618 digits), as a result of a search executed by a GIMPS server network.[14][15][16] This was the fourth Mersenne prime discovered by Cooper and his team in the past ten years.
On September 2, 2016, the Great Internet Mersenne Prime Search finished verifying all tests below M37,156,667, thus officially confirming its position as the 45th Mersenne prime.[17]
On January 3, 2018, it was announced that Jonathan Pace, a 51-year-old electrical engineer living in Germantown, Tennessee, had found a 50th Mersenne prime, 277,232,917 − 1 (a number with 23,249,425 digits), as a result of a search executed by a GIMPS server network.[18] The discovery was made by a computer in the offices of a church in the same town.[19][20]
On December 21, 2018, it was announced that The Great Internet Mersenne Prime Search (GIMPS) discovered the largest known prime number, 282,589,933 − 1, having 24,862,048 digits. A computer volunteered by Patrick Laroche from Ocala, Florida made the find on December 7, 2018.[21]
In late 2020, GIMPS began using a new technique to rule out potential Mersenne primes called the Probable prime (PRP) test, based on development from Robert Gerbicz in 2017, and a simple way to verify tests developed by Krzysztof Pietrzak in 2018. Due to the low error rate and ease of proof, this nearly halved the computing time to rule out potential primes over the Lucas-Lehmer test (as two users would no longer have to perform the same test to confirm the other's result), although exponents passing the PRP test still require one to confirm their primality.[22]
Theorems about Mersenne numbers
1. If a and p are natural numbers such that ap − 1 is prime, then a = 2 or p = 1.
• Proof: a ≡ 1 (mod a − 1). Then ap ≡ 1 (mod a − 1), so ap − 1 ≡ 0 (mod a − 1). Thus a − 1 | ap − 1. However, ap − 1 is prime, so a − 1 = ap − 1 or a − 1 = ±1. In the former case, a = ap, hence a = 0, 1 (which is a contradiction, as neither −1 nor 0 is prime) or p = 1. In the latter case, a = 2 or a = 0. If a = 0, however, 0p − 1 = 0 − 1 = −1 which is not prime. Therefore, a = 2.
2. If 2p − 1 is prime, then p is prime.
• Proof: Suppose that p is composite, hence can be written p = ab with a and b > 1. Then 2p − 1 = 2ab − 1 = (2a)b − 1 = (2a − 1)((2a)b−1 + (2a)b−2 + ... + 2a + 1) so 2p − 1 is composite. By contraposition, if 2p − 1 is prime then p is prime.
3. If p is an odd prime, then every prime q that divides 2p − 1 must be 1 plus a multiple of 2p. This holds even when 2p − 1 is prime.
• For example, 25 − 1 = 31 is prime, and 31 = 1 + 3 × (2 × 5). A composite example is 211 − 1 = 23 × 89, where 23 = 1 + (2 × 11) and 89 = 1 + 4 × (2 × 11).
• Proof: By Fermat's little theorem, q is a factor of 2q−1 − 1. Since q is a factor of 2p − 1, for all positive integers c, q is also a factor of 2pc − 1. Since p is prime and q is not a factor of 21 − 1, p is also the smallest positive integer x such that q is a factor of 2x − 1. As a result, for all positive integers x, q is a factor of 2x − 1 if and only if p is a factor of x. Therefore, since q is a factor of 2q−1 − 1, p is a factor of q − 1 so q ≡ 1 (mod p). Furthermore, since q is a factor of 2p − 1, which is odd, q is odd. Therefore, q ≡ 1 (mod 2p).
• This fact leads to a proof of Euclid's theorem, which asserts the infinitude of primes, distinct from the proof written by Euclid: for every odd prime p, all primes dividing 2p − 1 are larger than p; thus there are always larger primes than any particular prime.
• It follows from this fact that for every prime p > 2, there is at least one prime of the form 2kp+1 less than or equal to Mp, for some integer k.
4. If p is an odd prime, then every prime q that divides 2p − 1 is congruent to ±1 (mod 8).
• Proof: 2p+1 ≡ 2 (mod q), so 21/2(p+1) is a square root of 2 mod q. By quadratic reciprocity, every prime modulus in which the number 2 has a square root is congruent to ±1 (mod 8).
5. A Mersenne prime cannot be a Wieferich prime.
• Proof: We show if p = 2m − 1 is a Mersenne prime, then the congruence 2p−1 ≡ 1 (mod p2) does not hold. By Fermat's little theorem, m | p − 1. Therefore, one can write p − 1 = mλ. If the given congruence is satisfied, then p2 | 2mλ − 1, therefore 0 ≡ 2mλ − 1/2m − 1 = 1 + 2m + 22m + ... + 2(λ − 1)m ≡ −λ mod (2m − 1). Hence 2m − 1 | λ, and therefore λ ≥ 2m − 1. This leads to p − 1 ≥ m(2m − 1), which is impossible since m ≥ 2.
6. If m and n are natural numbers then m and n are coprime if and only if 2m − 1 and 2n − 1 are coprime. Consequently, a prime number divides at most one prime-exponent Mersenne number.[23] That is, the set of pernicious Mersenne numbers is pairwise coprime.
7. If p and 2p + 1 are both prime (meaning that p is a Sophie Germain prime), and p is congruent to 3 (mod 4), then 2p + 1 divides 2p − 1.[24]
• Example: 11 and 23 are both prime, and 11 = 2 × 4 + 3, so 23 divides 211 − 1.
• Proof: Let q be 2p + 1. By Fermat's little theorem, 22p ≡ 1 (mod q), so either 2p ≡ 1 (mod q) or 2p ≡ −1 (mod q). Supposing latter true, then 2p+1 = (21/2(p + 1))2 ≡ −2 (mod q), so −2 would be a quadratic residue mod q. However, since p is congruent to 3 (mod 4), q is congruent to 7 (mod 8) and therefore 2 is a quadratic residue mod q. Also since q is congruent to 3 (mod 4), −1 is a quadratic nonresidue mod q, so −2 is the product of a residue and a nonresidue and hence it is a nonresidue, which is a contradiction. Hence, the former congruence must be true and 2p + 1 divides Mp.
8. All composite divisors of prime-exponent Mersenne numbers are strong pseudoprimes to the base 2.
9. With the exception of 1, a Mersenne number cannot be a perfect power. That is, and in accordance with Mihăilescu's theorem, the equation 2m − 1 = nk has no solutions where m, n, and k are integers with m > 1 and k > 1.
List of known Mersenne primes
Main article: List of Mersenne primes and perfect numbers
As of 2023, the 51 known Mersenne primes are 2p − 1 for the following p:
2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 37156667, 42643801, 43112609, 57885161, 74207281, 77232917, 82589933. (sequence A000043 in the OEIS)
Factorization of composite Mersenne numbers
Since they are prime numbers, Mersenne primes are divisible only by 1 and themselves. However, not all Mersenne numbers are Mersenne primes. Mersenne numbers are very good test cases for the special number field sieve algorithm, so often the largest number factorized with this algorithm has been a Mersenne number. As of June 2019, 21,193 − 1 is the record-holder,[25] having been factored with a variant of the special number field sieve that allows the factorization of several numbers at once. See integer factorization records for links to more information. The special number field sieve can factorize numbers with more than one large factor. If a number has only one very large factor then other algorithms can factorize larger numbers by first finding small factors and then running a primality test on the cofactor. As of September 2022, the largest completely factored number (with probable prime factors allowed) is 212,720,787 − 1 = 1,119,429,257 × 175,573,124,547,437,977 × 8,480,999,878,421,106,991 × q, where q is a 3,829,294-digit probable prime. It was discovered by a GIMPS participant with nickname "Funky Waddle".[26][27] As of September 2022, the Mersenne number M1277 is the smallest composite Mersenne number with no known factors; it has no prime factors below 268,[28] and is very unlikely to have any factors below 1065 (~2216).[29]
The table below shows factorizations for the first 20 composite Mersenne numbers (sequence A244453 in the OEIS).
p Mp Factorization of Mp
11 2047 23 × 89
23 8388607 47 × 178,481
29 536870911 233 × 1,103 × 2,089
37 137438953471 223 × 616,318,177
41 2199023255551 13,367 × 164,511,353
43 8796093022207 431 × 9,719 × 2,099,863
47 140737488355327 2,351 × 4,513 × 13,264,529
53 9007199254740991 6,361 × 69,431 × 20,394,401
59 576460752303423487 179,951 × 3,203,431,780,337 (13 digits)
67 147573952589676412927 193,707,721 × 761,838,257,287 (12 digits)
71 2361183241434822606847 228,479 × 48,544,121 × 212,885,833
73 9444732965739290427391 439 × 2,298,041 × 9,361,973,132,609 (13 digits)
79 604462909807314587353087 2,687 × 202,029,703 × 1,113,491,139,767 (13 digits)
83 967140655691...033397649407 167 × 57,912,614,113,275,649,087,721 (23 digits)
97 158456325028...187087900671 11,447 × 13,842,607,235,828,485,645,766,393 (26 digits)
101 253530120045...993406410751 7,432,339,208,719 (13 digits) × 341,117,531,003,194,129 (18 digits)
103 101412048018...973625643007 2,550,183,799 × 3,976,656,429,941,438,590,393 (22 digits)
109 649037107316...312041152511 745,988,807 × 870,035,986,098,720,987,332,873 (24 digits)
113 103845937170...992658440191 3,391 × 23,279 × 65,993 × 1,868,569 × 1,066,818,132,868,207 (16 digits)
131 272225893536...454145691647 263 × 10,350,794,431,055,162,386,718,619,237,468,234,569 (38 digits)
The number of factors for the first 500 Mersenne numbers can be found at (sequence A046800 in the OEIS).
Mersenne numbers in nature and elsewhere
In the mathematical problem Tower of Hanoi, solving a puzzle with an n-disc tower requires Mn steps, assuming no mistakes are made.[30] The number of rice grains on the whole chessboard in the wheat and chessboard problem is M64.[31]
The asteroid with minor planet number 8191 is named 8191 Mersenne after Marin Mersenne, because 8191 is a Mersenne prime (3 Juno, 7 Iris, 31 Euphrosyne and 127 Johanna having been discovered and named during the 19th century).[32]
In geometry, an integer right triangle that is primitive and has its even leg a power of 2 ( ≥ 4 ) generates a unique right triangle such that its inradius is always a Mersenne number. For example, if the even leg is 2n + 1 then because it is primitive it constrains the odd leg to be 4n − 1, the hypotenuse to be 4n + 1 and its inradius to be 2n − 1.[33]
Mersenne–Fermat primes
A Mersenne–Fermat number is defined as 2pr − 1/2pr − 1 − 1 with p prime, r natural number, and can be written as MF(p, r). When r = 1, it is a Mersenne number. When p = 2, it is a Fermat number. The only known Mersenne–Fermat primes with r > 1 are
MF(2, 2), MF(2, 3), MF(2, 4), MF(2, 5), MF(3, 2), MF(3, 3), MF(7, 2), and MF(59, 2).[34]
In fact, MF(p, r) = Φpr(2), where Φ is the cyclotomic polynomial.
Generalizations
Main article: Generalized Mersenne prime
The simplest generalized Mersenne primes are prime numbers of the form f(2n), where f(x) is a low-degree polynomial with small integer coefficients.[35] An example is 264 − 232 + 1, in this case, n = 32, and f(x) = x2 − x + 1; another example is 2192 − 264 − 1, in this case, n = 64, and f(x) = x3 − x − 1.
It is also natural to try to generalize primes of the form 2n − 1 to primes of the form bn − 1 (for b ≠ 2 and n > 1). However (see also theorems above), bn − 1 is always divisible by b − 1, so unless the latter is a unit, the former is not a prime. This can be remedied by allowing b to be an algebraic integer instead of an integer:
Complex numbers
In the ring of integers (on real numbers), if b − 1 is a unit, then b is either 2 or 0. But 2n − 1 are the usual Mersenne primes, and the formula 0n − 1 does not lead to anything interesting (since it is always −1 for all n > 0). Thus, we can regard a ring of "integers" on complex numbers instead of real numbers, like Gaussian integers and Eisenstein integers.
Gaussian Mersenne primes
If we regard the ring of Gaussian integers, we get the case b = 1 + i and b = 1 − i, and can ask (WLOG) for which n the number (1 + i)n − 1 is a Gaussian prime which will then be called a Gaussian Mersenne prime.[36]
(1 + i)n − 1 is a Gaussian prime for the following n:
2, 3, 5, 7, 11, 19, 29, 47, 73, 79, 113, 151, 157, 163, 167, 239, 241, 283, 353, 367, 379, 457, 997, 1367, 3041, 10141, 14699, 27529, 49207, 77291, 85237, 106693, 160423, 203789, 364289, 991961, 1203793, 1667321, 3704053, 4792057, ... (sequence A057429 in the OEIS)
Like the sequence of exponents for usual Mersenne primes, this sequence contains only (rational) prime numbers.
As for all Gaussian primes, the norms (that is, squares of absolute values) of these numbers are rational primes:
5, 13, 41, 113, 2113, 525313, 536903681, 140737471578113, ... (sequence A182300 in the OEIS).
Eisenstein Mersenne primes
One may encounter cases where such a Mersenne prime is also an Eisenstein prime, being of the form b = 1 + ω and b = 1 − ω. In these cases, such numbers are called Eisenstein Mersenne primes.
(1 + ω)n − 1 is an Eisenstein prime for the following n:
2, 5, 7, 11, 17, 19, 79, 163, 193, 239, 317, 353, 659, 709, 1049, 1103, 1759, 2029, 5153, 7541, 9049, 10453, 23743, 255361, 534827, 2237561, ... (sequence A066408 in the OEIS)
The norms (that is, squares of absolute values) of these Eisenstein primes are rational primes:
7, 271, 2269, 176419, 129159847, 1162320517, ... (sequence A066413 in the OEIS)
Repunit primes
Main article: Repunit
The other way to deal with the fact that bn − 1 is always divisible by b − 1, it is to simply take out this factor and ask which values of n make
${\frac {b^{n}-1}{b-1}}$
be prime. (The integer b can be either positive or negative.) If, for example, we take b = 10, we get n values of:
2, 19, 23, 317, 1031, 49081, 86453, 109297, 270343, ... (sequence A004023 in the OEIS),
corresponding to primes 11, 1111111111111111111, 11111111111111111111111, ... (sequence A004022 in the OEIS).
These primes are called repunit primes. Another example is when we take b = −12, we get n values of:
2, 5, 11, 109, 193, 1483, 11353, 21419, 21911, 24071, 106859, 139739, ... (sequence A057178 in the OEIS),
corresponding to primes −11, 19141, 57154490053, ....
It is a conjecture that for every integer b which is not a perfect power, there are infinitely many values of n such that bn − 1/b − 1 is prime. (When b is a perfect power, it can be shown that there is at most one n value such that bn − 1/b − 1 is prime)
Least n such that bn − 1/b − 1 is prime are (starting with b = 2, 0 if no such n exists)
2, 3, 2, 3, 2, 5, 3, 0, 2, 17, 2, 5, 3, 3, 2, 3, 2, 19, 3, 3, 2, 5, 3, 0, 7, 3, 2, 5, 2, 7, 0, 3, 13, 313, 2, 13, 3, 349, 2, 3, 2, 5, 5, 19, 2, 127, 19, 0, 3, 4229, 2, 11, 3, 17, 7, 3, 2, 3, 2, 7, 3, 5, 0, 19, 2, 19, 5, 3, 2, 3, 2, ... (sequence A084740 in the OEIS)
For negative bases b, they are (starting with b = −2, 0 if no such n exists)
3, 2, 2, 5, 2, 3, 2, 3, 5, 5, 2, 3, 2, 3, 3, 7, 2, 17, 2, 3, 3, 11, 2, 3, 11, 0, 3, 7, 2, 109, 2, 5, 3, 11, 31, 5, 2, 3, 53, 17, 2, 5, 2, 103, 7, 5, 2, 7, 1153, 3, 7, 21943, 2, 3, 37, 53, 3, 17, 2, 7, 2, 3, 0, 19, 7, 3, 2, 11, 3, 5, 2, ... (sequence A084742 in the OEIS) (notice this OEIS sequence does not allow n = 2)
Least base b such that bprime(n) − 1/b − 1 is prime are
2, 2, 2, 2, 5, 2, 2, 2, 10, 6, 2, 61, 14, 15, 5, 24, 19, 2, 46, 3, 11, 22, 41, 2, 12, 22, 3, 2, 12, 86, 2, 7, 13, 11, 5, 29, 56, 30, 44, 60, 304, 5, 74, 118, 33, 156, 46, 183, 72, 606, 602, 223, 115, 37, 52, 104, 41, 6, 338, 217, ... (sequence A066180 in the OEIS)
For negative bases b, they are
3, 2, 2, 2, 2, 2, 2, 2, 2, 7, 2, 16, 61, 2, 6, 10, 6, 2, 5, 46, 18, 2, 49, 16, 70, 2, 5, 6, 12, 92, 2, 48, 89, 30, 16, 147, 19, 19, 2, 16, 11, 289, 2, 12, 52, 2, 66, 9, 22, 5, 489, 69, 137, 16, 36, 96, 76, 117, 26, 3, ... (sequence A103795 in the OEIS)
Other generalized Mersenne primes
Another generalized Mersenne number is
${\frac {a^{n}-b^{n}}{a-b}}$
with a, b any coprime integers, a > 1 and −a < b < a. (Since an − bn is always divisible by a − b, the division is necessary for there to be any chance of finding prime numbers.)[lower-alpha 1] We can ask which n makes this number prime. It can be shown that such n must be primes themselves or equal to 4, and n can be 4 if and only if a + b = 1 and a2 + b2 is prime.[lower-alpha 2] It is a conjecture that for any pair (a, b) such that a and b are not both perfect rth powers for any r and −4ab is not a perfect fourth power, there are infinitely many values of n such that an − bn/a − b is prime.[lower-alpha 3] However, this has not been proved for any single value of (a, b).
For more information, see [37][38][39][40][41][42][43][44][45][46]
a b numbers n such that an − bn/a − b is prime
(some large terms are only probable primes, these n are checked up to 100000 for |b| ≤ 5 or |b| = a − 1, 20000 for 5 < |b| < a − 1)
OEIS sequence
2 1 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 37156667, 42643801, 43112609, 57885161, ..., 74207281, ..., 77232917, ..., 82589933, ... A000043
2 −1 3, 4*, 5, 7, 11, 13, 17, 19, 23, 31, 43, 61, 79, 101, 127, 167, 191, 199, 313, 347, 701, 1709, 2617, 3539, 5807, 10501, 10691, 11279, 12391, 14479, 42737, 83339, 95369, 117239, 127031, 138937, 141079, 267017, 269987, 374321, 986191, 4031399, ..., 13347311, 13372531, ... A000978
3 2 2, 3, 5, 17, 29, 31, 53, 59, 101, 277, 647, 1061, 2381, 2833, 3613, 3853, 3929, 5297, 7417, 90217, 122219, 173191, 256199, 336353, 485977, 591827, 1059503, ... A057468
3 1 3, 7, 13, 71, 103, 541, 1091, 1367, 1627, 4177, 9011, 9551, 36913, 43063, 49681, 57917, 483611, 877843, ... A028491
3 −1 2*, 3, 5, 7, 13, 23, 43, 281, 359, 487, 577, 1579, 1663, 1741, 3191, 9209, 11257, 12743, 13093, 17027, 26633, 104243, 134227, 152287, 700897, 1205459, ... A007658
3 −2 3, 4*, 7, 11, 83, 149, 223, 599, 647, 1373, 8423, 149497, 388897, ... A057469
4 3 2, 3, 7, 17, 59, 283, 311, 383, 499, 521, 541, 599, 1193, 1993, 2671, 7547, 24019, 46301, 48121, 68597, 91283, 131497, 148663, 184463, 341233, ... A059801
4 1 2 (no others)
4 −1 2*, 3 (no others)
4 −3 3, 5, 19, 37, 173, 211, 227, 619, 977, 1237, 2437, 5741, 13463, 23929, 81223, 121271, ... A128066
5 4 3, 43, 59, 191, 223, 349, 563, 709, 743, 1663, 5471, 17707, 19609, 35449, 36697, 45259, 91493, 246497, 265007, 289937, ... A059802
5 3 13, 19, 23, 31, 47, 127, 223, 281, 2083, 5281, 7411, 7433, 19051, 27239, 35863, 70327, ... A121877
5 2 2, 5, 7, 13, 19, 37, 59, 67, 79, 307, 331, 599, 1301, 12263, 12589, 18443, 20149, 27983, ... A082182
5 1 3, 7, 11, 13, 47, 127, 149, 181, 619, 929, 3407, 10949, 13241, 13873, 16519, 201359, 396413, 1888279, ... A004061
5 −1 5, 67, 101, 103, 229, 347, 4013, 23297, 30133, 177337, 193939, 266863, 277183, 335429, ... A057171
5 −2 2*, 3, 17, 19, 47, 101, 1709, 2539, 5591, 6037, 8011, 19373, 26489, 27427, ... A082387
5 −3 2*, 3, 5, 7, 17, 19, 109, 509, 661, 709, 1231, 12889, 13043, 26723, 43963, 44789, ... A122853
5 −4 4*, 5, 7, 19, 29, 61, 137, 883, 1381, 1823, 5227, 25561, 29537, 300893, ... A128335
6 5 2, 5, 11, 13, 23, 61, 83, 421, 1039, 1511, 31237, 60413, 113177, 135647, 258413, ... A062572
6 1 2, 3, 7, 29, 71, 127, 271, 509, 1049, 6389, 6883, 10613, 19889, 79987, 608099, ... A004062
6 −1 2*, 3, 11, 31, 43, 47, 59, 107, 811, 2819, 4817, 9601, 33581, 38447, 41341, 131891, 196337, ... A057172
6 −5 3, 4*, 5, 17, 397, 409, 643, 1783, 2617, 4583, 8783, ... A128336
7 6 2, 3, 7, 29, 41, 67, 1327, 1399, 2027, 69371, 86689, 355039, ... A062573
7 5 3, 5, 7, 113, 397, 577, 7573, 14561, 58543, ... A128344
7 4 2, 5, 11, 61, 619, 2879, 2957, 24371, 69247, ... A213073
7 3 3, 7, 19, 109, 131, 607, 863, 2917, 5923, 12421, ... A128024
7 2 3, 7, 19, 79, 431, 1373, 1801, 2897, 46997, ... A215487
7 1 5, 13, 131, 149, 1699, 14221, 35201, 126037, 371669, 1264699, ... A004063
7 −1 3, 17, 23, 29, 47, 61, 1619, 18251, 106187, 201653, ... A057173
7 −2 2*, 5, 23, 73, 101, 401, 419, 457, 811, 1163, 1511, 8011, ... A125955
7 −3 3, 13, 31, 313, 3709, 7933, 14797, 30689, 38333, ... A128067
7 −4 2*, 3, 5, 19, 41, 47, 8231, 33931, 43781, 50833, 53719, 67211, ... A218373
7 −5 2*, 11, 31, 173, 271, 547, 1823, 2111, 5519, 7793, 22963, 41077, 49739, ... A128337
7 −6 3, 53, 83, 487, 743, ... A187805
8 7 7, 11, 17, 29, 31, 79, 113, 131, 139, 4357, 44029, 76213, 83663, 173687, 336419, 615997, ... A062574
8 5 2, 19, 1021, 5077, 34031, 46099, 65707, ... A128345
8 3 2, 3, 7, 19, 31, 67, 89, 9227, 43891, ... A128025
8 1 3 (no others)
8 −1 2* (no others)
8 −3 2*, 5, 163, 191, 229, 271, 733, 21059, 25237, ... A128068
8 −5 2*, 7, 19, 167, 173, 223, 281, 21647, ... A128338
8 −7 4*, 7, 13, 31, 43, 269, 353, 383, 619, 829, 877, 4957, 5711, 8317, 21739, 24029, 38299, ... A181141
9 8 2, 7, 29, 31, 67, 149, 401, 2531, 19913, 30773, 53857, 170099, ... A059803
9 7 3, 5, 7, 4703, 30113, ... A273010
9 5 3, 11, 17, 173, 839, 971, 40867, 45821, ... A128346
9 4 2 (no others)
9 2 2, 3, 5, 13, 29, 37, 1021, 1399, 2137, 4493, 5521, ... A173718
9 1 (none)
9 −1 3, 59, 223, 547, 773, 1009, 1823, 3803, 49223, 193247, 703393, ... A057175
9 −2 2*, 3, 7, 127, 283, 883, 1523, 4001, ... A125956
9 −4 2*, 3, 5, 7, 11, 17, 19, 41, 53, 109, 167, 2207, 3623, 5059, 5471, 7949, 21211, 32993, 60251, ... A211409
9 −5 3, 5, 13, 17, 43, 127, 229, 277, 6043, 11131, 11821, ... A128339
9 −7 2*, 3, 107, 197, 2843, 3571, 4451, ..., 31517, ... A301369
9 −8 3, 7, 13, 19, 307, 619, 2089, 7297, 75571, 76103, 98897, ... A187819
10 9 2, 3, 7, 11, 19, 29, 401, 709, 2531, 15787, 66949, 282493, ... A062576
10 7 2, 31, 103, 617, 10253, 10691, ... A273403
10 3 2, 3, 5, 37, 599, 38393, 51431, ... A128026
10 1 2, 19, 23, 317, 1031, 49081, 86453, 109297, 270343, ... A004023
10 −1 5, 7, 19, 31, 53, 67, 293, 641, 2137, 3011, 268207, ... A001562
10 −3 2*, 3, 19, 31, 101, 139, 167, 1097, 43151, 60703, 90499, ... A128069
10 −7 2*, 3, 5, 11, 19, 1259, 1399, 2539, 2843, 5857, 10589, ...
10 −9 4*, 7, 67, 73, 1091, 1483, 10937, ... A217095
11 10 3, 5, 19, 311, 317, 1129, 4253, 7699, 18199, 35153, 206081, ... A062577
11 9 5, 31, 271, 929, 2789, 4153, ... A273601
11 8 2, 7, 11, 17, 37, 521, 877, 2423, ... A273600
11 7 5, 19, 67, 107, 593, 757, 1801, 2243, 2383, 6043, 10181, 11383, 15629, ... A273599
11 6 2, 3, 11, 163, 191, 269, 1381, 1493, ... A273598
11 5 5, 41, 149, 229, 263, 739, 3457, 20269, 98221, ... A128347
11 4 3, 5, 11, 17, 71, 89, 827, 22307, 45893, 63521, ... A216181
11 3 3, 5, 19, 31, 367, 389, 431, 2179, 10667, 13103, 90397, ... A128027
11 2 2, 5, 11, 13, 331, 599, 18839, 23747, 24371, 29339, 32141, 67421, ... A210506
11 1 17, 19, 73, 139, 907, 1907, 2029, 4801, 5153, 10867, 20161, 293831, ... A005808
11 −1 5, 7, 179, 229, 439, 557, 6113, 223999, 327001, ... A057177
11 −2 3, 5, 17, 67, 83, 101, 1373, 6101, 12119, 61781, ... A125957
11 −3 3, 103, 271, 523, 23087, 69833, ... A128070
11 −4 2*, 7, 53, 67, 71, 443, 26497, ... A224501
11 −5 7, 11, 181, 421, 2297, 2797, 4129, 4139, 7151, 29033, ... A128340
11 −6 2*, 5, 7, 107, 383, 17359, 21929, 26393, ...
11 −7 7, 1163, 4007, 10159, ...
11 −8 2*, 3, 13, 31, 59, 131, 223, 227, 1523, ...
11 −9 2*, 3, 17, 41, 43, 59, 83, ...
11 −10 53, 421, 647, 1601, 35527, ... A185239
12 11 2, 3, 7, 89, 101, 293, 4463, 70067, ... A062578
12 7 2, 3, 7, 13, 47, 89, 139, 523, 1051, ... A273814
12 5 2, 3, 31, 41, 53, 101, 421, 1259, 4721, 45259, ... A128348
12 1 2, 3, 5, 19, 97, 109, 317, 353, 701, 9739, 14951, 37573, 46889, 769543, ... A004064
12 −1 2*, 5, 11, 109, 193, 1483, 11353, 21419, 21911, 24071, 106859, 139739, ... A057178
12 −5 2*, 3, 5, 13, 347, 977, 1091, 4861, 4967, 34679, ... A128341
12 −7 2*, 3, 7, 67, 79, 167, 953, 1493, 3389, 4871, ...
12 −11 47, 401, 509, 8609, ... A213216
*Note: if b < 0 and n is even, then the numbers n are not included in the corresponding OEIS sequence.
When a = b + 1, it is (b + 1)n − bn, a difference of two consecutive perfect nth powers, and if an − bn is prime, then a must be b + 1, because it is divisible by a − b.
Least n such that (b + 1)n − bn is prime are
2, 2, 2, 3, 2, 2, 7, 2, 2, 3, 2, 17, 3, 2, 2, 5, 3, 2, 5, 2, 2, 229, 2, 3, 3, 2, 3, 3, 2, 2, 5, 3, 2, 3, 2, 2, 3, 3, 2, 7, 2, 3, 37, 2, 3, 5, 58543, 2, 3, 2, 2, 3, 2, 2, 3, 2, 5, 3, 4663, 54517, 17, 3, 2, 5, 2, 3, 3, 2, 2, 47, 61, 19, ... (sequence A058013 in the OEIS)
Least b such that (b + 1)prime(n) − bprime(n) is prime are
1, 1, 1, 1, 5, 1, 1, 1, 5, 2, 1, 39, 6, 4, 12, 2, 2, 1, 6, 17, 46, 7, 5, 1, 25, 2, 41, 1, 12, 7, 1, 7, 327, 7, 8, 44, 26, 12, 75, 14, 51, 110, 4, 14, 49, 286, 15, 4, 39, 22, 109, 367, 22, 67, 27, 95, 80, 149, 2, 142, 3, 11, ... (sequence A222119 in the OEIS)
See also
• Repunit
• Fermat prime
• Power of two
• Erdős–Borwein constant
• Mersenne conjectures
• Mersenne twister
• Double Mersenne number
• Prime95 / MPrime
• Great Internet Mersenne Prime Search (GIMPS)
• Largest known prime number
• Wieferich prime
• Wagstaff prime
• Cullen prime
• Woodall prime
• Proth prime
• Solinas prime
• Gillies' conjecture
• Williams number
Notes
1. This number is the same as the Lucas number Un(a + b, ab), since a and b are the roots of the quadratic equation x2 − (a + b)x + ab = 0.
2. Since a4 − b4/a − b = (a + b)(a2 + b2). Thus, in this case the pair (a, b) must be (x + 1, −x) and x2 + (x + 1)2 must be prime. That is, x must be in OEIS: A027861.
3. When a and b are both perfect rth powers for some r > 1 or when −4ab is a perfect fourth power, it can be shown that there are at most two values of n with this property: in these cases, an − bn/a − b can be factored algebraically.
References
1. "GIMPS Project Discovers Largest Known Prime Number: 282,589,933-1". Mersenne Research, Inc. 21 December 2018. Retrieved 21 December 2018.
2. "GIMPS Milestones Report". Mersenne.org. Mersenne Research, Inc. Retrieved 5 December 2020.
3. Caldwell, Chris. "Heuristics: Deriving the Wagstaff Mersenne Conjecture".
4. Chris K. Caldwell, Mersenne Primes: History, Theorems and Lists
5. The Prime Pages, Mersenne's conjecture.
6. Hardy, G. H.; Wright, E. M. (1959). An Introduction to the Theory of Numbers (4th ed.). Oxford University Press.
7. Cole, F. N. (1 December 1903). "On the factoring of large numbers". Bulletin of the American Mathematical Society. 10 (3): 134–138. doi:10.1090/S0002-9904-1903-01079-9.
8. Bell, E.T. and Mathematical Association of America (1951). Mathematics, queen and servant of science. McGraw-Hill New York. p. 228.
9. "h2g2: Mersenne Numbers". BBC News. Archived from the original on December 5, 2014.
10. Horace S. Uhler (1952). "A Brief History of the Investigations on Mersenne Numbers and the Latest Immense Primes". Scripta Mathematica. 18: 122–131.
11. Brian Napper, The Mathematics Department and the Mark 1.
12. Maugh II, Thomas H. (2008-09-27). "UCLA mathematicians discover a 13-million-digit prime number". Los Angeles Times. Retrieved 2011-05-21.
13. Tia Ghose. "Largest Prime Number Discovered". Scientific American. Retrieved 2013-02-07.
14. Cooper, Curtis (7 January 2016). "Mersenne Prime Number discovery – 274207281 − 1 is Prime!". Mersenne Research, Inc. Retrieved 22 January 2016.
15. Brook, Robert (January 19, 2016). "Prime number with 22 million digits is the biggest ever found". New Scientist. Retrieved 19 January 2016.
16. Chang, Kenneth (21 January 2016). "New Biggest Prime Number = 2 to the 74 Mil ... Uh, It's Big". The New York Times. Retrieved 22 January 2016.
17. "Milestones". Archived from the original on 2016-09-03.
18. "Mersenne Prime Discovery - 2^77232917-1 is Prime!". www.mersenne.org. Retrieved 2018-01-03.
19. "Largest-known prime number found on church computer". christianchronicle.org. January 12, 2018.
20. "Found: A Special, Mind-Bogglingly Large Prime Number". January 5, 2018.
21. "GIMPS Discovers Largest Known Prime Number: 2^82,589,933-1". Retrieved 2019-01-01.
22. "GIMPS - The Math - PrimeNet". www.mersenne.org. Retrieved 29 June 2021.
23. Will Edgington's Mersenne Page Archived 2014-10-14 at the Wayback Machine
24. Caldwell, Chris K. "Proof of a result of Euler and Lagrange on Mersenne Divisors". Prime Pages.
25. Kleinjung, Thorsten; Bos, Joppe W.; Lenstra, Arjen K. (2014). "Mersenne Factorization Factory". Advances in Cryptology – ASIACRYPT 2014. Lecture Notes in Computer Science. Vol. 8874. pp. 358–377. doi:10.1007/978-3-662-45611-8_19. ISBN 978-3-662-45607-1.
26. Henri Lifchitz and Renaud Lifchitz. "PRP Top Records". Retrieved 2022-09-05.
27. "M12720787 Mersenne number exponent details". www.mersenne.ca. Retrieved 5 September 2022.
28. "Exponent Status for M1277". Retrieved 2021-07-21.
29. "M1277 Mersenne number exponent details". www.mersenne.ca. Retrieved 24 June 2022.
30. Petković, Miodrag (2009). Famous Puzzles of Great Mathematicians. AMS Bookstore. p. 197. ISBN 978-0-8218-4814-2.
31. Weisstein, Eric W. "Wheat and Chessboard Problem". Mathworld. Wolfram. Retrieved 2023-02-11.
32. Alan Chamberlin. "JPL Small-Body Database Browser". Ssd.jpl.nasa.gov. Retrieved 2011-05-21.
33. "OEIS A016131". The On-Line Encyclopedia of Integer Sequences.
34. "A research of Mersenne and Fermat primes". Archived from the original on 2012-05-29.
35. Solinas, Jerome A. (1 January 2011). "Generalized Mersenne Prime". In Tilborg, Henk C. A. van; Jajodia, Sushil (eds.). Encyclopedia of Cryptography and Security. Springer US. pp. 509–510. doi:10.1007/978-1-4419-5906-5_32. ISBN 978-1-4419-5905-8.
36. Chris Caldwell: The Prime Glossary: Gaussian Mersenne (part of the Prime Pages)
37. Zalnezhad, Ali; Zalnezhad, Hossein; Shabani, Ghasem; Zalnezhad, Mehdi (March 2015). "Relationships and Algorithm in order to Achieve the Largest Primes". arXiv:1503.07688 [math.NT].
38. (x, 1) and (x, −1) for x = 2 to 50
39. (x, 1) for x = 2 to 160
40. (x, −1) for x = 2 to 160
41. (x + 1, x) for x = 1 to 160
42. (x + 1, −x) for x = 1 to 40
43. (x + 2, x) for odd x = 1 to 107
44. (x, −1) for x = 2 to 200
45. PRP records, search for $(a^{n}-b^{n})/c$, that is, (a, b)
46. PRP records, search for $(a^{n}+b^{n})/c$, that is, (a, −b)
External links
Look up Mersenne prime in Wiktionary, the free dictionary.
• "Mersenne number", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• GIMPS home page
• GIMPS Milestones Report – status page gives various statistics on search progress, typically updated every week, including progress towards proving the ordering of the largest known Mersenne primes
• GIMPS, known factors of Mersenne numbers
• Mq = (8x)2 − (3qy)2 Property of Mersenne numbers with prime exponent that are composite (PDF)
• Mq = x2 + d·y2 math thesis (PS)
• Grime, James. "31 and Mersenne Primes". Numberphile. Brady Haran. Archived from the original on 2013-05-31. Retrieved 2013-04-06.
• Mersenne prime bibliography with hyperlinks to original publications
• report about Mersenne primes – detection in detail (in German)
• GIMPS wiki
• Will Edgington's Mersenne Page – contains factors for small Mersenne numbers
• Known factors of Mersenne numbers
• Decimal digits and English names of Mersenne primes
• Prime curios: 2305843009213693951
• http://www.leyland.vispa.com/numth/factorization/cunningham/2-.txt Archived 2014-11-05 at the Wayback Machine
• http://www.leyland.vispa.com/numth/factorization/cunningham/2+.txt Archived 2013-05-02 at the Wayback Machine
• OEIS sequence A250197 (Numbers n such that the left Aurifeuillian primitive part of 2^n+1 is prime) – Factorization of Mersenne numbers Mn (n up to 1280)
• Factorization of completely factored Mersenne numbers
• The Cunningham project, factorization of bn ± 1, b = 2, 3, 5, 6, 7, 10, 11, 12
• http://www.leyland.vispa.com/numth/factorization/cunningham/main.htm Archived 2016-03-04 at the Wayback Machine
• http://www.leyland.vispa.com/numth/factorization/anbn/main.htm Archived 2016-02-02 at the Wayback Machine
MathWorld links
• Weisstein, Eric W. "Mersenne number". MathWorld.
• Weisstein, Eric W. "Mersenne prime". MathWorld.
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• Moser's number
• Graham's number
• TREE(3)
• SSCG(3)
• BH(3)
• Rayo's number
• Transfinite numbers
Expression
methods
Notations
• Scientific notation
• Knuth's up-arrow notation
• Conway chained arrow notation
• Steinhaus–Moser notation
Operators
• Hyperoperation
• Tetration
• Pentation
• Ackermann function
• Grzegorczyk hierarchy
• Fast-growing hierarchy
Related
articles
(alphabetical
order)
• Busy beaver
• Extended real number line
• Indefinite and fictitious numbers
• Infinitesimal
• Largest known prime number
• List of numbers
• Long and short scales
• Number systems
• Number names
• Orders of magnitude
• Power of two
• Power of three
• Power of 10
• Sagan Unit
• Names
• History
| Wikipedia |
Largest known prime number
The largest known prime number (as of June 2023) is 282,589,933 − 1, a number which has 24,862,048 digits when written in base 10. It was found via a computer volunteered by Patrick Laroche of the Great Internet Mersenne Prime Search (GIMPS) in 2018.[1]
A prime number is a natural number greater than 1 with no divisors other than 1 and itself. According to Euclid's theorem there are infinitely many prime numbers, so there is no largest prime.
Many of the largest known primes are Mersenne primes, numbers that are one less than a power of two, because they can utilize a specialized primality test that is faster than the general one. As of June 2023, the six largest known primes are Mersenne primes.[2] The last seventeen record primes were Mersenne primes.[3][4] The binary representation of any Mersenne prime is composed of all ones, since the binary form of 2k − 1 is simply k ones.[5]
Current record
The record is currently held by 282,589,933 − 1 with 24,862,048 digits, found by GIMPS in December 2018.[1] The first and last 120 digits of its value are shown below:
148894445742041325547806458472397916603026273992795324185271289425213239361064475310309971132180337174752834401423587560 ...
(24,861,808 digits skipped)
... 062107557947958297531595208807192693676521782184472526640076912114355308311969487633766457823695074037951210325217902591[6]
This prime has been holding the record for 4 years and 7 months (as of August 2023), longer than any other record prime since M19937 (which held the record for 7 years, 1971–1978).
Prizes
There are several prizes offered by the Electronic Frontier Foundation (EFF) for record primes.[7] A prime with one million digits was found in 1999, earning the discoverer a US$50,000 prize.[8] In 2008, a ten-million digit prime won a US$100,000 prize and a Cooperative Computing Award from the EFF.[7] Time called this prime the 29th top invention of 2008.[9]
Both of these primes were discovered through the Great Internet Mersenne Prime Search (GIMPS), which coordinates long-range search efforts among tens of thousands of computers and thousands of volunteers. The $50,000 prize went to the discoverer and the $100,000 prize went to GIMPS. GIMPS will split the US$150,000 prize for the first prime of over 100 million digits with the winning participant. A further prize is offered for the first prime with at least one billion digits.[7]
GIMPS also offers a US$3,000 research discovery award for participants who discover a new Mersenne prime of less than 100 million digits.[10]
History of largest known prime numbers
The following table lists the progression of the largest known prime number in ascending order.[3] Here Mp = 2p − 1 is the Mersenne number with exponent p. The longest record-holder known was M19 = 524,287, which was the largest known prime for 144 years. No records are known prior to 1456.
Number Decimal expansion
(partial for numbers > M1000)
Digits Year found Discoverer
M13 8,191 4 1456 Anonymous
M17 131,071 6 1588 Pietro Cataldi
M19 524,287 6 1588 Pietro Cataldi
${\tfrac {2^{32}+1}{641}}$ 6,700,417 7 1732 Leonhard Euler?
Euler did not explicitly publish the primality of 6,700,417, but the techniques he had used to factorise 232 + 1 meant that he had already done most of the work needed to prove this, and some experts believe he knew of it.[11]
M31 2,147,483,647 10 1772 Leonhard Euler
${\tfrac {10^{18}+1}{1000001}}$ 999,999,000,001 12 1851 Included (but question-marked) in a list of primes by Looff. Given his uncertainty, some do not include this as a record.
${\tfrac {2^{64}+1}{274177}}$ 67,280,421,310,721 14 1855 Thomas Clausen (but no proof was provided).
M127 170,141,183,460,469,231,731,687,303,715,884,105,727 39 1876 Édouard Lucas
${\tfrac {2^{148}+1}{17}}$ 20,988,936,657,440,586,486,151,264,256,610,222,593,863,921 44 1951 Aimé Ferrier with a mechanical calculator; the largest record not set by computer.
180×(M127)2+1
5210644015679228794060694325390955853335898483908056458352183851018372555735221
79 1951 J. C. P. Miller & D. J. Wheeler[12]
Using Cambridge's EDSAC computer
M521
6864797660130609714981900799081393217269435300143305409394463459185543183397656052122559640661454554977296311391480858037121987999716643812574028291115057151
157 1952 Raphael M. Robinson
M607
531137992816767098689588206552468627329593117727031923199444138200403559860852242739162502265229285668889329486246501015346579337652707239409519978766587351943831270835393219031728127
183 1952 Raphael M. Robinson
M1279 104079321946...703168729087 386 1952 Raphael M. Robinson
M2203 147597991521...686697771007 664 1952 Raphael M. Robinson
M2281 446087557183...418132836351 687 1952 Raphael M. Robinson
M3217 259117086013...362909315071 969 1957 Hans Riesel
M4423 285542542228...902608580607 1,332 1961 Alexander Hurwitz
M9689 478220278805...826225754111 2,917 1963 Donald B. Gillies
M9941 346088282490...883789463551 2,993 1963 Donald B. Gillies
M11213 281411201369...087696392191 3,376 1963 Donald B. Gillies
M19937 431542479738...030968041471 6,002 1971 Bryant Tuckerman
M21701 448679166119...353511882751 6,533 1978 Laura A. Nickel and Landon Curt Noll[13]
M23209 402874115778...523779264511 6,987 1979 Landon Curt Noll[13]
M44497 854509824303...961011228671 13,395 1979 David Slowinski and Harry L. Nelson[13]
M86243 536927995502...709433438207 25,962 1982 David Slowinski[13]
M132049 512740276269...455730061311 39,751 1983 David Slowinski[13]
M216091 746093103064...103815528447 65,050 1985 David Slowinski[13]
$391581\times 2^{216193}-1$ 148140632376...836387377151 65,087 1989 A group, "Amdahl Six": John Brown, Landon Curt Noll, B. K. Parady, Gene Ward Smith, Joel F. Smith, Sergio E. Zarantonello.[14][15]
Largest non-Mersenne prime that was the largest known prime when it was discovered.
M756839 174135906820...328544677887 227,832 1992 David Slowinski and Paul Gage[13]
M859433 129498125604...243500142591 258,716 1994 David Slowinski and Paul Gage[13]
M1257787 412245773621...976089366527 378,632 1996 David Slowinski and Paul Gage[13]
M1398269 814717564412...868451315711 420,921 1996 GIMPS, Joel Armengaud
M2976221 623340076248...743729201151 895,932 1997 GIMPS, Gordon Spence
M3021377 127411683030...973024694271 909,526 1998 GIMPS, Roland Clarkson
M6972593 437075744127...142924193791 2,098,960 1999 GIMPS, Nayan Hajratwala
M13466917 924947738006...470256259071 4,053,946 2001 GIMPS, Michael Cameron
M20996011 125976895450...762855682047 6,320,430 2003 GIMPS, Michael Shafer
M24036583 299410429404...882733969407 7,235,733 2004 GIMPS, Josh Findley
M25964951 122164630061...280577077247 7,816,230 2005 GIMPS, Martin Nowak
M30402457 315416475618...411652943871 9,152,052 2005 GIMPS, University of Central Missouri professors Curtis Cooper and Steven Boone
M32582657 124575026015...154053967871 9,808,358 2006 GIMPS, Curtis Cooper and Steven Boone
M43112609 316470269330...166697152511 12,978,189 2008 GIMPS, Edson Smith
M57885161 581887266232...071724285951 17,425,170 2013 GIMPS, Curtis Cooper
M74207281 300376418084...391086436351 22,338,618 2016 GIMPS, Curtis Cooper
M77232917 467333183359...069762179071 23,249,425 2017 GIMPS, Jonathan Pace
M82589933 148894445742...325217902591 24,862,048 2018 GIMPS, Patrick Laroche
GIMPS found the fifteen latest records (all of them Mersenne primes) on ordinary computers operated by participants around the world.
The twenty largest known prime numbers
A list of the 5,000 largest known primes is maintained by the PrimePages,[16] of which the twenty largest are listed below.[17]
RankNumberDiscoveredDigitsFormRef
1 282589933 − 1 2018-12-07 24,862,048 Mersenne [1]
2 277232917 − 1 2017-12-26 23,249,425 Mersenne [18]
3 274207281 − 1 2016-01-07 22,338,618 Mersenne [19]
4 257885161 − 1 2013-01-25 17,425,170 Mersenne [20]
5 243112609 − 1 2008-08-23 12,978,189 Mersenne [21]
6 242643801 − 1 2009-06-04 12,837,064 Mersenne [22]
7 Φ3(−4658591048576) 2023-05-31 11,887,192 Cyclotomic polynomial [23]
8 237156667 − 1 2008-09-06 11,185,272 Mersenne [21]
9 232582657 − 1 2006-09-04 9,808,358 Mersenne [24]
10 10223 × 231172165 + 1 2016-10-31 9,383,761 Proth [25]
11 230402457 − 1 2005-12-15 9,152,052 Mersenne [26]
12 225964951 − 1 2005-02-18 7,816,230 Mersenne [27]
13 224036583 − 1 2004-05-15 7,235,733 Mersenne [28]
14 19637361048576 + 1 2022-09-24 6,598,776 Generalized Fermat [29]
15 19517341048576 + 1 2022-08-09 6,595,985 Generalized Fermat [30]
16 202705 × 221320516 + 1 2021-12-01 6,418,121 Proth [31]
17 220996011 − 1 2003-11-17 6,320,430 Mersenne [32]
18 10590941048576 + 1 2018-10-31 6,317,602 Generalized Fermat [33]
19 3 × 220928756 − 1 2023-07-05 6,300,184 Generalized Fermat [34]
20 9194441048576 + 1 2017-08-29 6,253,210 Generalized Fermat [35]
See also
• List of largest known primes and probable primes
References
1. "GIMPS Project Discovers Largest Known Prime Number: 282,589,933-1". Mersenne Research, Inc. 21 December 2018. Retrieved 21 December 2018.
2. "The largest known primes – Database Search Output". Prime Pages. Retrieved 19 March 2023.
3. Caldwell, Chris. "The Largest Known Prime by Year: A Brief History". Prime Pages. Retrieved 19 March 2023.
4. The last non-Mersenne to be the largest known prime, was 391,581 ⋅ 2216,193 − 1; see also The Largest Known Prime by year: A Brief History originally by Caldwell.
5. "Perfect Numbers". Penn State University. Retrieved 6 October 2019. An interesting side note is about the binary representations of those numbers...
6. "51st Known Mersenne Prime Discovered".
7. "Record 12-Million-Digit Prime Number Nets $100,000 Prize". Electronic Frontier Foundation. Electronic Frontier Foundation. October 14, 2009. Retrieved November 26, 2011.
8. Electronic Frontier Foundation, Big Prime Nets Big Prize.
9. "Best Inventions of 2008 - 29. The 46th Mersenne Prime". Time. Time Inc. October 29, 2008. Archived from the original on November 2, 2008. Retrieved January 17, 2012.
10. "GIMPS by Mersenne Research, Inc". mersenne.org. Retrieved 21 November 2022.
11. Edward Sandifer, C. (19 November 2014). How Euler Did Even More. ISBN 9780883855843.
12. J. Miller, Large Prime Numbers. Nature 168, 838 (1951).
13. Landon Curt Noll, Large Prime Number Found by SGI/Cray Supercomputer.
14. Letters to the Editor. The American Mathematical Monthly 97, no. 3 (1990), p. 214. Accessed May 22, 2020.
15. Proof-code: Z, The Prime Pages.
16. "The Prime Database: The List of Largest Known Primes Home Page". t5k.org/primes. Retrieved 19 March 2023.
17. "The Top Twenty: Largest Known Primes". Retrieved 19 March 2023.
18. "GIMPS Project Discovers Largest Known Prime Number: 277,232,917-1". mersenne.org. Great Internet Mersenne Prime Search. Retrieved 3 January 2018.
19. "GIMPS Project Discovers Largest Known Prime Number: 274,207,281-1". mersenne.org. Great Internet Mersenne Prime Search. Retrieved 29 September 2017.
20. "GIMPS Discovers 48th Mersenne Prime, 257,885,161-1 is now the Largest Known Prime". mersenne.org. Great Internet Mersenne Prime Search. 5 February 2013. Retrieved 29 September 2017.
21. "GIMPS Discovers 45th and 46th Mersenne Primes, 243,112,609-1 is now the Largest Known Prime". mersenne.org. Great Internet Mersenne Prime Search. 15 September 2008. Retrieved 29 September 2017.
22. "GIMPS Discovers 47th Mersenne Prime, 242,643,801-1 is newest, but not the largest, known Mersenne Prime". mersenne.org. Great Internet Mersenne Prime Search. 12 April 2009. Retrieved 29 September 2017.
23. "PrimePage Primes: Phi(3, - 465859^1048576)". t5k.org.
24. "GIMPS Discovers 44th Mersenne Prime, 232,582,657-1 is now the Largest Known Prime". mersenne.org. Great Internet Mersenne Prime Search. 11 September 2006. Retrieved 29 September 2017.
25. "PrimeGrid's Seventeen or Bust Subproject" (PDF). primegrid.com. PrimeGrid. Retrieved 30 September 2017.
26. "GIMPS Discovers 43rd Mersenne Prime, 230,402,457-1 is now the Largest Known Prime". mersenne.org. Great Internet Mersenne Prime Search. 24 December 2005. Retrieved 29 September 2017.
27. "GIMPS Discovers 42nd Mersenne Prime, 225,964,951-1 is now the Largest Known Prime". mersenne.org. Great Internet Mersenne Prime Search. 27 February 2005. Retrieved 29 September 2017.
28. "GIMPS Discovers 41st Mersenne Prime, 224,036,583-1 is now the Largest Known Prime". mersenne.org. Great Internet Mersenne Prime Search. 28 May 2004. Retrieved 29 September 2017.
29. "PrimeGrid's Generalized Fermat Prime Search" (PDF). primegrid.com. PrimeGrid. Retrieved 7 October 2022.
30. "PrimeGrid's Generalized Fermat Prime Search" (PDF). primegrid.com. PrimeGrid. Retrieved 17 September 2022.
31. "PrimeGrid's Extended Sierpinski Problem Prime Search" (PDF). primegrid.com. PrimeGrid. Retrieved 28 December 2021.
32. "GIMPS Discovers 40th Mersenne Prime, 220,996,011-1 is now the Largest Known Prime". mersenne.org. Great Internet Mersenne Prime Search. 2 December 2003. Retrieved 29 September 2017.
33. "PrimeGrid's 321 Prime Search" (PDF). primegrid.com. PrimeGrid. Retrieved 17 July 2023.
34. "PrimeGrid's Generalized Fermat Prime Search" (PDF). primegrid.com. PrimeGrid. Retrieved 7 November 2018.
35. "PrimeGrid's Generalized Fermat Prime Search" (PDF). primegrid.com. PrimeGrid. Retrieved 30 September 2017.
External links
• Press release about the largest known prime 282,589,933−1
• Press release about the former largest known prime 277,232,917−1
• Press release about the former largest known prime 274,207,281−1
Prime number classes
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First 60 primes
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Expression
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Operators
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Related
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(alphabetical
order)
• Busy beaver
• Extended real number line
• Indefinite and fictitious numbers
• Infinitesimal
• Largest known prime number
• List of numbers
• Long and short scales
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• Power of two
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| Wikipedia |
Order-4 heptagonal tiling
In geometry, the order-4 heptagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {7,4}.
Order-4 heptagonal tiling
Poincaré disk model of the hyperbolic plane
TypeHyperbolic regular tiling
Vertex configuration74
Schläfli symbol{7,4}
r{7,7}
Wythoff symbol4 | 7 2
2 | 7 7
Coxeter diagram
Symmetry group[7,4], (*742)
[7,7], (*772)
DualOrder-7 square tiling
PropertiesVertex-transitive, edge-transitive, face-transitive
Symmetry
This tiling represents a hyperbolic kaleidoscope of 7 mirrors meeting as edges of a regular heptagon. This symmetry by orbifold notation is called *2222222 with 7 order-2 mirror intersections. In Coxeter notation can be represented as [1+,7,1+,4], removing two of three mirrors (passing through the heptagon center) in the [7,4] symmetry.
The kaleidoscopic domains can be seen as bicolored heptagons, representing mirror images of the fundamental domain. This coloring represents the uniform tiling t1{7,7} and as a quasiregular tiling is called a heptaheptagonal tiling.
Related polyhedra and tiling
Uniform heptagonal/square tilings
Symmetry: [7,4], (*742) [7,4]+, (742) [7+,4], (7*2) [7,4,1+], (*772)
{7,4} t{7,4} r{7,4} 2t{7,4}=t{4,7} 2r{7,4}={4,7} rr{7,4} tr{7,4} sr{7,4} s{7,4} h{4,7}
Uniform duals
V74 V4.14.14 V4.7.4.7 V7.8.8 V47 V4.4.7.4 V4.8.14 V3.3.4.3.7 V3.3.7.3.7 V77
Uniform heptaheptagonal tilings
Symmetry: [7,7], (*772) [7,7]+, (772)
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
{7,7} t{7,7}
r{7,7} 2t{7,7}=t{7,7} 2r{7,7}={7,7} rr{7,7} tr{7,7} sr{7,7}
Uniform duals
V77 V7.14.14 V7.7.7.7 V7.14.14 V77 V4.7.4.7 V4.14.14 V3.3.7.3.7
This tiling is topologically related as a part of sequence of regular tilings with heptagonal faces, starting with the heptagonal tiling, with Schläfli symbol {6,n}, and Coxeter diagram , progressing to infinity.
{7,3}
{7,4}
{7,5}
{7,6}
{7,7}
This tiling is also topologically related as a part of sequence of regular polyhedra and tilings with four faces per vertex, starting with the octahedron, with Schläfli symbol {n,4}, and Coxeter diagram , with n progressing to infinity.
*n42 symmetry mutation of regular tilings: {n,4}
Spherical Euclidean Hyperbolic tilings
24 34 44 54 64 74 84 ...∞4
References
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 19, The Hyperbolic Archimedean Tessellations)
• "Chapter 10: Regular honeycombs in hyperbolic space". The Beauty of Geometry: Twelve Essays. Dover Publications. 1999. ISBN 0-486-40919-8. LCCN 99035678.
See also
Wikimedia Commons has media related to Order-4 heptagonal tiling.
• Square tiling
• Tilings of regular polygons
• List of uniform planar tilings
• List of regular polytopes
External links
• Weisstein, Eric W. "Hyperbolic tiling". MathWorld.
• Weisstein, Eric W. "Poincaré hyperbolic disk". MathWorld.
• Hyperbolic and Spherical Tiling Gallery
• KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
• Hyperbolic Planar Tessellations, Don Hatch
Tessellation
Periodic
• Pythagorean
• Rhombille
• Schwarz triangle
• Rectangle
• Domino
• Uniform tiling and honeycomb
• Coloring
• Convex
• Kisrhombille
• Wallpaper group
• Wythoff
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• Ammann–Beenker
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• List
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Other
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By vertex type
Spherical
• 2n
• 33.n
• V33.n
• 42.n
• V42.n
Regular
• 2∞
• 36
• 44
• 63
Semi-
regular
• 32.4.3.4
• V32.4.3.4
• 33.42
• 33.∞
• 34.6
• V34.6
• 3.4.6.4
• (3.6)2
• 3.122
• 42.∞
• 4.6.12
• 4.82
Hyper-
bolic
• 32.4.3.5
• 32.4.3.6
• 32.4.3.7
• 32.4.3.8
• 32.4.3.∞
• 32.5.3.5
• 32.5.3.6
• 32.6.3.6
• 32.6.3.8
• 32.7.3.7
• 32.8.3.8
• 33.4.3.4
• 32.∞.3.∞
• 34.7
• 34.8
• 34.∞
• 35.4
• 37
• 38
• 3∞
• (3.4)3
• (3.4)4
• 3.4.62.4
• 3.4.7.4
• 3.4.8.4
• 3.4.∞.4
• 3.6.4.6
• (3.7)2
• (3.8)2
• 3.142
• 3.162
• (3.∞)2
• 3.∞2
• 42.5.4
• 42.6.4
• 42.7.4
• 42.8.4
• 42.∞.4
• 45
• 46
• 47
• 48
• 4∞
• (4.5)2
• (4.6)2
• 4.6.12
• 4.6.14
• V4.6.14
• 4.6.16
• V4.6.16
• 4.6.∞
• (4.7)2
• (4.8)2
• 4.8.10
• V4.8.10
• 4.8.12
• 4.8.14
• 4.8.16
• 4.8.∞
• 4.102
• 4.10.12
• 4.122
• 4.12.16
• 4.142
• 4.162
• 4.∞2
• (4.∞)2
• 54
• 55
• 56
• 5∞
• 5.4.6.4
• (5.6)2
• 5.82
• 5.102
• 5.122
• (5.∞)2
• 64
• 65
• 66
• 68
• 6.4.8.4
• (6.8)2
• 6.82
• 6.102
• 6.122
• 6.162
• 73
• 74
• 77
• 7.62
• 7.82
• 7.142
• 83
• 84
• 86
• 88
• 8.62
• 8.122
• 8.162
• ∞3
• ∞4
• ∞5
• ∞∞
• ∞.62
• ∞.82
| Wikipedia |
Order-4 octagonal tiling
In geometry, the order-4 octagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {8,4}. Its checkerboard coloring can be called a octaoctagonal tiling, and Schläfli symbol of r{8,8}.
Order-4 octagonal tiling
Poincaré disk model of the hyperbolic plane
TypeHyperbolic regular tiling
Vertex configuration84
Schläfli symbol{8,4}
r{8,8}
Wythoff symbol4 | 8 2
Coxeter diagram
or
Symmetry group[8,4], (*842)
[8,8], (*882)
DualOrder-8 square tiling
PropertiesVertex-transitive, edge-transitive, face-transitive
Uniform constructions
There are four uniform constructions of this tiling, three of them as constructed by mirror removal from the [8,8] kaleidoscope. Removing the mirror between the order 2 and 4 points, [8,8,1+], gives [(8,8,4)], (*884) symmetry. Removing two mirrors as [8,4*], leaves remaining mirrors *4444 symmetry.
Four uniform constructions of 8.8.8.8
Uniform
Coloring
Symmetry [8,4]
(*842)
[8,8]
(*882)
=
[(8,4,8)] = [8,8,1+]
(*884)
=
=
[1+,8,8,1+]
(*4444)
=
Symbol {8,4} r{8,8} r(8,4,8) = r{8,8}1⁄2 r{8,4}1⁄8 = r{8,8}1⁄4
Coxeter
diagram
=
=
= =
=
Symmetry
This tiling represents a hyperbolic kaleidoscope of 8 mirrors meeting as edges of a regular hexagon. This symmetry by orbifold notation is called (*22222222) or (*28) with 8 order-2 mirror intersections. In Coxeter notation can be represented as [8*,4], removing two of three mirrors (passing through the octagon center) in the [8,4] symmetry. Adding a bisecting mirror through 2 vertices of an octagonal fundamental domain defines a trapezohedral *4422 symmetry. Adding 4 bisecting mirrors through the vertices defines *444 symmetry. Adding 4 bisecting mirrors through the edge defines *4222 symmetry. Adding all 8 bisectors leads to full *842 symmetry.
*444
*4222
*832
The kaleidoscopic domains can be seen as bicolored octagonal tiling, representing mirror images of the fundamental domain. This coloring represents the uniform tiling r{8,8}, a quasiregular tiling and it can be called a octaoctagonal tiling.
Related polyhedra and tiling
This tiling is topologically related as a part of sequence of regular tilings with octagonal faces, starting with the octagonal tiling, with Schläfli symbol {8,n}, and Coxeter diagram , progressing to infinity.
*n42 symmetry mutation of regular tilings: {n,4}
Spherical Euclidean Hyperbolic tilings
24 34 44 54 64 74 84 ...∞4
Regular tilings: {n,8}
Spherical Hyperbolic tilings
{2,8}
{3,8}
{4,8}
{5,8}
{6,8}
{7,8}
{8,8}
...
{∞,8}
This tiling is also topologically related as a part of sequence of regular polyhedra and tilings with four faces per vertex, starting with the octahedron, with Schläfli symbol {n,4}, and Coxeter diagram , with n progressing to infinity.
{3,4}
{4,4}
{5,4}
{6,4}
{7,4}
{8,4}
...
{∞,4}
Uniform octagonal/square tilings
[8,4], (*842)
(with [8,8] (*882), [(4,4,4)] (*444) , [∞,4,∞] (*4222) index 2 subsymmetries)
(And [(∞,4,∞,4)] (*4242) index 4 subsymmetry)
=
=
=
=
=
=
=
=
=
=
=
{8,4} t{8,4}
r{8,4} 2t{8,4}=t{4,8} 2r{8,4}={4,8} rr{8,4} tr{8,4}
Uniform duals
V84 V4.16.16 V(4.8)2 V8.8.8 V48 V4.4.4.8 V4.8.16
Alternations
[1+,8,4]
(*444)
[8+,4]
(8*2)
[8,1+,4]
(*4222)
[8,4+]
(4*4)
[8,4,1+]
(*882)
[(8,4,2+)]
(2*42)
[8,4]+
(842)
=
=
=
=
=
=
h{8,4} s{8,4} hr{8,4} s{4,8} h{4,8} hrr{8,4} sr{8,4}
Alternation duals
V(4.4)4 V3.(3.8)2 V(4.4.4)2 V(3.4)3 V88 V4.44 V3.3.4.3.8
Uniform octaoctagonal tilings
Symmetry: [8,8], (*882)
=
=
=
=
=
=
=
=
=
=
=
=
=
=
{8,8} t{8,8}
r{8,8} 2t{8,8}=t{8,8} 2r{8,8}={8,8} rr{8,8} tr{8,8}
Uniform duals
V88 V8.16.16 V8.8.8.8 V8.16.16 V88 V4.8.4.8 V4.16.16
Alternations
[1+,8,8]
(*884)
[8+,8]
(8*4)
[8,1+,8]
(*4242)
[8,8+]
(8*4)
[8,8,1+]
(*884)
[(8,8,2+)]
(2*44)
[8,8]+
(882)
= = = =
=
=
=
h{8,8} s{8,8} hr{8,8} s{8,8} h{8,8} hrr{8,8} sr{8,8}
Alternation duals
V(4.8)8 V3.4.3.8.3.8 V(4.4)4 V3.4.3.8.3.8 V(4.8)8 V46 V3.3.8.3.8
See also
Wikimedia Commons has media related to Order-4 octagonal tiling.
• Square tiling
• Tilings of regular polygons
• List of uniform planar tilings
• List of regular polytopes
References
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 19, The Hyperbolic Archimedean Tessellations)
• "Chapter 10: Regular honeycombs in hyperbolic space". The Beauty of Geometry: Twelve Essays. Dover Publications. 1999. ISBN 0-486-40919-8. LCCN 99035678.
External links
• Weisstein, Eric W. "Hyperbolic tiling". MathWorld.
• Weisstein, Eric W. "Poincaré hyperbolic disk". MathWorld.
• Hyperbolic and Spherical Tiling Gallery
• KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
• Hyperbolic Planar Tessellations, Don Hatch
Tessellation
Periodic
• Pythagorean
• Rhombille
• Schwarz triangle
• Rectangle
• Domino
• Uniform tiling and honeycomb
• Coloring
• Convex
• Kisrhombille
• Wallpaper group
• Wythoff
Aperiodic
• Ammann–Beenker
• Aperiodic set of prototiles
• List
• Einstein problem
• Socolar–Taylor
• Gilbert
• Penrose
• Pentagonal
• Pinwheel
• Quaquaversal
• Rep-tile and Self-tiling
• Sphinx
• Socolar
• Truchet
Other
• Anisohedral and Isohedral
• Architectonic and catoptric
• Circle Limit III
• Computer graphics
• Honeycomb
• Isotoxal
• List
• Packing
• Problems
• Domino
• Wang
• Heesch's
• Squaring
• Dividing a square into similar rectangles
• Prototile
• Conway criterion
• Girih
• Regular Division of the Plane
• Regular grid
• Substitution
• Voronoi
• Voderberg
By vertex type
Spherical
• 2n
• 33.n
• V33.n
• 42.n
• V42.n
Regular
• 2∞
• 36
• 44
• 63
Semi-
regular
• 32.4.3.4
• V32.4.3.4
• 33.42
• 33.∞
• 34.6
• V34.6
• 3.4.6.4
• (3.6)2
• 3.122
• 42.∞
• 4.6.12
• 4.82
Hyper-
bolic
• 32.4.3.5
• 32.4.3.6
• 32.4.3.7
• 32.4.3.8
• 32.4.3.∞
• 32.5.3.5
• 32.5.3.6
• 32.6.3.6
• 32.6.3.8
• 32.7.3.7
• 32.8.3.8
• 33.4.3.4
• 32.∞.3.∞
• 34.7
• 34.8
• 34.∞
• 35.4
• 37
• 38
• 3∞
• (3.4)3
• (3.4)4
• 3.4.62.4
• 3.4.7.4
• 3.4.8.4
• 3.4.∞.4
• 3.6.4.6
• (3.7)2
• (3.8)2
• 3.142
• 3.162
• (3.∞)2
• 3.∞2
• 42.5.4
• 42.6.4
• 42.7.4
• 42.8.4
• 42.∞.4
• 45
• 46
• 47
• 48
• 4∞
• (4.5)2
• (4.6)2
• 4.6.12
• 4.6.14
• V4.6.14
• 4.6.16
• V4.6.16
• 4.6.∞
• (4.7)2
• (4.8)2
• 4.8.10
• V4.8.10
• 4.8.12
• 4.8.14
• 4.8.16
• 4.8.∞
• 4.102
• 4.10.12
• 4.122
• 4.12.16
• 4.142
• 4.162
• 4.∞2
• (4.∞)2
• 54
• 55
• 56
• 5∞
• 5.4.6.4
• (5.6)2
• 5.82
• 5.102
• 5.122
• (5.∞)2
• 64
• 65
• 66
• 68
• 6.4.8.4
• (6.8)2
• 6.82
• 6.102
• 6.122
• 6.162
• 73
• 74
• 77
• 7.62
• 7.82
• 7.142
• 83
• 84
• 86
• 88
• 8.62
• 8.122
• 8.162
• ∞3
• ∞4
• ∞5
• ∞∞
• ∞.62
• ∞.82
| Wikipedia |
2 + 2 = 5
"Two plus two equals five" (2 + 2 = 5) is a mathematically incorrect phrase used in the 1949 dystopian novel Nineteen Eighty-Four by George Orwell. It appears as a possible statement of Ingsoc (English Socialism) philosophy, like the dogma "War is Peace", which the Party expects the citizens of Oceania to believe is true. In writing his secret diary in the year 1984, the protagonist Winston Smith ponders if the Inner Party might declare that "two plus two equals five" is a fact. Smith further ponders whether or not belief in such a consensus reality makes the lie true.[1]
About the falsity of "two plus two equals five", in Room 101, the interrogator O'Brien tells the thought criminal Smith that control over physical reality is unimportant to the Party, provided the citizens of Oceania subordinate their real-world perceptions to the political will of the Party; and that, by way of doublethink: "Sometimes, Winston. [Sometimes it is four fingers.] Sometimes they are five. Sometimes they are three. Sometimes they are all of them at once".[1]
As a theme and as a subject in the arts, the anti-intellectual slogan 2 + 2 = 5 pre-dates Orwell and has produced literature, such as Deux et deux font cinq (Two and Two Make Five), written in 1895 by Alphonse Allais, which is a collection of absurdist short stories;[2] and the 1920 imagist art manifesto 2 × 2 = 5 by the poet Vadim Shershenevich, in the 20th century.[3]
Self-evident truth and self-evident falsehood
In the 17th century, in the Meditations on First Philosophy, in which the Existence of God and the Immortality of the Soul are Demonstrated (1641), René Descartes said that the standard of truth is self-evidence of clear and distinct ideas. Despite the logician Descartes' understanding of "self-evident truth", the philosopher Descartes considered that the self-evident truth of "two plus two equals four" might not exist beyond the human mind; that there might not exist correspondence between abstract ideas and concrete reality.[4]
In establishing the mundane reality of the self-evident truth of 2 + 2 = 4, in De Neutralibus et Mediis Libellus (1652) Johann Wigand said: "That twice two are four; a man may not lawfully make a doubt of it, because that manner of knowledge is grauen [graven] into mannes [man's] nature."[5]
In the comedy-of-manners play Dom Juan, or The Feast with the Statue (1665), by Molière, the libertine protagonist, Dom Juan, is asked in what values he believes, and answers that he believes "two plus two equals four".[6]
In the 18th century, the self-evident falsehood of 2 + 2 = 5 was attested in the Cyclopædia, or an Universal Dictionary of Arts and Sciences (1728), by Ephraim Chambers: "Thus, a Proposition would be absurd, that should affirm, that two and two make five; or that should deny 'em to make four."[7] In 1779, Samuel Johnson likewise said that "You may have a reason why two and two should make five, but they will still make but four."[5]
In the 19th century, in a personal letter to his future wife, Anabella Milbanke, Lord Byron said: "I know that two and two make four—& should be glad to prove it, too, if I could—though I must say if, by any sort of process, I could convert 2 & 2 into five, it would give me much greater pleasure."[8]
In Gilbert and Sullivan's Princess Ida (1884), the Princess comments that "The narrow-minded pedant still believes/That two and two make four! Why, we can prove,/We women—household drudges as we are –/That two and two make five—or three—or seven;/Or five-and-twenty, if the case demands!"[9]
Politics, literature, propaganda
France
In the late 18th century, in the pamphlet What is the Third Estate? (1789), about the legalistic denial of political rights to the common-folk majority of France, Emmanuel-Joseph Sieyès, said: "Consequently, if it be claimed that, under the French constitution, 200,000 individuals, out of 26 million citizens, constitute two-thirds of the common will, only one comment is possible: It is a claim that two and two make five."
Using the illogic of "two and two make five", Sieyès mocked the demagoguery of the Estates-General for assigning disproportionate voting power to the political minorities of France—the Clergy (First Estate) and the French nobility (Second Estate)—in relation to the Third Estate, the numeric and political majority of the citizens of France.[10]
In the 19th century, in the novel Séraphîta (1834), about the nature of androgyny, Honoré de Balzac said:
Thus, you will never find, in all Nature, two identical objects; in the natural order, therefore, two and two can never make four, for, to attain that result, we must combine units that are exactly alike, and you know that it is impossible to find two leaves alike on the same tree, or two identical individuals in the same species of tree. That axiom of your numeration, false in visible nature, is false likewise in the invisible universe of your abstractions, where the same variety is found in your ideas, which are the objects of the visible world extended by their interrelations; indeed, the differences are more striking there than elsewhere.[11]
In the pamphlet "Napoléon le Petit" (1852), about the limitations of the Second French Empire (1852–1870), such as majority political support for the monarchist coup d'Ḗtat, which installed Napoleon III (r. 1852–1870), and the French peoples' discarding from national politics the liberal values that informed the anti-monarchist Revolution, Victor Hugo said: "Now, get seven million, five hundred thousand votes to declare that two-and-two-make-five, that the straight line is the longest road, that the whole is less than its part; get it declared by eight millions, by ten millions, by a hundred millions of votes, you will not have advanced a step."[12]
In The Plague (1947), French philosopher Albert Camus declared that times came in history when those who dared to say that 2 + 2 = 4 rather than 2 + 2 = 5 were put to death.
Russia
In the late 19th century, the Russian press used the phrase 2 + 2 = 5 to describe the moral confusion of social decline at the turn of a century, because political violence characterised much of the ideological conflict among proponents of humanist democracy and defenders of tsarist autocracy in Russia.[13] In The Reaction in Germany (1842), Mikhail Bakunin said that the political compromises of the French Positivists, at the start of the July Revolution (1830), confirmed their middle-of-the-road mediocrity: "The Left says, 2 times 2 are 4; the Right, 2 times 2 are 6; and the Juste-milieu says, 2 times 2 are 5".[14][15][16]
In Notes from Underground (1864), by Feodor Dostoevsky, the anonymous protagonist accepts the falsehood of "two plus two equals five", and considers the implications (ontological and epistemological) of rejecting the truth of "two times two makes four", and proposed that the intellectualism of free will—Man's inherent capability to choose or to reject logic and illogic—is the cognitive ability that makes humanity human: "I admit that twice two makes four is an excellent thing, but, if we are to give everything its due, twice two makes five is sometimes a very charming thing, too."[17]
In the literary vignette "Prayer" (1881), Ivan Turgenev said that: "Whatever a man prays for, he prays for a miracle. Every prayer reduces itself to this: 'Great God, grant that twice two be not four'."[18] In God and the State (1882), Bakunin dismissed deism: "Imagine a philosophical vinegar sauce of the most opposed systems, a mixture of Fathers of the Church, scholastic philosophers, Descartes and Pascal, Kant and Scottish psychologists, all this a superstructure on the divine and innate ideas of Plato, and covered up with a layer of Hegelian immanence, accompanied, of course, by an ignorance, as contemptuous as it is complete, of natural science, and proving, just as two times two make five, the existence of a personal god."[19] Moreover, the slogan "two plus two equals five", is the title of the collection of absurdist short stories Deux et deux font cinq (Two and Two Make Five, 1895), by Alphonse Allais;[2] and the title of the imagist art manifesto 2 x 2 = 5 (1920), by the poet Vadim Shershenevich.[3]
In 1931, the artist Yakov Guminer supported Stalin's shortened production schedule for the economy of the Soviet Union with a propaganda poster that announced the "Arithmetic of an Alternative Plan: 2 + 2 plus the Enthusiasm of the Workers = 5" after Stalin's announcement, in 1930, that the first five-year plan (1928–1933) instead would be completed in 1932, in four years' time.[20]
George Orwell
George Orwell used the idea of 2 + 2 = 5 in an essay of January 1939 in The Adelphi; "Review of Power: A New Social Analysis by Bertrand Russell":[21]
It is quite possible that we are descending into an age in which two plus two will make five when the Leader says so.
In propaganda work for the BBC (British Broadcasting Corporation) during the Second World War (1939–1945), Orwell applied the illogic of 2 + 2 = 5 to counter the reality-denying psychology of Nazi propaganda, which he addressed in the essay "Looking Back on the Spanish War" (1943), indicating that:
Nazi theory, indeed, specifically denies that such a thing as "the truth" exists. There is, for instance, no such thing as "Science". There is only "German Science", "Jewish Science", etc. The implied objective of this line of thought is a nightmare world in which the Leader, or some ruling clique, controls not only the future, but the past. If the Leader says of such and such an event, "It never happened"—well, it never happened. If he says that "two and two are five"—well, two and two are five. This prospect frightens me much more than bombs—and, after our experiences of the last few years [the Blitz, 1940–41] that is not a frivolous statement.[22]
In addressing Nazi anti-intellectualism, Orwell's reference might have been Hermann Göring's hyperbolic praise of Adolf Hitler: "If the Führer wants it, two and two makes five!"[23] In the political novel Nineteen Eighty-Four (1949), concerning the Party's philosophy of government for Oceania, Orwell said:
In the end, the Party would announce that two and two made five, and you would have to believe it. It was inevitable that they should make that claim sooner or later: the logic of their position demanded it. Not merely the validity of experience, but the very existence of external reality, was tacitly denied by their philosophy. The heresy of heresies was common sense. And what was terrifying was not that they would kill you for thinking otherwise, but that they might be right. For, after all, how do we know that two and two make four? Or that the force of gravity works? Or that the past is unchangeable? If both the past and the external world exist only in the mind, and if the mind itself is controllable—what then?[24]
Contemporary usage
Politics and religion
In The Cult of the Amateur: How Today's Internet is Killing Our Culture (2007), the media critic Andrew Keen uses the slogan "two plus two equals five" to criticise the Wikipedia policy allowing any user to edit the encyclopedia — that the enthusiasm of the amateur for user generated content, peer production, and Web 2.0 technology leads to an encyclopedia of common knowledge, and not an encyclopedia of expert knowledge; that the "wisdom of the crowd" will distort what society considers to be the truth.[25]
Former mathematician Kareem Carr has said "If someone says 2+2=5, the correct response is, 'What are your definitions and axioms?' not a rant about the decline of Western civilization".[26]
See also
• One Plus One Equals Three
• 1=2
References
1. Orwell, George (1949). Part Three, Chapter Two – via george-orwell.org. {{cite book}}: |work= ignored (help)
2. Allais, Alphonse (1895). Deux et deux font cinq (2 + 2 = 5) (in French). Ollendorff.
3. Шершеневич, Вадим (1920). 2 × 2 = 5: листы имажиниста (in Russian).
4. "Descartes' Meditations Home Page". Wright.edu. 27 July 2005. Retrieved 1 February 2012.
5. Wilson, F. P. (1970). The Oxford Dictionary of English Proverbs (3rd ed.). Oxford: Clarendon Press. p. 849. ISBN 0198691181.
6. "Moliere Don Juan Adapted by Timothy Mooney". Moliere-in-english.com. Archived from the original on 1 March 2012. Retrieved 1 February 2012.
7. Chambers, Ephraim (1728). Cyclopaedia; or, an Universal Dictionary of Arts and Sciences . . .Volume the First. London: James and John Knapton et al. p. 11. Retrieved 25 December 2016.
8. Byron, George Gordon (1974) [Written 1813–1814]. Leslie A. Marchand (ed.). Alas! the Love of Women: 1813–1814. Belknap Press. p. 159. ISBN 9780674089426.
9. Bradley, Ian (2016). The Complete Annotated Gilbert and Sullivan. Oxford. p. 555.
10. Keith M. Baker; John W. Boyer; Julius Kirshner (15 May 1987). University of Chicago Readings in Western Civilization, Volume 7: The Old Regime and the French Revolution. University of Chicago Press. p. 154. ISBN 978-0-226-06950-0.
11. Seraphita by Honoré de Balzac.
12. Long, Roderick T. "Victor Hugo on the Limits of Democracy". Archived from the original on 15 January 2012. Retrieved 5 December 2011.
13. e.g. Novoe vremia newspaper ("New Times"), 31 October 1900.
14. Bakunin, Mikhail Aleksandrovich (1974). Selected writings. Grove Press : distributed by Random House. ISBN 9780394178738.
15. Bakunin, Michail (1842). "The Reaction in Germany From the Notebooks of a Frenchman". The Anarchist Library – via Internet Archive.
16. Meyers, Jeffrey (2010). Orwell: Life and Art. University of Illinois Press. p. 149. ISBN 9780252090226.
17. "Notes from the Underground: Part I: Chapter IX by Fyodor Dostoevsky". Classic Reader. Retrieved 22 May 2018.
18. Turgenev, Ivan Dream Tales and Poems, p. 102
19. The Communist Manifesto and Other Revolutionary Writings. Dover Publications. 2003. p. 199. ISBN 0486424650.
20. Stalin, Joseph (June 1930). Political Report of the Central Committee to the Sixteenth Congress of the C.P.S.U.(B.). Pravda, No. 177. Retrieved 10 October 2014.
21. Orwell, George (January 1939). "Review of Power: A New Social Analysis by Bertrand Russell". Retrieved 30 December 2020.
22. Orwell, George. "Looking Back on the Spanish War". orwell.ru.
23. "Hermann Göring". Museum of Tolerance Multimedia Learning Center. Archived from the original on 27 December 2004. Retrieved 18 February 2012.
24. Orwell, George Nineteen Eighty-Four (1949), ISBN 0-452-28423-6
25. Keen, Andrew (2007). The Cult of the Amateur. Doubleday. pp. 39–40, 44.
26. Delbert, Caroline (7 August 2020). "Why Some People Think 2+2=5 ...and why they're right". Popular Mechanics. Retrieved 16 January 2021.
Further reading
• Euler, Houston (1990), Reviewed by Andy Olson, "The history of 2 + 2 = 5", Mathematics Magazine, 63 (5): 338–339, doi:10.2307/2690909, JSTOR 2690909
• Krueger, L. E. & Hallford, E. W. (1984), "Why 2 + 2 = 5 looks so wrong: On the odd-even rule in sum verification", Memory & Cognition, 12 (2): 171–180, doi:10.3758/bf03198431, PMID 6727639
External links
• "Two Plus Two Equals Red", Time Magazine, Monday, 30 Jun 1947
George Orwell's Nineteen Eighty-Four
Characters
• Winston Smith
• Julia
• O'Brien
• Big Brother
• Emmanuel Goldstein
Places
• Political geography
• Ministries
Groups
• Thought Police
Concepts
• Newspeak
• Doublethink
• 2 + 2 = 5
• Thoughtcrime
• Telescreen
• Memory hole
• The Theory and Practice of Oligarchical Collectivism (Goldstein's book)
• Two Minutes Hate
• Hate Week
Adaptations
Film
• 1956 film
• 1984 film
Television
• US program
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Stage
• 2005 opera
• 2013 play
Related
• In popular media
• The Ministry of Truth (Lynskey book)
• "1984" (advertisement)
• Room 101
• radio series
• British TV series
• Australian TV series
• Groupthink
• Category
| Wikipedia |
16-cell honeycomb honeycomb
In the geometry of hyperbolic 5-space, the 16-cell honeycomb honeycomb is one of five paracompact regular space-filling tessellations (or honeycombs). It is called paracompact because it has infinite vertex figures, with all vertices as ideal points at infinity. With Schläfli symbol {3,3,4,3,3}, it has three 16-cell honeycombs around each cell. It is self-dual.
16-cell honeycomb honeycomb
(No image)
TypeHyperbolic regular honeycomb
Schläfli symbol{3,3,4,3,3}
Coxeter diagram
5-faces {3,3,4,3}
4-faces {3,3,4}
Cells {3,3}
Faces {3}
Cell figure {3}
Face figure {3,3}
Edge figure {4,3,3}
Vertex figure {3,4,3,3}
Dualself-dual
Coxeter groupX5, [3,3,4,3,3]
PropertiesRegular
Related honeycombs
It is related to the regular Euclidean 4-space 16-cell honeycomb, {3,3,4,3}.
See also
• List of regular polytopes
References
• Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. ISBN 0-486-61480-8. (Tables I and II: Regular polytopes and honeycombs, pp. 294–296)
• Coxeter, The Beauty of Geometry: Twelve Essays, Dover Publications, 1999 ISBN 0-486-40919-8 (Chapter 10: Regular honeycombs in hyperbolic space, Summary tables II,III,IV,V, p212-213)
| Wikipedia |
3-3 duoprism
In the geometry of 4 dimensions, the 3-3 duoprism or triangular duoprism is a four-dimensional convex polytope. It can be constructed as the Cartesian product of two triangles and is the simplest of an infinite family of four-dimensional polytopes constructed as Cartesian products of two polygons, the duoprisms.
3-3 duoprism
Schlegel diagram
TypeUniform duoprism
Schläfli symbol{3}×{3} = {3}2
Coxeter diagram
Cells6 triangular prisms
Faces9 squares,
6 triangles
Edges18
Vertices9
Vertex figure
Tetragonal disphenoid
Symmetry[[3,2,3]] = [6,2+,6], order 72
Dual3-3 duopyramid
Propertiesconvex, vertex-uniform, facet-transitive
It has 9 vertices, 18 edges, 15 faces (9 squares, and 6 triangles), in 6 triangular prism cells. It has Coxeter diagram , and symmetry [[3,2,3]], order 72. Its vertices and edges form a $3\times 3$ rook's graph.
Hypervolume
The hypervolume of a uniform 3-3 duoprism, with edge length a, is $V_{4}={3 \over 16}a^{4}$. This is the square of the area of an equilateral triangle, $A={{\sqrt {3}} \over 4}a^{2}$.
Graph
The graph of vertices and edges of the 3-3 duoprism has 9 vertices and 18 edges. Like the Berlekamp–van Lint–Seidel graph and the unknown solution to Conway's 99-graph problem, every edge is part of a unique triangle and every non-adjacent pair of vertices is the diagonal of a unique square. It is a toroidal graph, a locally linear graph, a strongly regular graph with parameters (9,4,1,2), the $3\times 3$ rook's graph, and the Paley graph of order 9.[1] This graph is also the Cayley graph of the group $G=\langle a,b:a^{3}=b^{3}=1,\ ab=ba\rangle \simeq C_{3}\times C_{3}$ with generating set $S=\{a,a^{2},b,b^{2}\}$.
Images
Orthogonal projections
Net 3D perspective projection with 2 different rotations
Symmetry
In 5-dimensions, some uniform 5-polytopes have 3-3 duoprism vertex figures, some with unequal edge-lengths and therefore lower symmetry:
Symmetry [[3,2,3]], order 72 [3,2], order 12
Coxeter
diagram
Schlegel
diagram
Name t2α5 t03α5 t03γ5 t03β5
The birectified 16-cell honeycomb also has a 3-3 duoprism vertex figure. There are three constructions for the honeycomb with two lower symmetries.
Symmetry [3,2,3], order 36 [3,2], order 12 [3], order 6
Coxeter
diagram
Skew
orthogonal
projection
Related complex polygons
The regular complex polytope 3{4}2, , in $\mathbb {C} ^{2}$ has a real representation as a 3-3 duoprism in 4-dimensional space. 3{4}2 has 9 vertices, and 6 3-edges. Its symmetry is 3[4]2, order 18. It also has a lower symmetry construction, , or 3{}×3{}, with symmetry 3[2]3, order 9. This is the symmetry if the red and blue 3-edges are considered distinct.[2]
Perspective projection
Orthogonal projection with coinciding central vertices
Orthogonal projection, offset view to avoid overlapping elements.
Related polytopes
k22 figures in n dimensions
Space Finite Euclidean Hyperbolic
n 4 5 6 7 8
Coxeter
group
A2A2 E6 ${\tilde {E}}_{6}$=E6+ ${\bar {T}}_{7}$=E6++
Coxeter
diagram
Symmetry [[32,2,-1]] [[32,2,0]] [[32,2,1]] [[32,2,2]] [[32,2,3]]
Order 72 1440 103,680 ∞
Graph ∞ ∞
Name −122 022 122 222 322
3-3 duopyramid
3-3 duopyramid
TypeUniform dual duopyramid
Schläfli symbol{3}+{3} = 2{3}
Coxeter diagram
Cells9 tetragonal disphenoids
Faces18 isosceles triangles
Edges15 (9+6)
Vertices6 (3+3)
Symmetry[[3,2,3]] = [6,2+,6], order 72
Dual3-3 duoprism
Propertiesconvex, vertex-uniform, facet-transitive
The dual of a 3-3 duoprism is called a 3-3 duopyramid or triangular duopyramid. It has 9 tetragonal disphenoid cells, 18 triangular faces, 15 edges, and 6 vertices.
It can be seen in orthogonal projection as a 6-gon circle of vertices, and edges connecting all pairs, just like a 5-simplex seen in projection.
orthogonal projection
Related complex polygon
The regular complex polygon 2{4}3 has 6 vertices in $\mathbb {C} ^{2}$ with a real representation in $\mathbb {R} ^{4}$ matching the same vertex arrangement of the 3-3 duopyramid. It has 9 2-edges corresponding to the connecting edges of the 3-3 duopyramid, while the 6 edges connecting the two triangles are not included. It can be seen in a hexagonal projection with 3 sets of colored edges. This arrangement of vertices and edges makes a complete bipartite graph with each vertex from one triangle is connected to every vertex on the other. It is also called a Thomsen graph or 4-cage.[3]
The 2{4}3 with 6 vertices in blue and red connected by 9 2-edges as a complete bipartite graph.
It has 3 sets of 3 edges, seen here with colors.
See also
• 3-4 duoprism
• Tesseract (4-4 duoprism)
• 5-5 duoprism
• Convex regular 4-polytope
• Duocylinder
Notes
1. Makhnev, A. A.; Minakova, I. M. (January 2004), "On automorphisms of strongly regular graphs with parameters $\lambda =1$, $\mu =2$", Discrete Mathematics and Applications, 14 (2), doi:10.1515/156939204872374, MR 2069991, S2CID 118034273
2. Coxeter, H. S. M.; Regular Complex Polytopes, Cambridge University Press, (1974).
3. Regular Complex Polytopes, p.110, p.114
References
• Regular Polytopes, H. S. M. Coxeter, Dover Publications, Inc., 1973, New York, p. 124.
• Coxeter, The Beauty of Geometry: Twelve Essays, Dover Publications, 1999, ISBN 0-486-40919-8 (Chapter 5: Regular Skew Polyhedra in three and four dimensions and their topological analogues)
• Coxeter, H. S. M. Regular Skew Polyhedra in Three and Four Dimensions. Proc. London Math. Soc. 43, 33-62, 1937.
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 26)
• Norman Johnson Uniform Polytopes, Manuscript (1991)
• N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
• Catalogue of Convex Polychora, section 6, George Olshevsky.
• Apollonian Ball Packings and Stacked Polytopes Discrete & Computational Geometry, June 2016, Volume 55, Issue 4, pp 801–826
External links
• The Fourth Dimension Simply Explained—describes duoprisms as "double prisms" and duocylinders as "double cylinders"
• Polygloss – glossary of higher-dimensional terms
• Exploring Hyperspace with the Geometric Product
| Wikipedia |
3-4-3-12 tiling
In geometry of the Euclidean plane, the 3-4-3-12 tiling is one of 20 2-uniform tilings of the Euclidean plane by regular polygons, containing regular triangles, squares, and dodecagons, arranged in two vertex configuration: 3.4.3.12 and 3.12.12.[1][2][3][4]
3-4-3-12 tiling
Type2-uniform tiling
Vertex configuration
3.4.3.12 and 3.12.12
Symmetryp4m, [4,4], (*442)
Rotation symmetryp4, [4,4]+, (442)
Properties2-uniform, 3-isohedral, 3-isotoxal
The 3.12.12 vertex figure alone generates a truncated hexagonal tiling, while the 3.4.3.12 only exists in this 2-uniform tiling. There are 2 3-uniform tilings that contain both of these vertex figures among one more.
It has square symmetry, p4m, [4,4], (*442). It is also called a demiregular tiling by some authors.
Circle Packing
This 2-uniform tiling can be used as a circle packing. Cyan circles are in contact with 3 other circles (1 cyan, 2 pink), corresponding to the V3.122 planigon, and pink circles are in contact with 4 other circles (2 cyan, 2 pink), corresponding to the V3.4.3.12 planigon. It is homeomorphic to the ambo operation on the tiling, with the cyan and pink gap polygons corresponding to the cyan and pink circles (one dimensional duals to the respective planigons). Both images coincide.
Circle Packing Ambo
Dual tiling
The dual tiling has kite ('ties') and isosceles triangle faces, defined by face configurations: V3.4.3.12 and V3.12.12. The kites meet in sets of 4 around a center vertex, and the triangles are in pairs making planigon rhombi. Every four kites and four isosceles triangles make a square of side length $2+{\sqrt {3}}$.
Dual tiling
V3.4.3.12
Semiplanigon
V3.12.12
Planigon
This is one of the only dual uniform tilings which only uses planigons (and semiplanigons) containing a 30° angle. Conversely, 3.4.3.12; 3.122 is one of the only uniform tilings in which every vertex is contained on a dodecagon.
Related tilings
It has 2 related 3-uniform tilings that include both 3.4.3.12 and 3.12.12 vertex figures:
3.4.3.12, 3.12.12, 3.4.6.4
3.4.3.12, 3.12.12, 3.3.4.12
V3.4.3.12, V3.12.12, V3.4.6.4
V3.4.3.12, V3.12.12, V3.3.4.12
This tiling can be seen in a series as a lattice of 4n-gons starting from the square tiling. For 16-gons (n=4), the gaps can be filled with isogonal octagons and isosceles triangles.
4 8 12 16 20
Square tiling
Q
Truncated square tiling
tQ
3-4-3-12 tiling
Twice-truncated square tiling
ttQ
20-gons, squares
trapezoids, triangles
Notes
1. Critchlow, pp. 62–67
2. Grünbaum and Shephard 1986, pp. 65–67
3. In Search of Demiregular Tilings #1
4. Chavey (1989)
References
• Keith Critchlow, Order in Space: A design source book, 1970, pp. 62–67
• Ghyka, M. The Geometry of Art and Life, (1946), 2nd edition, New York: Dover, 1977. Demiregular tiling
• Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X. pp. 35–43
• Grünbaum, Branko; Shephard, G. C. (1987). Tilings and Patterns. W. H. Freeman. ISBN 0-7167-1193-1. p. 65
• Sacred Geometry Design Sourcebook: Universal Dimensional Patterns, Bruce Rawles, 1997. pp. 36–37
External links
• Chavey, D. (1989). "Tilings by Regular Polygons—II: A Catalog of Tilings". Computers & Mathematics with Applications. 17: 147–165. doi:10.1016/0898-1221(89)90156-9.
• Dutch, Steve. "Uniform Tilings". Archived from the original on 2006-09-09. Retrieved 2006-09-09.
• Weisstein, Eric W. "Demiregular tessellation". MathWorld.
• In Search of Demiregular Tilings, Helmer Aslaksen
• n-uniform tilings Brian Galebach, 2-Uniform Tiling 2 of 20
| Wikipedia |
3-4-6-12 tiling
In geometry of the Euclidean plane, the 3-4-6-12 tiling is one of 20 2-uniform tilings of the Euclidean plane by regular polygons, containing regular triangles, squares, hexagons and dodecagons, arranged in two vertex configuration: 3.4.6.4 and 4.6.12.[1][2][3][4]
3-4-6-12 tiling
Type2-uniform tiling
Vertex configuration
3.4.6.4 and 4.6.12
Symmetryp6m, [6,3], (*632)
Rotation symmetryp6, [6,3]+, (632)
Properties2-uniform, 4-isohedral, 4-isotoxal
It has hexagonal symmetry, p6m, [6,3], (*632). It is also called a demiregular tiling by some authors.
Geometry
Its two vertex configurations are shared with two 1-uniform tilings:
rhombitrihexagonal tiling truncated trihexagonal tiling
3.4.6.4
4.6.12
It can be seen as a type of diminished rhombitrihexagonal tiling, with dodecagons replacing periodic sets of hexagons and surrounding squares and triangles. This is similar to the Johnson solid, a diminished rhombicosidodecahedron, which is a rhombicosidodecahedron with faces removed, leading to new decagonal faces. The dual of this variant is shown to the right (deltoidal hexagonal insets).
Related k-uniform tilings of regular polygons
The hexagons can be dissected into 6 triangles, and the dodecagons can be dissected into triangles, hexagons and squares.
Dissected polygons
Hexagon Dodecagon
(each has 2 orientations)
Dual Processes (Dual 'Insets')
3-uniform tilings
48 26 18 (2-uniform)
[36; 32.4.3.4; 32.4.12]
[3.42.6; (3.4.6.4)2]
[36; 32.4.3.4]
V[36; 32.4.3.4; 32.4.12]
V[3.42.6; (3.4.6.4)2]
V[36; 32.4.3.4]
3-uniform duals
Circle Packing
This 2-uniform tiling can be used as a circle packing. Cyan circles are in contact with 3 other circles (2 cyan, 1 pink), corresponding to the V4.6.12 planigon, and pink circles are in contact with 4 other circles (1 cyan, 2 pink), corresponding to the V3.4.6.4 planigon. It is homeomorphic to the ambo operation on the tiling, with the cyan and pink gap polygons corresponding to the cyan and pink circles (mini-vertex configuration polygons; one dimensional duals to the respective planigons). Both images coincide.
C[3.4.6.12] a[3.4.6.12]
Dual tiling
The dual tiling has right triangle and kite faces, defined by face configurations: V3.4.6.4 and V4.6.12, and can be seen combining the deltoidal trihexagonal tiling and kisrhombille tilings.
Dual tiling
V3.4.6.4
V4.6.12
Deltoidal trihexagonal tiling
Kisrhombille tiling
Notes
1. Critchlow, pp. 62–67
2. Grünbaum and Shephard 1986, pp. 65–67
3. In Search of Demiregular Tilings #4
4. Chavey (1989)
References
• Keith Critchlow, Order in Space: A design source book, 1970, pp. 62–67
• Ghyka, M. The Geometry of Art and Life, (1946), 2nd edition, New York: Dover, 1977. Demiregular tiling #15
• Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X. pp. 35–43
• Grünbaum, Branko; Shephard, G. C. (1987). Tilings and Patterns. W. H. Freeman. ISBN 0-7167-1193-1. p. 65
• Sacred Geometry Design Sourcebook: Universal Dimensional Patterns, Bruce Rawles, 1997. pp. 36–37
External links
• Chavey, D. (1989). "Tilings by Regular Polygons—II: A Catalog of Tilings". Computers & Mathematics with Applications. 17: 147–165. doi:10.1016/0898-1221(89)90156-9.
• Dutch, Steve. "Uniform Tilings". Archived from the original on 2006-09-09. Retrieved 2006-09-09.
• Weisstein, Eric W. "Demiregular tessellation". MathWorld.
• In Search of Demiregular Tilings, Helmer Aslaksen
• n-uniform tilings Brian Galebach, 2-Uniform Tiling 1 of 20
| Wikipedia |
3-4 duoprism
In geometry of 4 dimensions, a 3-4 duoprism, the second smallest p-q duoprism, is a 4-polytope resulting from the Cartesian product of a triangle and a square.
Uniform 3-4 duoprisms
Schlegel diagrams
TypePrismatic uniform polychoron
Schläfli symbol{3}×{4}
Coxeter-Dynkin diagram
Cells3 square prisms,
4 triangular prisms
Faces3+12 squares,
4 triangles
Edges24
Vertices12
Vertex figure
Digonal disphenoid
Symmetry[3,2,4], order 48
Dual3-4 duopyramid
Propertiesconvex, vertex-uniform
The 3-4 duoprism exists in some of the uniform 5-polytopes in the B5 family.
Images
Net
3D projection with 3 different rotations
Skew orthogonal projections with primary triangles and squares colored
Related complex polygons
The quasiregular complex polytope 3{}×4{}, , in $\mathbb {C} ^{2}$ has a real representation as a 3-4 duoprism in 4-dimensional space. It has 12 vertices, and 4 3-edges and 3 4-edges. Its symmetry is 3[2]4, order 12.[1]
Related polytopes
The birectified 5-cube, has a uniform 3-4 duoprism vertex figure:
3-4 duopyramid
3-4 duopyramid
Typeduopyramid
Schläfli symbol{3}+{4}
Coxeter-Dynkin diagram
Cells12 digonal disphenoids
Faces24 isosceles triangles
Edges19 (12+3+4)
Vertices7 (3+4)
Symmetry[3,2,4], order 48
Dual3-4 duoprism
Propertiesconvex, facet-transitive
The dual of a 3-4 duoprism is called a 3-4 duopyramid. It has 12 digonal disphenoid cells, 24 isosceles triangular faces, 12 edges, and 7 vertices.
Orthogonal projection
Vertex-centered perspective
See also
• Polytope and polychoron
• Convex regular polychoron
• Duocylinder
• Tesseract
Notes
1. Coxeter, H. S. M.; Regular Complex Polytopes, Cambridge University Press, (1974).
References
• Regular Polytopes, H. S. M. Coxeter, Dover Publications, Inc., 1973, New York, p. 124.
• Coxeter, The Beauty of Geometry: Twelve Essays, Dover Publications, 1999, ISBN 0-486-40919-8 (Chapter 5: Regular Skew Polyhedra in three and four dimensions and their topological analogues)
• Coxeter, H. S. M. Regular Skew Polyhedra in Three and Four Dimensions. Proc. London Math. Soc. 43, 33–62, 1937.
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 26)
• Norman Johnson Uniform Polytopes, Manuscript (1991)
• N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
• Catalogue of Convex Polychora, section 6, George Olshevsky.
External links
• The Fourth Dimension Simply Explained—describes duoprisms as "double prisms" and duocylinders as "double cylinders"
• Polygloss - glossary of higher-dimensional terms
• Exploring Hyperspace with the Geometric Product
| Wikipedia |
Exponentiation
In mathematics, exponentiation is an operation involving two numbers, the base and the exponent or power. Exponentiation is written as bn, where b is the base and n is the power; this is pronounced as "b (raised) to the (power of) n".[1] When n is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, bn is the product of multiplying n bases:[1]
$b^{n}=\underbrace {b\times b\times \dots \times b\times b} _{n{\text{ times}}}.$
bn
notation
base b and exponent n
Arithmetic operations
Addition (+)
$\scriptstyle \left.{\begin{matrix}\scriptstyle {\text{term}}\,+\,{\text{term}}\\\scriptstyle {\text{summand}}\,+\,{\text{summand}}\\\scriptstyle {\text{addend}}\,+\,{\text{addend}}\\\scriptstyle {\text{augend}}\,+\,{\text{addend}}\end{matrix}}\right\}\,=\,$ $\scriptstyle {\text{sum}}$
Subtraction (−)
$\scriptstyle \left.{\begin{matrix}\scriptstyle {\text{term}}\,-\,{\text{term}}\\\scriptstyle {\text{minuend}}\,-\,{\text{subtrahend}}\end{matrix}}\right\}\,=\,$ $\scriptstyle {\text{difference}}$
Multiplication (×)
$\scriptstyle \left.{\begin{matrix}\scriptstyle {\text{factor}}\,\times \,{\text{factor}}\\\scriptstyle {\text{multiplier}}\,\times \,{\text{multiplicand}}\end{matrix}}\right\}\,=\,$ $\scriptstyle {\text{product}}$
Division (÷)
$\scriptstyle \left.{\begin{matrix}\scriptstyle {\frac {\scriptstyle {\text{dividend}}}{\scriptstyle {\text{divisor}}}}\\[1ex]\scriptstyle {\frac {\scriptstyle {\text{numerator}}}{\scriptstyle {\text{denominator}}}}\end{matrix}}\right\}\,=\,$ $\scriptstyle \left\{{\begin{matrix}\scriptstyle {\text{fraction}}\\\scriptstyle {\text{quotient}}\\\scriptstyle {\text{ratio}}\end{matrix}}\right.$
Exponentiation (^)
$\scriptstyle {\text{base}}^{\text{exponent}}\,=\,$ $\scriptstyle {\text{power}}$
nth root (√)
$\scriptstyle {\sqrt[{\text{degree}}]{\scriptstyle {\text{radicand}}}}\,=\,$ $\scriptstyle {\text{root}}$
Logarithm (log)
$\scriptstyle \log _{\text{base}}({\text{anti-logarithm}})\,=\,$ $\scriptstyle {\text{logarithm}}$
The exponent is usually shown as a superscript to the right of the base. In that case, bn is called "b raised to the nth power", "b (raised) to the power of n", "the nth power of b", "b to the nth power",[2] or most briefly as "b to the nth".
Starting from the basic fact stated above that, for any positive integer $n$, $b^{n}$ is $n$ occurrences of $b$ all multiplied by each other, several other properties of exponentiation directly follow. In particular:[nb 1]
${\begin{aligned}b^{n+m}&=\underbrace {b\times \dots \times b} _{n+m{\text{ times}}}\\[1ex]&=\underbrace {b\times \dots \times b} _{n{\text{ times}}}\times \underbrace {b\times \dots \times b} _{m{\text{ times}}}\\[1ex]&=b^{n}\times b^{m}\end{aligned}}$
In other words, when multiplying a base raised to one exponent by the same base raised to another exponent, the exponents add. From this basic rule that exponents add, we can derive that $b^{0}$ must be equal to 1 for any $b\neq 0$, as follows. For any $n$, $b^{0}\times b^{n}=b^{0+n}=b^{n}$. Dividing both sides by $b^{n}$ gives $b^{0}=b^{n}/b^{n}=1$.
The fact that $b^{1}=b$ can similarly be derived from the same rule. For example, $(b^{1})^{3}=b^{1}\times b^{1}\times b^{1}=b^{1+1+1}=b^{3}$. Taking the cube root of both sides gives $b^{1}=b$.
The rule that multiplying makes exponents add can also be used to derive the properties of negative integer exponents. Consider the question of what $b^{-1}$ should mean. In order to respect the "exponents add" rule, it must be the case that $b^{-1}\times b^{1}=b^{-1+1}=b^{0}=1$. Dividing both sides by $b^{1}$ gives $b^{-1}=1/b^{1}$, which can be more simply written as $b^{-1}=1/b$, using the result from above that $b^{1}=b$. By a similar argument, $b^{-n}=1/b^{n}$.
The properties of fractional exponents also follow from the same rule. For example, suppose we consider ${\sqrt {b}}$ and ask if there is some suitable exponent, which we may call $r$, such that $b^{r}={\sqrt {b}}$. From the definition of the square root, we have that ${\sqrt {b}}\times {\sqrt {b}}=b$. Therefore, the exponent $r$ must be such that $b^{r}\times b^{r}=b$. Using the fact that multiplying makes exponents add gives $b^{r+r}=b$. The $b$ on the right-hand side can also be written as $b^{1}$, giving $b^{r+r}=b^{1}$. Equating the exponents on both sides, we have $r+r=1$. Therefore, $r={\frac {1}{2}}$, so ${\sqrt {b}}=b^{1/2}$.
The definition of exponentiation can be extended to allow any real or complex exponent. Exponentiation by integer exponents can also be defined for a wide variety of algebraic structures, including matrices.
Exponentiation is used extensively in many fields, including economics, biology, chemistry, physics, and computer science, with applications such as compound interest, population growth, chemical reaction kinetics, wave behavior, and public-key cryptography.
History of the notation
Etymology
The term power (Latin: potentia, potestas, dignitas) is a mistranslation[3][4] of the ancient Greek δύναμις (dúnamis, here: "amplification"[3]) used by the Greek mathematician Euclid for the square of a line,[5] following Hippocrates of Chios.[6]
Law of exponents
Main article: The Sand Reckoner § Naming large numbers
In The Sand Reckoner, Archimedes discovered and proved the law of exponents, 10a · 10b = 10a+b, necessary to manipulate powers of 10.
Māl and kaʿbah ("square" and "cube")
In the 9th century, the Persian mathematician Muhammad ibn Mūsā al-Khwārizmī used the terms مَال (māl, "possessions", "property") for a square—the Muslims, "like most mathematicians of those and earlier times, thought of a squared number as a depiction of an area, especially of land, hence property"[7]—and كَعْبَة (kaʿbah, "cube") for a cube, which later Islamic mathematicians represented in mathematical notation as the letters mīm (m) and kāf (k), respectively, by the 15th century, as seen in the work of Abū al-Hasan ibn Alī al-Qalasādī.[8]
Introducing exponents
Nicolas Chuquet used a form of exponential notation in the 15th century, for example 122 for 12x2.[9] This was later used by Henricus Grammateus and Michael Stifel in the 16th century. In the late 16th century, Jost Bürgi would use Roman numerals for exponents in a way similar to that of Chuquet, for example iii4 for 4x3.[10]
"Exponent"; "square" and "cube"
The word exponent was coined in 1544 by Michael Stifel.[11][12] In the 16th century, Robert Recorde used the terms square, cube, zenzizenzic (fourth power), sursolid (fifth), zenzicube (sixth), second sursolid (seventh), and zenzizenzizenzic (eighth).[7] Biquadrate has been used to refer to the fourth power as well.
Modern exponential notation
In 1636, James Hume used in essence modern notation, when in L'algèbre de Vietè he wrote Aiii for A3.[13] Early in the 17th century, the first form of our modern exponential notation was introduced by René Descartes in his text titled La Géométrie; there, the notation is introduced in Book I.[14]
I designate ... aa, or a2 in multiplying a by itself; and a3 in multiplying it once more again by a, and thus to infinity.
— René Descartes, La Géométrie
Some mathematicians (such as Descartes) used exponents only for powers greater than two, preferring to represent squares as repeated multiplication. Thus they would write polynomials, for example, as ax + bxx + cx3 + d.
"Indices"
Samuel Jeake introduced the term indices in 1696.[5] A historical synonym, involution, is now rare[15] and should not be confused with its more common meaning.
Variable exponents, non-integer exponents
In 1748, Leonhard Euler introduced variable exponents, and, implicitly, non-integer exponents by writing:
Consider exponentials or powers in which the exponent itself is a variable. It is clear that quantities of this kind are not algebraic functions, since in those the exponents must be constant.[16]
Terminology
The expression b2 = b · b is called "the square of b" or "b squared", because the area of a square with side-length b is b2.
Similarly, the expression b3 = b · b · b is called "the cube of b" or "b cubed", because the volume of a cube with side-length b is b3.
When it is a positive integer, the exponent indicates how many copies of the base are multiplied together. For example, 35 = 3 · 3 · 3 · 3 · 3 = 243. The base 3 appears 5 times in the multiplication, because the exponent is 5. Here, 243 is the 5th power of 3, or 3 raised to the 5th power.
The word "raised" is usually omitted, and sometimes "power" as well, so 35 can be simply read "3 to the 5th", or "3 to the 5". Therefore, the exponentiation bn can be expressed as "b to the power of n", "b to the nth power", "b to the nth", or most briefly as "b to the n".
A formula with nested exponentiation, such as 357 (which means 3(57) and not (35)7), is called a tower of powers, or simply a tower.[17] For example, writing $b^{c^{d}}$ is equivalent to writing $b^{\left(c^{d}\right)}$. This can be generalized to where writing $b^{c^{d^{f}}}$ means $b^{\left(c^{\left(d^{f}\right)}\right)}$. For example, ${\sqrt {100}}$ can be computed as $100^{\frac {1}{2}}$, which can be computed as $100^{2^{-1}}$, which is equal to $100^{\left(2^{-1}\right)}$, which is equal to 10.
Integer exponents
The exponentiation operation with integer exponents may be defined directly from elementary arithmetic operations.
Positive exponents
The definition of the exponentiation as an iterated multiplication can be formalized by using induction,[18] and this definition can be used as soon one has an associative multiplication:
The base case is
$b^{1}=b$
and the recurrence is
$b^{n+1}=b^{n}\cdot b.$
The associativity of multiplication implies that for any positive integers m and n,
$b^{m+n}=b^{m}\cdot b^{n},$
and
$(b^{m})^{n}=b^{mn}.$
Zero exponent
By definition, any nonzero number raised to the 0 power is 1:[19][1]
$b^{0}=1.$
This definition is the only possible that allows extending the formula
$b^{m+n}=b^{m}\cdot b^{n}$
to zero exponents. It may be used in every algebraic structure with a multiplication that has an identity.
Intuitionally, $b^{0}$ may be interpreted as the empty product of copies of b. So, the equality $b^{0}=1$ is a special case of the general convention for the empty product.
The case of 00 is more complicated. In contexts where only integer powers are considered, the value 1 is generally assigned to $0^{0},$ but, otherwise, the choice of whether to assign it a value and what value to assign may depend on context. For more details, see Zero to the power of zero.
Negative exponents
Exponentiation with negative exponents is defined by the following identity, which holds for any integer n and nonzero b:
$b^{-n}={\frac {1}{b^{n}}}$.[1]
Raising 0 to a negative exponent is undefined but, in some circumstances, it may be interpreted as infinity ($\infty $).
This definition of exponentiation with negative exponents is the only one that allows extending the identity $b^{m+n}=b^{m}\cdot b^{n}$ to negative exponents (consider the case $m=-n$).
The same definition applies to invertible elements in a multiplicative monoid, that is, an algebraic structure, with an associative multiplication and a multiplicative identity denoted 1 (for example, the square matrices of a given dimension). In particular, in such a structure, the inverse of an invertible element x is standardly denoted $x^{-1}.$
Identities and properties
The following identities, often called exponent rules, hold for all integer exponents, provided that the base is non-zero:[1]
${\begin{aligned}b^{m+n}&=b^{m}\cdot b^{n}\\\left(b^{m}\right)^{n}&=b^{m\cdot n}\\(b\cdot c)^{n}&=b^{n}\cdot c^{n}\end{aligned}}$
Unlike addition and multiplication, exponentiation is not commutative. For example, 23 = 8 ≠ 32 = 9. Also unlike addition and multiplication, exponentiation is not associative. For example, (23)2 = 82 = 64, whereas 2(32) = 29 = 512. Without parentheses, the conventional order of operations for serial exponentiation in superscript notation is top-down (or right-associative), not bottom-up[20][21][22][23] (or left-associative). That is,
$b^{p^{q}}=b^{\left(p^{q}\right)},$
which, in general, is different from
$\left(b^{p}\right)^{q}=b^{pq}.$
Powers of a sum
The powers of a sum can normally be computed from the powers of the summands by the binomial formula
$(a+b)^{n}=\sum _{i=0}^{n}{\binom {n}{i}}a^{i}b^{n-i}=\sum _{i=0}^{n}{\frac {n!}{i!(n-i)!}}a^{i}b^{n-i}.$
However, this formula is true only if the summands commute (i.e. that ab = ba), which is implied if they belong to a structure that is commutative. Otherwise, if a and b are, say, square matrices of the same size, this formula cannot be used. It follows that in computer algebra, many algorithms involving integer exponents must be changed when the exponentiation bases do not commute. Some general purpose computer algebra systems use a different notation (sometimes ^^ instead of ^) for exponentiation with non-commuting bases, which is then called non-commutative exponentiation.
Combinatorial interpretation
See also: Exponentiation over sets
For nonnegative integers n and m, the value of nm is the number of functions from a set of m elements to a set of n elements (see cardinal exponentiation). Such functions can be represented as m-tuples from an n-element set (or as m-letter words from an n-letter alphabet). Some examples for particular values of m and n are given in the following table:
nm The nm possible m-tuples of elements from the set {1, ..., n}
05 = 0 none
14 = 1 (1, 1, 1, 1)
23 = 8 (1, 1, 1), (1, 1, 2), (1, 2, 1), (1, 2, 2), (2, 1, 1), (2, 1, 2), (2, 2, 1), (2, 2, 2)
32 = 9 (1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)
41 = 4 (1), (2), (3), (4)
50 = 1 ()
Powers of ten
See also: Scientific notation
Main article: Power of 10
In the base ten (decimal) number system, integer powers of 10 are written as the digit 1 followed or preceded by a number of zeroes determined by the sign and magnitude of the exponent. For example, 103 = 1000 and 10−4 = 0.0001.
Exponentiation with base 10 is used in scientific notation to denote large or small numbers. For instance, 299792458 m/s (the speed of light in vacuum, in metres per second) can be written as 2.99792458×108 m/s and then approximated as 2.998×108 m/s.
SI prefixes based on powers of 10 are also used to describe small or large quantities. For example, the prefix kilo means 103 = 1000, so a kilometre is 1000 m.
Powers of two
Main article: Power of two
The first negative powers of 2 are commonly used, and have special names, e.g.: half and quarter.
Powers of 2 appear in set theory, since a set with n members has a power set, the set of all of its subsets, which has 2n members.
Integer powers of 2 are important in computer science. The positive integer powers 2n give the number of possible values for an n-bit integer binary number; for example, a byte may take 28 = 256 different values. The binary number system expresses any number as a sum of powers of 2, and denotes it as a sequence of 0 and 1, separated by a binary point, where 1 indicates a power of 2 that appears in the sum; the exponent is determined by the place of this 1: the nonnegative exponents are the rank of the 1 on the left of the point (starting from 0), and the negative exponents are determined by the rank on the right of the point.
Powers of one
Every power of one equals: 1n = 1. This is true even if n is negative.
The first power of a number is the number itself: $n^{1}=n.$
Powers of zero
If the exponent n is positive (n > 0), the nth power of zero is zero: 0n = 0.
If the exponent n is negative (n < 0), the nth power of zero 0n is undefined, because it must equal $1/0^{-n}$ with −n > 0, and this would be $1/0$ according to above.
The expression 00 is either defined as 1, or it is left undefined.
Powers of negative one
If n is an even integer, then (−1)n = 1. This is because a negative number multiplied by another negative number cancels out, and gives a positive number.
If n is an odd integer, then (−1)n = −1. This is because there will be a remaining (-1) after removing all (-1) pairs.
Because of this, powers of −1 are useful for expressing alternating sequences. For a similar discussion of powers of the complex number i, see § nth roots of a complex number.
Large exponents
The limit of a sequence of powers of a number greater than one diverges; in other words, the sequence grows without bound:
bn → ∞ as n → ∞ when b > 1
This can be read as "b to the power of n tends to +∞ as n tends to infinity when b is greater than one".
Powers of a number with absolute value less than one tend to zero:
bn → 0 as n → ∞ when |b| < 1
Any power of one is always one:
bn = 1 for all n if b = 1
Powers of –1 alternate between 1 and –1 as n alternates between even and odd, and thus do not tend to any limit as n grows.
If b < –1, bn alternates between larger and larger positive and negative numbers as n alternates between even and odd, and thus does not tend to any limit as n grows.
If the exponentiated number varies while tending to 1 as the exponent tends to infinity, then the limit is not necessarily one of those above. A particularly important case is
(1 + 1/n)n → e as n → ∞
See § The exponential function below.
Other limits, in particular those of expressions that take on an indeterminate form, are described in § Limits of powers below.
Power functions
Real functions of the form $f(x)=cx^{n}$, where $c\neq 0$, are sometimes called power functions.[24] When $n$ is an integer and $n\geq 1$, two primary families exist: for $n$ even, and for $n$ odd. In general for $c>0$, when $n$ is even $f(x)=cx^{n}$ will tend towards positive infinity with increasing $x$, and also towards positive infinity with decreasing $x$. All graphs from the family of even power functions have the general shape of $y=cx^{2}$, flattening more in the middle as $n$ increases.[25] Functions with this kind of symmetry ($f(-x)=f(x)$) are called even functions.
When $n$ is odd, $f(x)$'s asymptotic behavior reverses from positive $x$ to negative $x$. For $c>0$, $f(x)=cx^{n}$ will also tend towards positive infinity with increasing $x$, but towards negative infinity with decreasing $x$. All graphs from the family of odd power functions have the general shape of $y=cx^{3}$, flattening more in the middle as $n$ increases and losing all flatness there in the straight line for $n=1$. Functions with this kind of symmetry ($f(-x)=-f(x)$) are called odd functions.
For $c<0$, the opposite asymptotic behavior is true in each case.[25]
Table of powers of decimal digits
nn2n3n4n5n6n7n8n9n10
1111111111
2481632641282565121024
392781243729218765611968359049
416642561024409616384655362621441048576
5251256253125156257812539062519531259765625
636216129677764665627993616796161007769660466176
749343240116807117649823543576480140353607282475249
8645124096327682621442097152167772161342177281073741824
9817296561590495314414782969430467213874204893486784401
10100100010000100000100000010000000100000000100000000010000000000
Rational exponents
If x is a nonnegative real number, and n is a positive integer, $x^{1/n}$ or ${\sqrt[{n}]{x}}$ denotes the unique positive real nth root of x, that is, the unique positive real number y such that $y^{n}=x.$
If x is a positive real number, and ${\frac {p}{q}}$ is a rational number, with p and q > 0 integers, then $ x^{p/q}$ is defined as
$x^{\frac {p}{q}}=\left(x^{p}\right)^{\frac {1}{q}}=(x^{\frac {1}{q}})^{p}.$
The equality on the right may be derived by setting $y=x^{\frac {1}{q}},$ and writing $(x^{\frac {1}{q}})^{p}=y^{p}=\left((y^{p})^{q}\right)^{\frac {1}{q}}=\left((y^{q})^{p}\right)^{\frac {1}{q}}=(x^{p})^{\frac {1}{q}}.$
If r is a positive rational number, $0^{r}=0,$ by definition.
All these definitions are required for extending the identity $(x^{r})^{s}=x^{rs}$ to rational exponents.
On the other hand, there are problems with the extension of these definitions to bases that are not positive real numbers. For example, a negative real number has a real nth root, which is negative, if n is odd, and no real root if n is even. In the latter case, whichever complex nth root one chooses for $x^{\frac {1}{n}},$ the identity $(x^{a})^{b}=x^{ab}$ cannot be satisfied. For example,
$\left((-1)^{2}\right)^{\frac {1}{2}}=1^{\frac {1}{2}}=1\neq (-1)^{2\cdot {\frac {1}{2}}}=(-1)^{1}=-1.$
See § Real exponents and § Non-integer powers of complex numbers for details on the way these problems may be handled.
Real exponents
For positive real numbers, exponentiation to real powers can be defined in two equivalent ways, either by extending the rational powers to reals by continuity (§ Limits of rational exponents, below), or in terms of the logarithm of the base and the exponential function (§ Powers via logarithms, below). The result is always a positive real number, and the identities and properties shown above for integer exponents remain true with these definitions for real exponents. The second definition is more commonly used, since it generalizes straightforwardly to complex exponents.
On the other hand, exponentiation to a real power of a negative real number is much more difficult to define consistently, as it may be non-real and have several values (see § Real exponents with negative bases). One may choose one of these values, called the principal value, but there is no choice of the principal value for which the identity
$\left(b^{r}\right)^{s}=b^{rs}$
is true; see § Failure of power and logarithm identities. Therefore, exponentiation with a basis that is not a positive real number is generally viewed as a multivalued function.
Limits of rational exponents
Since any irrational number can be expressed as the limit of a sequence of rational numbers, exponentiation of a positive real number b with an arbitrary real exponent x can be defined by continuity with the rule[26]
$b^{x}=\lim _{r(\in \mathbb {Q} )\to x}b^{r}\quad (b\in \mathbb {R} ^{+},\,x\in \mathbb {R} ),$
where the limit is taken over rational values of r only. This limit exists for every positive b and every real x.
For example, if x = π, the non-terminating decimal representation π = 3.14159... and the monotonicity of the rational powers can be used to obtain intervals bounded by rational powers that are as small as desired, and must contain $b^{\pi }:$
$\left[b^{3},b^{4}\right],\left[b^{3.1},b^{3.2}\right],\left[b^{3.14},b^{3.15}\right],\left[b^{3.141},b^{3.142}\right],\left[b^{3.1415},b^{3.1416}\right],\left[b^{3.14159},b^{3.14160}\right],\ldots $
So, the upper bounds and the lower bounds of the intervals form two sequences that have the same limit, denoted $b^{\pi }.$
This defines $b^{x}$ for every positive b and real x as a continuous function of b and x. See also Well-defined expression.[27]
The exponential function
Main article: Exponential function
The exponential function is often defined as $x\mapsto e^{x},$ where $e\approx 2.718$ is Euler's number. For avoiding circular reasoning, this definition cannot be used here. So, a definition of the exponential function, denoted $\exp(x),$ and of Euler's number are given, which rely only on exponentiation with positive integer exponents. Then a proof is sketched that, if one uses the definition of exponentiation given in preceding sections, one has
$\exp(x)=e^{x}.$
There are many equivalent ways to define the exponential function, one of them being
$\exp(x)=\lim _{n\rightarrow \infty }\left(1+{\frac {x}{n}}\right)^{n}.$
One has $\exp(0)=1,$ and the exponential identity $\exp(x+y)=\exp(x)\exp(y)$ holds as well, since
$\exp(x)\exp(y)=\lim _{n\rightarrow \infty }\left(1+{\frac {x}{n}}\right)^{n}\left(1+{\frac {y}{n}}\right)^{n}=\lim _{n\rightarrow \infty }\left(1+{\frac {x+y}{n}}+{\frac {xy}{n^{2}}}\right)^{n},$
and the second-order term ${\frac {xy}{n^{2}}}$ does not affect the limit, yielding $\exp(x)\exp(y)=\exp(x+y)$.
Euler's number can be defined as $e=\exp(1)$. It follows from the preceding equations that $\exp(x)=e^{x}$ when x is an integer (this results from the repeated-multiplication definition of the exponentiation). If x is real, $\exp(x)=e^{x}$ results from the definitions given in preceding sections, by using the exponential identity if x is rational, and the continuity of the exponential function otherwise.
The limit that defines the exponential function converges for every complex value of x, and therefore it can be used to extend the definition of $\exp(z)$, and thus $e^{z},$ from the real numbers to any complex argument z. This extended exponential function still satisfies the exponential identity, and is commonly used for defining exponentiation for complex base and exponent.
Powers via logarithms
The definition of ex as the exponential function allows defining bx for every positive real numbers b, in terms of exponential and logarithm function. Specifically, the fact that the natural logarithm ln(x) is the inverse of the exponential function ex means that one has
$b=\exp(\ln b)=e^{\ln b}$
for every b > 0. For preserving the identity $(e^{x})^{y}=e^{xy},$ one must have
$b^{x}=\left(e^{\ln b}\right)^{x}=e^{x\ln b}$
So, $e^{x\ln b}$ can be used as an alternative definition of bx for any positive real b. This agrees with the definition given above using rational exponents and continuity, with the advantage to extend straightforwardly to any complex exponent.
Complex exponents with a positive real base
If b is a positive real number, exponentiation with base b and complex exponent z is defined by means of the exponential function with complex argument (see the end of § The exponential function, above) as
$b^{z}=e^{(z\ln b)},$
where $\ln b$ denotes the natural logarithm of b.
This satisfies the identity
$b^{z+t}=b^{z}b^{t},$
In general, $ \left(b^{z}\right)^{t}$ is not defined, since bz is not a real number. If a meaning is given to the exponentiation of a complex number (see § Non-integer powers of complex numbers, below), one has, in general,
$\left(b^{z}\right)^{t}\neq b^{zt},$
unless z is real or t is an integer.
Euler's formula,
$e^{iy}=\cos y+i\sin y,$
allows expressing the polar form of $b^{z}$ in terms of the real and imaginary parts of z, namely
$b^{x+iy}=b^{x}(\cos(y\ln b)+i\sin(y\ln b)),$
where the absolute value of the trigonometric factor is one. This results from
$b^{x+iy}=b^{x}b^{iy}=b^{x}e^{iy\ln b}=b^{x}(\cos(y\ln b)+i\sin(y\ln b)).$
Non-integer powers of complex numbers
In the preceding sections, exponentiation with non-integer exponents has been defined for positive real bases only. For other bases, difficulties appear already with the apparently simple case of nth roots, that is, of exponents $1/n,$ where n is a positive integer. Although the general theory of exponentiation with non-integer exponents applies to nth roots, this case deserves to be considered first, since it does not need to use complex logarithms, and is therefore easier to understand.
nth roots of a complex number
Every nonzero complex number z may be written in polar form as
$z=\rho e^{i\theta }=\rho (\cos \theta +i\sin \theta ),$
where $\rho $ is the absolute value of z, and $\theta $ is its argument. The argument is defined up to an integer multiple of 2π; this means that, if $\theta $ is the argument of a complex number, then $\theta +2k\pi $ is also an argument of the same complex number for every integer $k$.
The polar form of the product of two complex numbers is obtained by multiplying the absolute values and adding the arguments. It follows that the polar form of an nth root of a complex number can be obtained by taking the nth root of the absolute value and dividing its argument by n:
$\left(\rho e^{i\theta }\right)^{\frac {1}{n}}={\sqrt[{n}]{\rho }}\,e^{\frac {i\theta }{n}}.$
If $2\pi $ is added to $\theta $, the complex number is not changed, but this adds $2i\pi /n$ to the argument of the nth root, and provides a new nth root. This can be done n times, and provides the n nth roots of the complex number.
It is usual to choose one of the n nth root as the principal root. The common choice is to choose the nth root for which $-\pi <\theta \leq \pi ,$ that is, the nth root that has the largest real part, and, if there are two, the one with positive imaginary part. This makes the principal nth root a continuous function in the whole complex plane, except for negative real values of the radicand. This function equals the usual nth root for positive real radicands. For negative real radicands, and odd exponents, the principal nth root is not real, although the usual nth root is real. Analytic continuation shows that the principal nth root is the unique complex differentiable function that extends the usual nth root to the complex plane without the nonpositive real numbers.
If the complex number is moved around zero by increasing its argument, after an increment of $2\pi ,$ the complex number comes back to its initial position, and its nth roots are permuted circularly (they are multiplied by $e^{2i\pi /n$). This shows that it is not possible to define a nth root function that is continuous in the whole complex plane.
Roots of unity
Main article: Root of unity
The nth roots of unity are the n complex numbers such that wn = 1, where n is a positive integer. They arise in various areas of mathematics, such as in discrete Fourier transform or algebraic solutions of algebraic equations (Lagrange resolvent).
The n nth roots of unity are the n first powers of $\omega =e^{\frac {2\pi i}{n}}$, that is $1=\omega ^{0}=\omega ^{n},\omega =\omega ^{1},\omega ^{2},\omega ^{n-1}.$ The nth roots of unity that have this generating property are called primitive nth roots of unity; they have the form $\omega ^{k}=e^{\frac {2k\pi i}{n}},$ with k coprime with n. The unique primitive square root of unity is $-1;$ the primitive fourth roots of unity are $i$ and $-i.$
The nth roots of unity allow expressing all nth roots of a complex number z as the n products of a given nth roots of z with a nth root of unity.
Geometrically, the nth roots of unity lie on the unit circle of the complex plane at the vertices of a regular n-gon with one vertex on the real number 1.
As the number $e^{\frac {2k\pi i}{n}}$ is the primitive nth root of unity with the smallest positive argument, it is called the principal primitive nth root of unity, sometimes shortened as principal nth root of unity, although this terminology can be confused with the principal value of $1^{1/n}$ which is 1.[28][29][30]
Complex exponentiation
Defining exponentiation with complex bases leads to difficulties that are similar to those described in the preceding section, except that there are, in general, infinitely many possible values for $z^{w$. So, either a principal value is defined, which is not continuous for the values of z that are real and nonpositive, or $z^{w$ is defined as a multivalued function.
In all cases, the complex logarithm is used to define complex exponentiation as
$z^{w}=e^{w\log z},$
where $\log z$ is the variant of the complex logarithm that is used, which is, a function or a multivalued function such that
$e^{\log z}=z$
for every z in its domain of definition.
Principal value
The principal value of the complex logarithm is the unique continuous function, commonly denoted $\log ,$ such that, for every nonzero complex number z,
$e^{\log z}=z,$
and the argument of z satisfies
$-\pi <\operatorname {Arg} z\leq \pi .$
The principal value of the complex logarithm is not defined for $z=0,$ it is discontinuous at negative real values of z, and it is holomorphic (that is, complex differentiable) elsewhere. If z is real and positive, the principal value of the complex logarithm is the natural logarithm: $\log z=\ln z.$
The principal value of $z^{w}$ is defined as $z^{w}=e^{w\log z},$ where $\log z$ is the principal value of the logarithm.
The function $(z,w)\to z^{w}$ is holomorphic except in the neighbourhood of the points where z is real and nonpositive.
If z is real and positive, the principal value of $z^{w}$ equals its usual value defined above. If $w=1/n,$ where n is an integer, this principal value is the same as the one defined above.
Multivalued function
In some contexts, there is a problem with the discontinuity of the principal values of $\log z$ and $z^{w}$ at the negative real values of z. In this case, it is useful to consider these functions as multivalued functions.
If $\log z$ denotes one of the values of the multivalued logarithm (typically its principal value), the other values are $2ik\pi +\log z,$ where k is any integer. Similarly, if $z^{w}$ is one value of the exponentiation, then the other values are given by
$e^{w(2ik\pi +\log z)}=z^{w}e^{2ik\pi w},$
where k is any integer.
Different values of k give different values of $z^{w}$ unless w is a rational number, that is, there is an integer d such that dw is an integer. This results from the periodicity of the exponential function, more specifically, that $e^{a}=e^{b}$ if and only if $a-b$ is an integer multiple of $2\pi i.$
If $w={\frac {m}{n}}$ is a rational number with m and n coprime integers with $n>0,$ then $z^{w}$ has exactly n values. In the case $m=1,$ these values are the same as those described in § nth roots of a complex number. If w is an integer, there is only one value that agrees with that of § Integer exponents.
The multivalued exponentiation is holomorphic for $z\neq 0,$ in the sense that its graph consists of several sheets that define each a holomorphic function in the neighborhood of every point. If z varies continuously along a circle around 0, then, after a turn, the value of $z^{w}$ has changed of sheet.
Computation
The canonical form $x+iy$ of $z^{w}$ can be computed from the canonical form of z and w. Although this can be described by a single formula, it is clearer to split the computation in several steps.
• Polar form of z. If $z=a+ib$ is the canonical form of z (a and b being real), then its polar form is
$z=\rho e^{i\theta }=\rho (\cos \theta +i\sin \theta ),$
where $\rho ={\sqrt {a^{2}+b^{2}}}$ and $\theta =\operatorname {atan2} (a,b)$ (see atan2 for the definition of this function).
• Logarithm of z. The principal value of this logarithm is $\log z=\ln \rho +i\theta ,$ where $\ln $ denotes the natural logarithm. The other values of the logarithm are obtained by adding $2ik\pi $ for any integer k.
• Canonical form of $w\log z.$ If $w=c+di$ with c and d real, the values of $w\log z$ are
$w\log z=(c\ln \rho -d\theta -2dk\pi )+i(d\ln \rho +c\theta +2ck\pi ),$
the principal value corresponding to $k=0.$
• Final result. Using the identities $e^{x+y}=e^{x}e^{y}$ and $e^{y\ln x}=x^{y},$ one gets
$z^{w}=\rho ^{c}e^{-d(\theta +2k\pi )}\left(\cos(d\ln \rho +c\theta +2ck\pi )+i\sin(d\ln \rho +c\theta +2ck\pi )\right),$
with $k=0$ for the principal value.
Examples
• $i^{i}$
The polar form of i is $i=e^{i\pi /2},$ and the values of $\log i$ are thus
$\log i=i\left({\frac {\pi }{2}}+2k\pi \right).$
It follows that
$i^{i}=e^{i\log i}=e^{-{\frac {\pi }{2}}}e^{-2k\pi }.$
So, all values of $i^{i}$ are real, the principal one being
$e^{-{\frac {\pi }{2}}}\approx 0.2079.$
• $(-2)^{3+4i}$
Similarly, the polar form of −2 is $-2=2e^{i\pi }.$ So, the above described method gives the values
${\begin{aligned}(-2)^{3+4i}&=2^{3}e^{-4(\pi +2k\pi )}(\cos(4\ln 2+3(\pi +2k\pi ))+i\sin(4\ln 2+3(\pi +2k\pi )))\\&=-2^{3}e^{-4(\pi +2k\pi )}(\cos(4\ln 2)+i\sin(4\ln 2)).\end{aligned}}$
In this case, all the values have the same argument $4\ln 2,$ and different absolute values.
In both examples, all values of $z^{w}$ have the same argument. More generally, this is true if and only if the real part of w is an integer.
Failure of power and logarithm identities
Some identities for powers and logarithms for positive real numbers will fail for complex numbers, no matter how complex powers and complex logarithms are defined as single-valued functions. For example:
• The identity log(bx) = x ⋅ log b holds whenever b is a positive real number and x is a real number. But for the principal branch of the complex logarithm one has
$\log((-i)^{2})=\log(-1)=i\pi \neq 2\log(-i)=2\log(e^{-i\pi /2})=2\,{\frac {-i\pi }{2}}=-i\pi $
Regardless of which branch of the logarithm is used, a similar failure of the identity will exist. The best that can be said (if only using this result) is that:
$\log w^{z}\equiv z\log w{\pmod {2\pi i}}$
This identity does not hold even when considering log as a multivalued function. The possible values of log(wz) contain those of z ⋅ log w as a proper subset. Using Log(w) for the principal value of log(w) and m, n as any integers the possible values of both sides are:
${\begin{aligned}\left\{\log w^{z}\right\}&=\left\{z\cdot \operatorname {Log} w+z\cdot 2\pi in+2\pi im\mid m,n\in \mathbb {Z} \right\}\\\left\{z\log w\right\}&=\left\{z\operatorname {Log} w+z\cdot 2\pi in\mid n\in \mathbb {Z} \right\}\end{aligned}}$
• The identities (bc)x = bxcx and (b/c)x = bx/cx are valid when b and c are positive real numbers and x is a real number. But, for the principal values, one has
$(-1\cdot -1)^{\frac {1}{2}}=1\neq (-1)^{\frac {1}{2}}(-1)^{\frac {1}{2}}=-1$
and
$\left({\frac {1}{-1}}\right)^{\frac {1}{2}}=(-1)^{\frac {1}{2}}=i\neq {\frac {1^{\frac {1}{2}}}{(-1)^{\frac {1}{2}}}}={\frac {1}{i}}=-i$
On the other hand, when x is an integer, the identities are valid for all nonzero complex numbers.
If exponentiation is considered as a multivalued function then the possible values of (−1 ⋅ −1)1/2 are {1, −1}. The identity holds, but saying {1} = {(−1 ⋅ −1)1/2} is wrong.
• The identity (ex)y = exy holds for real numbers x and y, but assuming its truth for complex numbers leads to the following paradox, discovered in 1827 by Clausen:[31] For any integer n, we have:
1. $e^{1+2\pi in}=e^{1}e^{2\pi in}=e\cdot 1=e$
2. $\left(e^{1+2\pi in}\right)^{1+2\pi in}=e\qquad $ (taking the $(1+2\pi in)$-th power of both sides)
3. $e^{1+4\pi in-4\pi ^{2}n^{2}}=e\qquad $ (using $\left(e^{x}\right)^{y}=e^{xy}$ and expanding the exponent)
4. $e^{1}e^{4\pi in}e^{-4\pi ^{2}n^{2}}=e\qquad $ (using $e^{x+y}=e^{x}e^{y}$)
5. $e^{-4\pi ^{2}n^{2}}=1\qquad $ (dividing by e)
but this is false when the integer n is nonzero.
The error is the following: by definition, $e^{y}$ is a notation for $\exp(y),$ a true function, and $x^{y}$ is a notation for $\exp(y\log x),$ which is a multi-valued function. Thus the notation is ambiguous when x = e. Here, before expanding the exponent, the second line should be
$\exp \left((1+2\pi in)\log \exp(1+2\pi in)\right)=\exp(1+2\pi in).$
Therefore, when expanding the exponent, one has implicitly supposed that $\log \exp z=z$ for complex values of z, which is wrong, as the complex logarithm is multivalued. In other words, the wrong identity (ex)y = exy must be replaced by the identity
$\left(e^{x}\right)^{y}=e^{y\log e^{x}},$
which is a true identity between multivalued functions.
Irrationality and transcendence
Main article: Gelfond–Schneider theorem
If b is a positive real algebraic number, and x is a rational number, then bx is an algebraic number. This results from the theory of algebraic extensions. This remains true if b is any algebraic number, in which case, all values of bx (as a multivalued function) are algebraic. If x is irrational (that is, not rational), and both b and x are algebraic, Gelfond–Schneider theorem asserts that all values of bx are transcendental (that is, not algebraic), except if b equals 0 or 1.
In other words, if x is irrational and $b\not \in \{0,1\},$ then at least one of b, x and bx is transcendental.
Integer powers in algebra
The definition of exponentiation with positive integer exponents as repeated multiplication may apply to any associative operation denoted as a multiplication.[nb 2] The definition of $x^{0}$ requires further the existence of a multiplicative identity.[32]
An algebraic structure consisting of a set together with an associative operation denoted multiplicatively, and a multiplicative identity denoted by 1 is a monoid. In such a monoid, exponentiation of an element x is defined inductively by
• $x^{0}=1,$
• $x^{n+1}=xx^{n}$ for every nonnegative integer n.
If n is a negative integer, $x^{n}$ is defined only if x has a multiplicative inverse.[33] In this case, the inverse of x is denoted $x^{-1},$ and $x^{n}$ is defined as $\left(x^{-1}\right)^{-n}.$
Exponentiation with integer exponents obeys the following laws, for x and y in the algebraic structure, and m and n integers:
${\begin{aligned}x^{0}&=1\\x^{m+n}&=x^{m}x^{n}\\(x^{m})^{n}&=x^{mn}\\(xy)^{n}&=x^{n}y^{n}\quad {\text{if }}xy=yx,{\text{and, in particular, if the multiplication is commutative.}}\end{aligned}}$
These definitions are widely used in many areas of mathematics, notably for groups, rings, fields, square matrices (which form a ring). They apply also to functions from a set to itself, which form a monoid under function composition. This includes, as specific instances, geometric transformations, and endomorphisms of any mathematical structure.
When there are several operations that may be repeated, it is common to indicate the repeated operation by placing its symbol in the superscript, before the exponent. For example, if f is a real function whose valued can be multiplied, $f^{n}$ denotes the exponentiation with respect of multiplication, and $f^{\circ n}$ may denote exponentiation with respect of function composition. That is,
$(f^{n})(x)=(f(x))^{n}=f(x)\,f(x)\cdots f(x),$
and
$(f^{\circ n})(x)=f(f(\cdots f(f(x))\cdots )).$
Commonly, $(f^{n})(x)$ is denoted $f(x)^{n},$ while $(f^{\circ n})(x)$ is denoted $f^{n}(x).$
In a group
A multiplicative group is a set with as associative operation denoted as multiplication, that has an identity element, and such that every element has an inverse.
So, if G is a group, $x^{n}$ is defined for every $x\in G$ and every integer n.
The set of all powers of an element of a group form a subgroup. A group (or subgroup) that consists of all powers of a specific element x is the cyclic group generated by x. If all the powers of x are distinct, the group is isomorphic to the additive group $\mathbb {Z} $ of the integers. Otherwise, the cyclic group is finite (it has a finite number of elements), and its number of elements is the order of x. If the order of x is n, then $x^{n}=x^{0}=1,$ and the cyclic group generated by x consists of the n first powers of x (starting indifferently from the exponent 0 or 1).
Order of elements play a fundamental role in group theory. For example, the order of an element in a finite group is always a divisor of the number of elements of the group (the order of the group). The possible orders of group elements are important in the study of the structure of a group (see Sylow theorems), and in the classification of finite simple groups.
Superscript notation is also used for conjugation; that is, gh = h−1gh, where g and h are elements of a group. This notation cannot be confused with exponentiation, since the superscript is not an integer. The motivation of this notation is that conjugation obeys some of the laws of exponentiation, namely $(g^{h})^{k}=g^{hk}$ and $(gh)^{k}=g^{k}h^{k}.$
In a ring
In a ring, it may occur that some nonzero elements satisfy $x^{n}=0$ for some integer n. Such an element is said to be nilpotent. In a commutative ring, the nilpotent elements form an ideal, called the nilradical of the ring.
If the nilradical is reduced to the zero ideal (that is, if $x\neq 0$ implies $x^{n}\neq 0$ for every positive integer n), the commutative ring is said reduced. Reduced rings important in algebraic geometry, since the coordinate ring of an affine algebraic set is always a reduced ring.
More generally, given an ideal I in a commutative ring R, the set of the elements of R that have a power in I is an ideal, called the radical of I. The nilradical is the radical of the zero ideal. A radical ideal is an ideal that equals its own radical. In a polynomial ring $k[x_{1},\ldots ,x_{n}]$ over a field k, an ideal is radical if and only if it is the set of all polynomials that are zero on an affine algebraic set (this is a consequence of Hilbert's Nullstellensatz).
Matrices and linear operators
If A is a square matrix, then the product of A with itself n times is called the matrix power. Also $A^{0}$ is defined to be the identity matrix,[34] and if A is invertible, then $A^{-n}=\left(A^{-1}\right)^{n}$.
Matrix powers appear often in the context of discrete dynamical systems, where the matrix A expresses a transition from a state vector x of some system to the next state Ax of the system.[35] This is the standard interpretation of a Markov chain, for example. Then $A^{2}x$ is the state of the system after two time steps, and so forth: $A^{n}x$ is the state of the system after n time steps. The matrix power $A^{n}$ is the transition matrix between the state now and the state at a time n steps in the future. So computing matrix powers is equivalent to solving the evolution of the dynamical system. In many cases, matrix powers can be expediently computed by using eigenvalues and eigenvectors.
Apart from matrices, more general linear operators can also be exponentiated. An example is the derivative operator of calculus, $d/dx$, which is a linear operator acting on functions $f(x)$ to give a new function $(d/dx)f(x)=f'(x)$. The n-th power of the differentiation operator is the n-th derivative:
$\left({\frac {d}{dx}}\right)^{n}f(x)={\frac {d^{n}}{dx^{n}}}f(x)=f^{(n)}(x).$
These examples are for discrete exponents of linear operators, but in many circumstances it is also desirable to define powers of such operators with continuous exponents. This is the starting point of the mathematical theory of semigroups.[36] Just as computing matrix powers with discrete exponents solves discrete dynamical systems, so does computing matrix powers with continuous exponents solve systems with continuous dynamics. Examples include approaches to solving the heat equation, Schrödinger equation, wave equation, and other partial differential equations including a time evolution. The special case of exponentiating the derivative operator to a non-integer power is called the fractional derivative which, together with the fractional integral, is one of the basic operations of the fractional calculus.
Finite fields
Main article: Finite field
A field is an algebraic structure in which multiplication, addition, subtraction, and division are defined and satisfy the properties that multiplication is associative and every nonzero element has a multiplicative inverse. This implies that exponentiation with integer exponents is well-defined, except for nonpositive powers of 0. Common examples are the complex numbers and their subfields, the rational numbers and the real numbers, which have been considered earlier in this article, and are all infinite.
A finite field is a field with a finite number of elements. This number of elements is either a prime number or a prime power; that is, it has the form $q=p^{k},$ where p is a prime number, and k is a positive integer. For every such q, there are fields with q elements. The fields with q elements are all isomorphic, which allows, in general, working as if there were only one field with q elements, denoted $\mathbb {F} _{q}.$
One has
$x^{q}=x$
for every $x\in \mathbb {F} _{q}.$
A primitive element in $\mathbb {F} _{q}$ is an element g such that the set of the q − 1 first powers of g (that is, $\{g^{1}=g,g^{2},\ldots ,g^{p-1}=g^{0}=1\}$) equals the set of the nonzero elements of $\mathbb {F} _{q}.$ There are $\varphi (p-1)$ primitive elements in $\mathbb {F} _{q},$ where $\varphi $ is Euler's totient function.
In $\mathbb {F} _{q},$ the Freshman's dream identity
$(x+y)^{p}=x^{p}+y^{p}$
is true for the exponent p. As $x^{p}=x$ in $\mathbb {F} _{q},$ It follows that the map
${\begin{aligned}F\colon {}&\mathbb {F} _{q}\to \mathbb {F} _{q}\\&x\mapsto x^{p}\end{aligned}}$
is linear over $\mathbb {F} _{q},$ and is a field automorphism, called the Frobenius automorphism. If $q=p^{k},$ the field $\mathbb {F} _{q}$ has k automorphisms, which are the k first powers (under composition) of F. In other words, the Galois group of $\mathbb {F} _{q}$ is cyclic of order k, generated by the Frobenius automorphism.
The Diffie–Hellman key exchange is an application of exponentiation in finite fields that is widely used for secure communications. It uses the fact that exponentiation is computationally inexpensive, whereas the inverse operation, the discrete logarithm, is computationally expensive. More precisely, if g is a primitive element in $\mathbb {F} _{q},$ then $g^{e}$ can be efficiently computed with exponentiation by squaring for any e, even if q is large, while there is no known algorithm allowing retrieving e from $g^{e}$ if q is sufficiently large.
Powers of sets
The Cartesian product of two sets S and T is the set of the ordered pairs $(x,y)$ such that $x\in S$ and $y\in T.$ This operation is not properly commutative nor associative, but has these properties up to canonical isomorphisms, that allow identifying, for example, $(x,(y,z)),$ $((x,y),z),$ and $(x,y,z).$
This allows defining the nth power $S^{n}$ of a set S as the set of all n-tuples $(x_{1},\ldots ,x_{n})$ of elements of S.
When S is endowed with some structure, it is frequent that $S^{n}$ is naturally endowed with a similar structure. In this case, the term "direct product" is generally used instead of "Cartesian product", and exponentiation denotes product structure. For example $\mathbb {R} ^{n}$ (where $\mathbb {R} $ denotes the real numbers) denotes the Cartesian product of n copies of $\mathbb {R} ,$ as well as their direct product as vector space, topological spaces, rings, etc.
Sets as exponents
See also: Function (mathematics) § Set exponentiation
A n-tuple $(x_{1},\ldots ,x_{n})$ of elements of S can be considered as a function from $\{1,\ldots ,n\}.$ This generalizes to the following notation.
Given two sets S and T, the set of all functions from T to S is denoted $S^{T}$. This exponential notation is justified by the following canonical isomorphisms (for the first one, see Currying):
$(S^{T})^{U}\cong S^{T\times U},$
$S^{T\sqcup U}\cong S^{T}\times S^{U},$
where $\times $ denotes the Cartesian product, and $\sqcup $ the disjoint union.
One can use sets as exponents for other operations on sets, typically for direct sums of abelian groups, vector spaces, or modules. For distinguishing direct sums from direct products, the exponent of a direct sum is placed between parentheses. For example, $\mathbb {R} ^{\mathbb {N} }$ denotes the vector space of the infinite sequences of real numbers, and $\mathbb {R} ^{(\mathbb {N} )}$ the vector space of those sequences that have a finite number of nonzero elements. The latter has a basis consisting of the sequences with exactly one nonzero element that equals 1, while the Hamel bases of the former cannot be explicitly described (because their existence involves Zorn's lemma).
In this context, 2 can represents the set $\{0,1\}.$ So, $2^{S}$ denotes the power set of S, that is the set of the functions from S to $\{0,1\},$ which can be identified with the set of the subsets of S, by mapping each function to the inverse image of 1.
This fits in with the exponentiation of cardinal numbers, in the sense that |ST| = |S||T|, where |X| is the cardinality of X.
In category theory
Main article: Cartesian closed category
In the category of sets, the morphisms between sets X and Y are the functions from X to Y. It results that the set of the functions from X to Y that is denoted $Y^{X}$ in the preceding section can also be denoted $\hom(X,Y).$ The isomorphism $(S^{T})^{U}\cong S^{T\times U}$ can be rewritten
$\hom(U,S^{T})\cong \hom(T\times U,S).$
This means the functor "exponentiation to the power T " is a right adjoint to the functor "direct product with T ".
This generalizes to the definition of exponentiation in a category in which finite direct products exist: in such a category, the functor $X\to X^{T}$ is, if it exists, a right adjoint to the functor $Y\to T\times Y.$ A category is called a Cartesian closed category, if direct products exist, and the functor $Y\to X\times Y$ has a right adjoint for every T.
Repeated exponentiation
Main articles: Tetration and Hyperoperation
Just as exponentiation of natural numbers is motivated by repeated multiplication, it is possible to define an operation based on repeated exponentiation; this operation is sometimes called hyper-4 or tetration. Iterating tetration leads to another operation, and so on, a concept named hyperoperation. This sequence of operations is expressed by the Ackermann function and Knuth's up-arrow notation. Just as exponentiation grows faster than multiplication, which is faster-growing than addition, tetration is faster-growing than exponentiation. Evaluated at (3, 3), the functions addition, multiplication, exponentiation, and tetration yield 6, 9, 27, and 7625597484987 (= 327 = 333 = 33) respectively.
Limits of powers
Zero to the power of zero gives a number of examples of limits that are of the indeterminate form 00. The limits in these examples exist, but have different values, showing that the two-variable function xy has no limit at the point (0, 0). One may consider at what points this function does have a limit.
More precisely, consider the function $f(x,y)=x^{y}$ defined on $D=\{(x,y)\in \mathbf {R} ^{2}:x>0\}$. Then D can be viewed as a subset of R2 (that is, the set of all pairs (x, y) with x, y belonging to the extended real number line R = [−∞, +∞], endowed with the product topology), which will contain the points at which the function f has a limit.
In fact, f has a limit at all accumulation points of D, except for (0, 0), (+∞, 0), (1, +∞) and (1, −∞).[37] Accordingly, this allows one to define the powers xy by continuity whenever 0 ≤ x ≤ +∞, −∞ ≤ y ≤ +∞, except for 00, (+∞)0, 1+∞ and 1−∞, which remain indeterminate forms.
Under this definition by continuity, we obtain:
• x+∞ = +∞ and x−∞ = 0, when 1 < x ≤ +∞.
• x+∞ = 0 and x−∞ = +∞, when 0 ≤ x < 1.
• 0y = 0 and (+∞)y = +∞, when 0 < y ≤ +∞.
• 0y = +∞ and (+∞)y = 0, when −∞ ≤ y < 0.
These powers are obtained by taking limits of xy for positive values of x. This method does not permit a definition of xy when x < 0, since pairs (x, y) with x < 0 are not accumulation points of D.
On the other hand, when n is an integer, the power xn is already meaningful for all values of x, including negative ones. This may make the definition 0n = +∞ obtained above for negative n problematic when n is odd, since in this case xn → +∞ as x tends to 0 through positive values, but not negative ones.
Efficient computation with integer exponents
Computing bn using iterated multiplication requires n − 1 multiplication operations, but it can be computed more efficiently than that, as illustrated by the following example. To compute 2100, apply Horner's rule to the exponent 100 written in binary:
$100=2^{2}+2^{5}+2^{6}=2^{2}(1+2^{3}(1+2))$.
Then compute the following terms in order, reading Horner's rule from right to left.
22 = 4
2 (22) = 23 = 8
(23)2 = 26 = 64
(26)2 = 212 = 4096
(212)2 = 224 = 16777216
2 (224) = 225 = 33554432
(225)2 = 250 = 1125899906842624
(250)2 = 2100 = 1267650600228229401496703205376
This series of steps only requires 8 multiplications instead of 99.
In general, the number of multiplication operations required to compute bn can be reduced to $\sharp n+\lfloor \log _{2}n\rfloor -1,$ by using exponentiation by squaring, where $\sharp n$ denotes the number of 1 in the binary representation of n. For some exponents (100 is not among them), the number of multiplications can be further reduced by computing and using the minimal addition-chain exponentiation. Finding the minimal sequence of multiplications (the minimal-length addition chain for the exponent) for bn is a difficult problem, for which no efficient algorithms are currently known (see Subset sum problem), but many reasonably efficient heuristic algorithms are available.[38] However, in practical computations, exponentiation by squaring is efficient enough, and much more easy to implement.
Iterated functions
Function composition is a binary operation that is defined on functions such that the codomain of the function written on the right is included in the domain of the function written on the left. It is denoted $g\circ f,$ and defined as
$(g\circ f)(x)=g(f(x))$
for every x in the domain of f.
If the domain of a function f equals its codomain, one may compose the function with itself an arbitrary number of time, and this defines the nth power of the function under composition, commonly called the nth iterate of the function. Thus $f^{n}$ denotes generally the nth iterate of f; for example, $f^{3}(x)$ means $f(f(f(x))).$[39]
When a multiplication is defined on the codomain of the function, this defines a multiplication on functions, the pointwise multiplication, which induces another exponentiation. When using functional notation, the two kinds of exponentiation are generally distinguished by placing the exponent of the functional iteration before the parentheses enclosing the arguments of the function, and placing the exponent of pointwise multiplication after the parentheses. Thus $f^{2}(x)=f(f(x)),$ and $f(x)^{2}=f(x)\cdot f(x).$ When functional notation is not used, disambiguation is often done by placing the composition symbol before the exponent; for example $f^{\circ 3}=f\circ f\circ f,$ and $f^{3}=f\cdot f\cdot f.$ For historical reasons, the exponent of a repeated multiplication is placed before the argument for some specific functions, typically the trigonometric functions. So, $\sin ^{2}x$ and $\sin ^{2}(x)$ both mean $\sin(x)\cdot \sin(x)$ and not $\sin(\sin(x)),$ which, in any case, is rarely considered. Historically, several variants of these notations were used by different authors.[40][41][42]
In this context, the exponent $-1$ denotes always the inverse function, if it exists. So $\sin ^{-1}x=\sin ^{-1}(x)=\arcsin x.$ For the multiplicative inverse fractions are generally used as in $1/\sin(x)={\frac {1}{\sin x}}.$
In programming languages
Programming languages generally express exponentiation either as an infix operator or as a function application, as they do not support superscripts. The most common operator symbol for exponentiation is the caret (^). The original version of ASCII included an uparrow symbol (↑), intended for exponentiation, but this was replaced by the caret in 1967, so the caret became usual in programming languages.[43] The notations include:
• x ^ y: AWK, BASIC, J, MATLAB, Wolfram Language (Mathematica), R, Microsoft Excel, Analytica, TeX (and its derivatives), TI-BASIC, bc (for integer exponents), Haskell (for nonnegative integer exponents), Lua and most computer algebra systems.
• x ** y. The Fortran character set did not include lowercase characters or punctuation symbols other than +-*/()&=.,' and so used ** for exponentiation[44][45] (the initial version used a xx b instead.[46]). Many other languages followed suit: Ada, Z shell, KornShell, Bash, COBOL, CoffeeScript, Fortran, FoxPro, Gnuplot, Groovy, JavaScript, OCaml, F#, Perl, PHP, PL/I, Python, Rexx, Ruby, SAS, Seed7, Tcl, ABAP, Mercury, Haskell (for floating-point exponents), Turing, VHDL.
• x ↑ y: Algol Reference language, Commodore BASIC, TRS-80 Level II/III BASIC.[47][48]
• x ^^ y: Haskell (for fractional base, integer exponents), D.
• x⋆y: APL.
In most programming languages with an infix exponentiation operator, it is right-associative, that is, a^b^c is interpreted as a^(b^c).[49] This is because (a^b)^c is equal to a^(b*c) and thus not as useful. In some languages, it is left-associative, notably in Algol, Matlab and the Microsoft Excel formula language.
Other programming languages use functional notation:
• (expt x y): Common Lisp.
• pown x y: F# (for integer base, integer exponent).
Still others only provide exponentiation as part of standard libraries:
• pow(x, y): C, C++ (in math library).
• Math.Pow(x, y): C#.
• math:pow(X, Y): Erlang.
• Math.pow(x, y): Java.
• [Math]::Pow(x, y): PowerShell.
In some statically typed languages that prioritize type safety such as Rust, exponentiation is performed via a multitude of methods:
• x.pow(y) for x and y as integers
• x.powf(y) for x and y as floating point numbers
• x.powi(y) for x as a float and y as an integer
See also
• Double exponential function
• Exponential decay
• Exponential field
• Exponential growth
• List of exponential topics
• Modular exponentiation
• Scientific notation
• Unicode subscripts and superscripts
• xy = yx
• Zero to the power of zero
Arithmetic expressions Polynomial expressions Algebraic expressions Closed-form expressions Analytic expressions Mathematical expressions
Constant YesYesYesYesYesYes
Elementary arithmetic operation YesAddition, subtraction, and multiplication onlyYesYesYesYes
Finite sum YesYesYesYesYesYes
Finite product YesYesYesYesYesYes
Finite continued fraction YesNoYesYesYesYes
Variable NoYesYesYesYesYes
Integer exponent NoYesYesYesYesYes
Integer nth root NoNoYesYesYesYes
Rational exponent NoNoYesYesYesYes
Integer factorial NoNoYesYesYesYes
Irrational exponent NoNoNoYesYesYes
Exponential function NoNoNoYesYesYes
Logarithm NoNoNoYesYesYes
Trigonometric function NoNoNoYesYesYes
Inverse trigonometric function NoNoNoYesYesYes
Hyperbolic function NoNoNoYesYesYes
Inverse hyperbolic function NoNoNoYesYesYes
Root of a polynomial that is not an algebraic solution NoNoNoNoYesYes
Gamma function and factorial of a non-integer NoNoNoNoYesYes
Bessel function NoNoNoNoYesYes
Special function NoNoNoNoYesYes
Infinite sum (series) (including power series) NoNoNoNoConvergent onlyYes
Infinite product NoNoNoNoConvergent onlyYes
Infinite continued fraction NoNoNoNoConvergent onlyYes
Limit NoNoNoNoNoYes
Derivative NoNoNoNoNoYes
Integral NoNoNoNoNoYes
Notes
1. There are three common notations for multiplication: $x\times y$ is most commonly used for explicit numbers and at a very elementary level; $xy$ is most common when variables are used; $x\cdot y$ is used for emphasizing that one talks of multiplication or when omitting the multiplication sign would be confusing.
2. More generally, power associativity is sufficient for the definition.
References
1. Nykamp, Duane. "Basic rules for exponentiation". Math Insight. Retrieved 2020-08-27.
2. Weisstein, Eric W. "Power". mathworld.wolfram.com. Retrieved 2020-08-27.
3. Rotman, Joseph J. (2015). Advanced Modern Algebra, Part 1. Graduate Studies in Mathematics. Vol. 165 (3rd ed.). Providence, RI: American Mathematical Society. p. 130, fn. 4. ISBN 978-1-4704-1554-9.
4. Szabó, Árpád (1978). The Beginnings of Greek Mathematics. Synthese Historical Library. Vol. 17. Translated by A.M. Ungar. Dordrecht: D. Reidel. p. 37. ISBN 90-277-0819-3.
5. O'Connor, John J.; Robertson, Edmund F., "Etymology of some common mathematical terms", MacTutor History of Mathematics Archive, University of St Andrews
6. Ball, W. W. Rouse (1915). A Short Account of the History of Mathematics (6th ed.). London: Macmillan. p. 38.
7. Quinion, Michael. "Zenzizenzizenzic". World Wide Words. Retrieved 2020-04-16.
8. O'Connor, John J.; Robertson, Edmund F., "Abu'l Hasan ibn Ali al Qalasadi", MacTutor History of Mathematics Archive, University of St Andrews
9. Cajori, Florian (1928). A History of Mathematical Notations. Vol. 1. The Open Court Company. p. 102.
10. Cajori, Florian (1928). A History of Mathematical Notations. Vol. 1. London: Open Court Publishing Company. p. 344.
11. Earliest Known Uses of Some of the Words of Mathematics
12. Stifel, Michael (1544). Arithmetica integra. Nuremberg: Johannes Petreius. p. 235v.
13. Cajori, Florian (1928). A History of Mathematical Notations. Vol. 1. The Open Court Company. p. 204.
14. Descartes, René (1637). "La Géométrie". Discourse de la méthode [...]. Leiden: Jan Maire. p. 299. Et aa, ou a2, pour multiplier a par soy mesme; Et a3, pour le multiplier encore une fois par a, & ainsi a l'infini (And aa, or a2, in order to multiply a by itself; and a3, in order to multiply it once more by a, and thus to infinity).
15. The most recent usage in this sense cited by the OED is from 1806 ("involution". Oxford English Dictionary (Online ed.). Oxford University Press. (Subscription or participating institution membership required.)).
16. Euler, Leonhard (1748). Introductio in analysin infinitorum (in Latin). Vol. I. Lausanne: Marc-Michel Bousquet. pp. 69, 98–99. Primum ergo considerandæ sunt quantitates exponentiales, seu Potestates, quarum Exponens ipse est quantitas variabilis. Perspicuum enim est hujusmodi quantitates ad Functiones algebraicas referri non posse, cum in his Exponentes non nisi constantes locum habeant.
17. Kauffman, Louis; J. Lomonaco, Samuel; Chen, Goong, eds. (2007-09-19). "4.6 Efficient decomposition of Hamiltonian". Mathematics of Quantum Computation and Quantum Technology. CRC Press. p. 105. ISBN 9781584889007. Archived from the original on 2022-02-26. Retrieved 2022-02-26.
18. Hodge, Jonathan K.; Schlicker, Steven; Sundstorm, Ted (2014). Abstract Algebra: an inquiry based approach. CRC Press. p. 94. ISBN 978-1-4665-6706-1.
19. Achatz, Thomas (2005). Technical Shop Mathematics (3rd ed.). Industrial Press. p. 101. ISBN 978-0-8311-3086-2.
20. Robinson, Raphael Mitchel (October 1958) [1958-04-07]. "A report on primes of the form k · 2n + 1 and on factors of Fermat numbers" (PDF). Proceedings of the American Mathematical Society. University of California, Berkeley, California, USA. 9 (5): 673–681 [677]. doi:10.1090/s0002-9939-1958-0096614-7. Archived (PDF) from the original on 2020-06-28. Retrieved 2020-06-28.
21. Bronstein, Ilja Nikolaevič; Semendjajew, Konstantin Adolfovič (1987) [1945]. "2.4.1.1. Definition arithmetischer Ausdrücke" [Definition of arithmetic expressions]. Written at Leipzig, Germany. In Grosche, Günter; Ziegler, Viktor; Ziegler, Dorothea (eds.). Taschenbuch der Mathematik [Pocketbook of mathematics] (in German). Vol. 1. Translated by Ziegler, Viktor. Weiß, Jürgen (23 ed.). Thun, Switzerland / Frankfurt am Main, Germany: Verlag Harri Deutsch (and B. G. Teubner Verlagsgesellschaft, Leipzig). pp. 115–120, 802. ISBN 3-87144-492-8.
22. Olver, Frank W. J.; Lozier, Daniel W.; Boisvert, Ronald F.; Clark, Charles W., eds. (2010). NIST Handbook of Mathematical Functions. National Institute of Standards and Technology (NIST), U.S. Department of Commerce, Cambridge University Press. ISBN 978-0-521-19225-5. MR 2723248.
23. Zeidler, Eberhard [in German]; Schwarz, Hans Rudolf; Hackbusch, Wolfgang; Luderer, Bernd [in German]; Blath, Jochen; Schied, Alexander; Dempe, Stephan; Wanka, Gert; Hromkovič, Juraj; Gottwald, Siegfried (2013) [2012]. Zeidler, Eberhard [in German] (ed.). Springer-Handbuch der Mathematik I (in German). Vol. I (1 ed.). Berlin / Heidelberg, Germany: Springer Spektrum, Springer Fachmedien Wiesbaden. p. 590. doi:10.1007/978-3-658-00285-5. ISBN 978-3-658-00284-8. (xii+635 pages)
24. Hass, Joel R.; Heil, Christopher E.; Weir, Maurice D.; Thomas, George B. (2018). Thomas' Calculus (14 ed.). Pearson. pp. 7–8. ISBN 9780134439020.
25. Anton, Howard; Bivens, Irl; Davis, Stephen (2012). Calculus: Early Transcendentals (9th ed.). John Wiley & Sons. p. 28. ISBN 9780470647691.
26. Denlinger, Charles G. (2011). Elements of Real Analysis. Jones and Bartlett. pp. 278–283. ISBN 978-0-7637-7947-4.
27. Tao, Terence (2016). "Limits of sequences". Analysis I. Texts and Readings in Mathematics. Vol. 37. pp. 126–154. doi:10.1007/978-981-10-1789-6_6. ISBN 978-981-10-1789-6.
28. Cormen, Thomas H.; Leiserson, Charles E.; Rivest, Ronald L.; Stein, Clifford (2001). Introduction to Algorithms (second ed.). MIT Press. ISBN 978-0-262-03293-3. Online resource Archived 2007-09-30 at the Wayback Machine
29. Cull, Paul; Flahive, Mary; Robson, Robby (2005). Difference Equations: From Rabbits to Chaos (Undergraduate Texts in Mathematics ed.). Springer. ISBN 978-0-387-23234-8. Defined on p. 351
30. "Principal root of unity", MathWorld.
31. Steiner, J.; Clausen, T.; Abel, Niels Henrik (1827). "Aufgaben und Lehrsätze, erstere aufzulösen, letztere zu beweisen" [Problems and propositions, the former to solve, the later to prove]. Journal für die reine und angewandte Mathematik. 2: 286–287.
32. Bourbaki, Nicolas (1970). Algèbre. Springer., I.2
33. Bloom, David M. (1979). Linear Algebra and Geometry. Cambridge University Press. p. 45. ISBN 978-0-521-29324-2.
34. Chapter 1, Elementary Linear Algebra, 8E, Howard Anton
35. Strang, Gilbert (1988), Linear algebra and its applications (3rd ed.), Brooks-Cole, Chapter 5.
36. E. Hille, R. S. Phillips: Functional Analysis and Semi-Groups. American Mathematical Society, 1975.
37. Nicolas Bourbaki, Topologie générale, V.4.2.
38. Gordon, D. M. (1998). "A Survey of Fast Exponentiation Methods" (PDF). Journal of Algorithms. 27: 129–146. CiteSeerX 10.1.1.17.7076. doi:10.1006/jagm.1997.0913.
39. Peano, Giuseppe (1903). Formulaire mathématique (in French). Vol. IV. p. 229.
40. Herschel, John Frederick William (1813) [1812-11-12]. "On a Remarkable Application of Cotes's Theorem". Philosophical Transactions of the Royal Society of London. London: Royal Society of London, printed by W. Bulmer and Co., Cleveland-Row, St. James's, sold by G. and W. Nicol, Pall-Mall. 103 (Part 1): 8–26 [10]. doi:10.1098/rstl.1813.0005. JSTOR 107384. S2CID 118124706.
41. Herschel, John Frederick William (1820). "Part III. Section I. Examples of the Direct Method of Differences". A Collection of Examples of the Applications of the Calculus of Finite Differences. Cambridge, UK: Printed by J. Smith, sold by J. Deighton & sons. pp. 1–13 [5–6]. Archived from the original on 2020-08-04. Retrieved 2020-08-04. (NB. Inhere, Herschel refers to his 1813 work and mentions Hans Heinrich Bürmann's older work.)
42. Cajori, Florian (1952) [March 1929]. A History of Mathematical Notations. Vol. 2 (3rd ed.). Chicago, USA: Open court publishing company. pp. 108, 176–179, 336, 346. ISBN 978-1-60206-714-1. Retrieved 2016-01-18.
43. Richard Gillam, Unicode Demystified: A Practical Programmer's Guide to the Encoding Standard, 2003, ISBN 0201700522, p. 33
44. Backus, John Warner; Beeber, R. J.; Best, Sheldon F.; Goldberg, Richard; Herrick, Harlan L.; Hughes, R. A.; Mitchell, L. B.; Nelson, Robert A.; Nutt, Roy; Sayre, David; Sheridan, Peter B.; Stern, Harold; Ziller, Irving (1956-10-15). Sayre, David (ed.). The FORTRAN Automatic Coding System for the IBM 704 EDPM: Programmer's Reference Manual (PDF). New York, USA: Applied Science Division and Programming Research Department, International Business Machines Corporation. p. 15. Archived (PDF) from the original on 2022-07-04. Retrieved 2022-07-04. (2+51+1 pages)
45. Brice Carnahan, James O. Wilkes, Introduction to Digital Computing and FORTRAN IV with MTS Applications, 1968, p. 2-2, 2-6
46. Backus, John Warner; Herrick, Harlan L.; Nelson, Robert A.; Ziller, Irving (1954-11-10). Backus, John Warner (ed.). Specifications for: The IBM Mathematical FORmula TRANSlating System, FORTRAN (PDF) (Preliminary report). New York, USA: Programming Research Group, Applied Science Division, International Business Machines Corporation. pp. 4, 6. Archived (PDF) from the original on 2022-03-29. Retrieved 2022-07-04. (29 pages)
47. Daneliuk, Timothy "Tim" A. (1982-08-09). "BASCOM - A BASIC compiler for TRS-80 I and II". InfoWorld. Software Reviews. Vol. 4, no. 31. Popular Computing, Inc. pp. 41–42. Archived from the original on 2020-02-07. Retrieved 2020-02-06.
48. "80 Contents". 80 Micro. 1001001, Inc. (45): 5. October 1983. ISSN 0744-7868. Retrieved 2020-02-06.
49. Robert W. Sebesta, Concepts of Programming Languages, 2010, ISBN 0136073476, p. 130, 324
Hyperoperations
Primary
• Successor (0)
• Addition (1)
• Multiplication (2)
• Exponentiation (3)
• Tetration (4)
• Pentation (5)
Inverse for left argument
• Predecessor (0)
• Subtraction (1)
• Division (2)
• Root extraction (3)
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Inverse for right argument
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| Wikipedia |
Ball (mathematics)
In mathematics, a ball is the solid figure bounded by a sphere; it is also called a solid sphere.[1] It may be a closed ball (including the boundary points that constitute the sphere) or an open ball (excluding them).
These concepts are defined not only in three-dimensional Euclidean space but also for lower and higher dimensions, and for metric spaces in general. A ball in n dimensions is called a hyperball or n-ball and is bounded by a hypersphere or (n−1)-sphere. Thus, for example, a ball in the Euclidean plane is the same thing as a disk, the area bounded by a circle. In Euclidean 3-space, a ball is taken to be the volume bounded by a 2-dimensional sphere. In a one-dimensional space, a ball is a line segment.
In other contexts, such as in Euclidean geometry and informal use, sphere is sometimes used to mean ball. In the field of topology the closed $n$-dimensional ball is often denoted as $B^{n}$ or $D^{n}$ while the open $n$-dimensional ball is $\operatorname {Int} B^{n}$ or $\operatorname {Int} D^{n}$.
In Euclidean space
In Euclidean n-space, an (open) n-ball of radius r and center x is the set of all points of distance less than r from x. A closed n-ball of radius r is the set of all points of distance less than or equal to r away from x.
In Euclidean n-space, every ball is bounded by a hypersphere. The ball is a bounded interval when n = 1, is a disk bounded by a circle when n = 2, and is bounded by a sphere when n = 3.
Volume
Main article: Volume of an n-ball
The n-dimensional volume of a Euclidean ball of radius r in n-dimensional Euclidean space is:[2]
$V_{n}(r)={\frac {\pi ^{\frac {n}{2}}}{\Gamma \left({\frac {n}{2}}+1\right)}}r^{n},$
where Γ is Leonhard Euler's gamma function (which can be thought of as an extension of the factorial function to fractional arguments). Using explicit formulas for particular values of the gamma function at the integers and half integers gives formulas for the volume of a Euclidean ball that do not require an evaluation of the gamma function. These are:
${\begin{aligned}V_{2k}(r)&={\frac {\pi ^{k}}{k!}}r^{2k}\,,\\[2pt]V_{2k+1}(r)&={\frac {2^{k+1}\pi ^{k}}{(2k+1)!!}}r^{2k+1}={\frac {2(k!)(4\pi )^{k}}{(2k+1)!}}r^{2k+1}\,.\end{aligned}}$
In the formula for odd-dimensional volumes, the double factorial (2k + 1)!! is defined for odd integers 2k + 1 as (2k + 1)!! = 1 ⋅ 3 ⋅ 5 ⋅ ⋯ ⋅ (2k − 1) ⋅ (2k + 1).
In general metric spaces
Let (M, d) be a metric space, namely a set M with a metric (distance function) d. The open (metric) ball of radius r > 0 centered at a point p in M, usually denoted by Br(p) or B(p; r), is defined by
$B_{r}(p)=\{x\in M\mid d(x,p)<r\},$
The closed (metric) ball, which may be denoted by Br[p] or B[p; r], is defined by
$B_{r}[p]=\{x\in M\mid d(x,p)\leq r\}.$
Note in particular that a ball (open or closed) always includes p itself, since the definition requires r > 0.
A unit ball (open or closed) is a ball of radius 1.
A subset of a metric space is bounded if it is contained in some ball. A set is totally bounded if, given any positive radius, it is covered by finitely many balls of that radius.
The open balls of a metric space can serve as a base, giving this space a topology, the open sets of which are all possible unions of open balls. This topology on a metric space is called the topology induced by the metric d.
Let Br(p) denote the closure of the open ball Br(p) in this topology. While it is always the case that Br(p) ⊆ Br(p) ⊆ Br[p], it is not always the case that Br(p) = Br[p]. For example, in a metric space X with the discrete metric, one has B1(p) = {p} and B1[p] = X, for any p ∈ X.
In normed vector spaces
Any normed vector space V with norm $\|\cdot \|$ is also a metric space with the metric $d(x,y)=\|x-y\|.$ In such spaces, an arbitrary ball $B_{r}(y)$ of points $x$ around a point $y$ with a distance of less than $r$ may be viewed as a scaled (by $r$) and translated (by $y$) copy of a unit ball $B_{1}(0).$ Such "centered" balls with $y=0$ are denoted with $B(r).$
The Euclidean balls discussed earlier are an example of balls in a normed vector space.
p-norm
In a Cartesian space Rn with the p-norm Lp, that is
$\left\|x\right\|_{p}=\left(|x_{1}|^{p}+|x_{2}|^{p}+\dots +|x_{n}|^{p}\right)^{1/p},$
an open ball around the origin with radius $r$ is given by the set
$B(r)=\left\{x\in \mathbb {R} ^{n}\,:\left\|x\right\|_{p}=\left(|x_{1}|^{p}+|x_{2}|^{p}+\dots +|x_{n}|^{p}\right)^{1/p}<r\right\}.$
For n = 2, in a 2-dimensional plane $\mathbb {R} ^{2}$, "balls" according to the L1-norm (often called the taxicab or Manhattan metric) are bounded by squares with their diagonals parallel to the coordinate axes; those according to the L∞-norm, also called the Chebyshev metric, have squares with their sides parallel to the coordinate axes as their boundaries. The L2-norm, known as the Euclidean metric, generates the well known disks within circles, and for other values of p, the corresponding balls are areas bounded by Lamé curves (hypoellipses or hyperellipses).
For n = 3, the L1- balls are within octahedra with axes-aligned body diagonals, the L∞-balls are within cubes with axes-aligned edges, and the boundaries of balls for Lp with p > 2 are superellipsoids. Obviously, p = 2 generates the inner of usual spheres.
General convex norm
More generally, given any centrally symmetric, bounded, open, and convex subset X of Rn, one can define a norm on Rn where the balls are all translated and uniformly scaled copies of X. Note this theorem does not hold if "open" subset is replaced by "closed" subset, because the origin point qualifies but does not define a norm on Rn.
In topological spaces
One may talk about balls in any topological space X, not necessarily induced by a metric. An (open or closed) n-dimensional topological ball of X is any subset of X which is homeomorphic to an (open or closed) Euclidean n-ball. Topological n-balls are important in combinatorial topology, as the building blocks of cell complexes.
Any open topological n-ball is homeomorphic to the Cartesian space Rn and to the open unit n-cube (hypercube) (0, 1)n ⊆ Rn. Any closed topological n-ball is homeomorphic to the closed n-cube [0, 1]n.
An n-ball is homeomorphic to an m-ball if and only if n = m. The homeomorphisms between an open n-ball B and Rn can be classified in two classes, that can be identified with the two possible topological orientations of B.
A topological n-ball need not be smooth; if it is smooth, it need not be diffeomorphic to a Euclidean n-ball.
Regions
See also: Spherical regions
A number of special regions can be defined for a ball:
• cap, bounded by one plane
• sector, bounded by a conical boundary with apex at the center of the sphere
• segment, bounded by a pair of parallel planes
• shell, bounded by two concentric spheres of differing radii
• wedge, bounded by two planes passing through a sphere center and the surface of the sphere
See also
• Ball – ordinary meaning
• Disk (mathematics)
• Formal ball, an extension to negative radii
• Neighbourhood (mathematics)
• Sphere, a similar geometric shape
• 3-sphere
• n-sphere, or hypersphere
• Alexander horned sphere
• Manifold
• Volume of an n-ball
• Octahedron – a 3-ball in the l1 metric.
References
1. Sūgakkai, Nihon (1993). Encyclopedic Dictionary of Mathematics. MIT Press. ISBN 9780262590204.
2. Equation 5.19.4, NIST Digital Library of Mathematical Functions. Release 1.0.6 of 2013-05-06.
• Smith, D. J.; Vamanamurthy, M. K. (1989). "How small is a unit ball?". Mathematics Magazine. 62 (2): 101–107. doi:10.1080/0025570x.1989.11977419. JSTOR 2690391.
• Dowker, J. S. (1996). "Robin Conditions on the Euclidean ball". Classical and Quantum Gravity. 13 (4): 585–610. arXiv:hep-th/9506042. Bibcode:1996CQGra..13..585D. doi:10.1088/0264-9381/13/4/003. S2CID 119438515.
• Gruber, Peter M. (1982). "Isometries of the space of convex bodies contained in a Euclidean ball". Israel Journal of Mathematics. 42 (4): 277–283. doi:10.1007/BF02761407.
| Wikipedia |
3-opt
In optimization, 3-opt is a simple local search algorithm for solving the travelling salesperson problem and related network optimization problems. Compared to the simpler 2-opt algorithm, it is slower but can generate higher-quality solutions.
3-opt analysis involves deleting 3 connections (or edges) in a network (or tour), to create 3 sub-tours. Then the 7 different ways of reconnecting the network are analysed to find the optimum one. This process is then repeated for a different set of 3 connections, until all possible combinations have been tried in a network. A single execution of 3-opt has a time complexity of $O(n^{3})$.[1] Iterated 3-opt has a higher time complexity.
This is the mechanism by which the 3-opt swap manipulates a given route:
def reverse_segment_if_better(tour, i, j, k):
"""If reversing tour[i:j] would make the tour shorter, then do it."""
# Given tour [...A-B...C-D...E-F...]
A, B, C, D, E, F = tour[i-1], tour[i], tour[j-1], tour[j], tour[k-1], tour[k % len(tour)]
d0 = distance(A, B) + distance(C, D) + distance(E, F)
d1 = distance(A, C) + distance(B, D) + distance(E, F)
d2 = distance(A, B) + distance(C, E) + distance(D, F)
d3 = distance(A, D) + distance(E, B) + distance(C, F)
d4 = distance(F, B) + distance(C, D) + distance(E, A)
if d0 > d1:
tour[i:j] = reversed(tour[i:j])
return -d0 + d1
elif d0 > d2:
tour[j:k] = reversed(tour[j:k])
return -d0 + d2
elif d0 > d4:
tour[i:k] = reversed(tour[i:k])
return -d0 + d4
elif d0 > d3:
tmp = tour[j:k] + tour[i:j]
tour[i:k] = tmp
return -d0 + d3
return 0
The principle is pretty simple. You compute the original distance $d_{0}$ and you compute the cost of each modification. If you find a better cost, apply the modification and return $\delta $ (relative cost). This is the complete 3-opt swap making use of the above mechanism:
def three_opt(tour):
"""Iterative improvement based on 3 exchange."""
while True:
delta = 0
for (a, b, c) in all_segments(len(tour)):
delta += reverse_segment_if_better(tour, a, b, c)
if delta >= 0:
break
return tour
def all_segments(n: int):
"""Generate all segments combinations"""
return ((i, j, k)
for i in range(n)
for j in range(i + 2, n)
for k in range(j + 2, n + (i > 0)))
For the given tour, you generate all segments combinations and for each combinations, you try to improve the tour by reversing segments. While you find a better result, you restart the process, otherwise finish.
See also
• 2-opt
• Local search (optimization)
• Lin–Kernighan heuristic
References
1. Blazinskas, Andrius; Misevicius, Alfonsas (2011). Combining 2-OPT, 3-OPT and 4-OPT with K-SWAP-KICK perturbations for the traveling salesman problem (PDF). 17th International Conference on Information and Software Technologies. Kaunas, Lithuania. S2CID 15324387.
• BOCK, F. (1958). "An algorithm for solving traveling-salesman and related network optimization problems". Operations Research. 6 (6).
• Lin, Shen (1965). "Computer Solutions of the Traveling Salesman Problem". Bell System Technical Journal. Institute of Electrical and Electronics Engineers (IEEE). 44 (10): 2245–2269. doi:10.1002/j.1538-7305.1965.tb04146.x. ISSN 0005-8580.
• Lin, S.; Kernighan, B. W. (1973). "An Effective Heuristic Algorithm for the Traveling-Salesman Problem". Operations Research. Institute for Operations Research and the Management Sciences (INFORMS). 21 (2): 498–516. doi:10.1287/opre.21.2.498. ISSN 0030-364X.
• Sipser, Michael (2006). Introduction to the theory of computation. Boston: Thomson Course Technology. ISBN 0-534-95097-3. OCLC 58544333.
| Wikipedia |
31 great circles of the spherical icosahedron
In geometry, the 31 great circles of the spherical icosahedron is an arrangement of 31 great circles in icosahedral symmetry.[1] It was first identified by Buckminster Fuller and is used in construction of geodesic domes.
Construction
The 31 great circles can be seen in 3 sets: 15, 10, and 6, each representing edges of a polyhedron projected onto a sphere. Fifteen great circles represent the edges of a disdyakis triacontahedron, the dual of a truncated icosidodecahedron. Six more great circles represent the edges of an icosidodecahedron, and the last ten great circles come from the edges of the uniform star dodecadodecahedron, making pentagrams with vertices at the edge centers of the icosahedron.
There are 62 points of intersection, positioned at the 12 vertices, and center of the 30 edges, and 20 faces of a regular icosahedron.
Images
The 31 great circles are shown here in 3 directions, with 5-fold, 3-fold, and 2-fold symmetry. There are 4 types of right spherical triangles by the intersected great circles, seen by color in the right image.
5-fold3-fold2-fold2-fold
See also
• 25 great circles of the spherical octahedron
References
1. "Fig. 457.40 Definition of Spherical Polyhedra in 31-Great-Circle Icosahedron System" (PDF). rwgrayprojects.
• R. Buckminster Fuller, Synergetics: Explorations in the Geometry of Thinking, 1982, pp 183–185.
• Edward Popko, Divided Spheres: Geodesics and the Orderly Subdivision of the Sphere, 2012, pp 22–25.
| Wikipedia |
Steiner's conic problem
In enumerative geometry, Steiner's conic problem is the problem of finding the number of smooth conics tangent to five given conics in the plane in general position. If the problem is considered in the complex projective plane CP2, the correct solution is 3264 (Bashelor, Ksir & Traves (2008)). The problem is named after Jakob Steiner who first posed it and who gave an incorrect solution in 1848.
History
Steiner (1848) claimed that the number of conics tangent to 5 given conics in general position is 7776 = 65, but later realized this was wrong. The correct number 3264 was found in about 1859 by Ernest de Jonquières who did not publish because of Steiner's reputation, and by Chasles (1864) using his theory of characteristics, and by Berner in 1865. However these results, like many others in classical intersection theory, do not seem to have been given complete proofs until the work of Fulton and Macpherson in about 1978.
Formulation and solution
The space of (possibly degenerate) conics in the complex projective plane CP2 can be identified with the complex projective space CP5 (since each conic is defined by a homogeneous degree-2 polynomial in three variables, with 6 complex coefficients, and multiplying such a polynomial by a non-zero complex number does not change the conic). Steiner observed that the conics tangent to a given conic form a degree 6 hypersurface in CP5. So the conics tangent to 5 given conics correspond to the intersection points of 5 degree 6 hypersurfaces, and by Bézout's theorem the number of intersection points of 5 generic degree 6 hypersurfaces is 65 = 7776, which was Steiner's incorrect solution. The reason this is wrong is that the five degree 6 hypersurfaces are not in general position and have a common intersection in the Veronese surface, corresponding to the set of double lines in the plane, all of which have double intersection points with the 5 conics. In particular the intersection of these 5 hypersurfaces is not even 0-dimensional but has a 2-dimensional component. So to find the correct answer, one has to somehow eliminate the plane of spurious degenerate conics from this calculation.
One way of eliminating the degenerate conics is to blow up CP5 along the Veronese surface. The Chow ring of the blowup is generated by H and E, where H is the total transform of a hyperplane and E is the exceptional divisor. The total transform of a degree 6 hypersurface is 6H, and Steiner calculated (6H)5 = 65P as H5=P (where P is the class of a point in the Chow ring). However the number of conics is not (6H)5 but (6H−2E)5 because the strict transform of the hypersurface of conics tangent to a given conic is 6H−2E.
Suppose that L = 2H−E is the strict transform of the conics tangent to a given line. Then the intersection numbers of H and L are given by H5=1P, H4L=2P, H3L2=4P, H2L3=4P, H1L4=2P, L5=1P. So we have (6H−2E)5 = (2H+2L)5 = 3264P.
Fulton & MacPherson (1978) gave a precise description of exactly what "general position" means (although their two propositions about this are not quite right, and are corrected in a note on page 29 of their paper). If the five conics have the properties that
• there is no line such that every one of the 5 conics is either tangent to it or passes through one of two fixed points on it (otherwise there is a "double line with 2 marked points" tangent to all 5 conics)
• no three of the conics pass through any point (otherwise there is a "double line with 2 marked points" tangent to all 5 conics passing through this triple intersection point)
• no two of the conics are tangent
• no three of the five conics are tangent to a line
• a pair of lines each tangent to two of the conics do not intersect on the fifth conic (otherwise this pair is a degenerate conic tangent to all 5 conics)
then the total number of conics C tangent to all 5 (counted with multiplicities) is 3264. Here the multiplicity is given by the product over all 5 conics Ci of (4 − number of intersection points of C and Ci). In particular if C intersects each of the five conics in exactly 3 points (one double point of tangency and two others) then the multiplicity is 1, and if this condition always holds then there are exactly 3264 conics tangent to the 5 given conics.
Over other algebraically closed fields the answer is similar, unless the field has characteristic 2 in which case the number of conics is 51 rather than 3264.
References
• Bashelor, Andrew; Ksir, Amy; Traves, Will (2008), "Enumerative algebraic geometry of conics" (PDF), Amer. Math. Monthly, 115 (8): 701–728, doi:10.1080/00029890.2008.11920584, JSTOR 27642583, MR 2456094
• Chasles, M. (1864), "Construction des coniques qui satisfont à cinque conditions", C. R. Acad. Sci. Paris, 58: 297–308
• Eisenbud, David; Joe, Harris (2016), 3264 and All That: A Second Course in Algebraic Geometry, C. U.P., ISBN 978-1107602724
• Fulton, William; MacPherson, Robert (1978), "Defining algebraic intersections", Algebraic geometry (Proc. Sympos., Univ. Tromsø, Tromsø, 1977), Lecture Notes in Math., vol. 687, Berlin: Springer, pp. 1–30, doi:10.1007/BFb0062926, ISBN 978-3-540-08954-4, MR 0527228
• Steiner, J. (1848), "Elementare Lösung einer geometrischen Aufgabe, und über einige damit in Beziehung stehende Eigenschaften der Kegelschnitte", J. Reine Angew. Math., 37: 161–192
External links
• Ghys, Étienne, TROIS MILLE DEUX CENT SOIXANTE-QUATRE… Comment Jean-Yves a récemment précisé un théorème de géométrie (in French)
• Welschinger, Jean-Yves (2006), "ÉNUMÉRATION DE FRACTIONS RATIONNELLES RÉELLES", Images des Mathématiques
| Wikipedia |
33344-33434 tiling
In geometry of the Euclidean plane, a 33344-33434 tiling is one of two of 20 2-uniform tilings of the Euclidean plane by regular polygons. They contains regular triangle and square faces, arranged in two vertex configuration: 3.3.3.4.4 and 3.3.4.3.4.[2]
33344-33434 tilings
Faced colored by their symmetry positions
Type2-uniform tiling
Designation[1][33.42; 32.4.3.4]1[33.42; 32.4.3.4]2
Vertex configurations3.3.4.3.4 and 3.3.3.4.4
Symmetryp4g, [4,4+], (4*2)pgg, [4+,4+], (22×)
Rotation symmetryp4, [4,4]+, (442)p2, [4+,4+]+, (2222)
Properties4-isohedral, 5-isotoxal3-isohedral, 6-isotoxal
The first has triangles in groups of 3 and square in groups of 1 and 2. It has 4 types of faces and 5 types of edges.
The second has triangles in groups of 4, and squares in groups of 2. It has 3 types of face and 6 types of edges.
Geometry
Its two vertex configurations are shared with two 1-uniform tilings:
3.3.4.3.4
3.3.3.4.4
snub square tiling
elongated triangular tiling
Circle Packings
These 2-uniform tilings can be used as a circle packings.
In the first 2-uniform tiling (whose dual resembles a key-lock pattern): cyan circles are in contact with 5 other circles (3 cyan, 2 pink), corresponding to the V33.42 planigon, and pink circles are also in contact with 5 other circles (4 cyan, 1 pink), corresponding to the V32.4.3.4 planigon. It is homeomorphic to the ambo operation on the tiling, with the cyan and pink gap polygons corresponding to the cyan and pink circles (mini-vertex configuration polygons; one dimensional duals to the respective planigons). Both images coincide.
In the second 2-uniform tiling (whose dual resembles jagged streams of water): cyan circles are in contact with 5 other circles (2 cyan, 3 pink), corresponding to the V33.42 planigon, and pink circles are also in contact with 5 other circles (3 cyan, 2 pink), corresponding to the V32.4.3.4 planigon. It is homeomorphic to the ambo operation on the tiling, with the cyan and pink gap polygons corresponding to the cyan and pink circles (mini-vertex configuration polygons; one dimensional duals to the respective planigons). Both images coincide.
Circle Packings of and Ambo Operations on Two Pentagonal Isoperimetric 2-dual-uniform tilings.
C[33.42; 32.4.3.4]1 a33.42; 32.4.3.4]1 C[33.42; 32.4.3.4]2 a[33.42; 32.4.3.4]2
Dual tilings
The dual tilings have right triangle and kite faces, defined by face configurations: V3.3.3.4.4 and V3.3.4.3.4, and can be seen combining the prismatic pentagonal tiling and Cairo pentagonal tilings.
Faces1-uniform2-uniform
V3.3.3.4.4V3.3.4.3.4V3.3.3.4.4 and V3.3.4.3.4
V3.3.3.4.4
80px
V3.3.4.3.4
prismatic pentagonal tiling
Cairo pentagonal tiling
Dual tiling I
Dual tiling II
Notes
1. Grünbaum, Branko; Shephard, G. C. (1987). Tilings and Patterns. W. H. Freeman. ISBN 0-7167-1193-1. p. 65-67
2. Chavey (1989)
References
• Keith Critchlow, Order in Space: A design source book, 1970, pp. 62–67
• Ghyka, M. The Geometry of Art and Life, (1946), 2nd edition, New York: Dover, 1977. Demiregular tiling #15
• Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X. pp. 35–43
• Sacred Geometry Design Sourcebook: Universal Dimensional Patterns, Bruce Rawles, 1997. pp. 36–37
• Introduction to Tessellations, Dale Seymour, Jill Britton, (1989), p.57, Fig 3-24 Tessellations of regular polygons that contain more than one type of vertex point
External links
• Chavey, D. (1989). "Tilings by Regular Polygons—II: A Catalog of Tilings". Computers & Mathematics with Applications. 17: 147–165. doi:10.1016/0898-1221(89)90156-9.
• Dutch, Steve. "Uniform Tilings". Archived from the original on 2006-09-09. Retrieved 2006-09-09.
• Weisstein, Eric W. "Demiregular tessellation". MathWorld.
• In Search of Demiregular Tilings, Helmer Aslaksen
• n-uniform tilings Brian Galebach, 2-Uniform Tiling 1 of 20
| Wikipedia |
Milü
Milü (Chinese: 密率; pinyin: mìlǜ; "close ratio"), also known as Zulü (Zu's ratio), is the name given to an approximation to π (pi) found by Chinese mathematician and astronomer Zu Chongzhi in the 5th century. Using Liu Hui's algorithm (which is based on the areas of regular polygons approximating a circle), Zu famously computed π to be between 3.1415926 and 3.1415927[1] and gave two rational approximations of π, 22/7 and 355/113, naming them respectively Yuelü (Chinese: 约率; pinyin: yuēlǜ; "approximate ratio") and Milü.[2]
Milü
Chinese密率
Transcriptions
Standard Mandarin
Hanyu Pinyinmì lǜ
Wade–Gilesmi4 lü4
Yue: Cantonese
Yale Romanizationmaht léut
Jyutpingmat6 leot2
355/113 is the best rational approximation of π with a denominator of four digits or fewer, being accurate to six decimal places. It is within 0.000009% of the value of π, or in terms of common fractions overestimates π by less than 1/3748629. The next rational number (ordered by size of denominator) that is a better rational approximation of π is 52163/16604, still only correct to six decimal places and hardly closer to π than 355/113. To be accurate to seven decimal places, one needs to go as far as 86953/27678. For eight, 102928/32763 is needed.[3]
The accuracy of Milü to the true value of π can be explained using the continued fraction expansion of π, the first few terms of which are [3; 7, 15, 1, 292, 1, 1, ...]. A property of continued fractions is that truncating the expansion of a given number at any point will give the "best rational approximation" to the number. To obtain Milü, truncate the continued fraction expansion of π immediately before the term 292; that is, π is approximated by the finite continued fraction [3; 7, 15, 1], which is equivalent to Milü. Since 292 is an unusually large term in a continued fraction expansion (corresponding to the next truncation introducing only a very small term, 1/292, to the overall fraction), this convergent will be especially close to the true value of π:[4]
$\pi =3+{\cfrac {1}{7+{\cfrac {1}{15+{\cfrac {1}{1+{\color {magenta}{\cfrac {1}{292+\cdots }}}}}}}}}\quad \approx \quad 3+{\cfrac {1}{7+{\cfrac {1}{15+{\cfrac {1}{1+{\color {magenta}0}}}}}}}={\frac {355}{113}}$
An easy mnemonic helps memorize this useful fraction by writing down each of the first three odd numbers twice: 1 1 3 3 5 5, then dividing the decimal number represented by the last 3 digits by the decimal number given by the first three digits. Alternatively, 1/π ≈ 113⁄355.
Zu's contemporary calendarist and mathematician He Chengtian invented a fraction interpolation method called "harmonization of the divisor of the day" (Chinese: zh:调日法; pinyin: diaorifa) to increase the accuracy of approximations of π by iteratively adding the numerators and denominators of fractions. Zu Chongzhi's approximation π ≈ 355/113 can be obtained with He Chengtian's method.[2]
See also
• Continued fraction expansion of π and its convergents
• History of approximations of π
• Pi Approximation Day
References
1. Literally, he found that if the diameter of a circle is 100,000,000 then the circumference is between 314,159,260 and 314,159,270. We do not know what method he used to do this calculation.
2. Martzloff, Jean-Claude (2006). A History of Chinese Mathematics. Springer. p. 281. ISBN 9783540337829.
3. "Fractional Approximations of Pi".
4. Weisstein, Eric W. "Pi Continued Fraction". mathworld.wolfram.com. Retrieved 2017-09-03.
External links
• Fractional Approximations of Pi
| Wikipedia |
3D projection
A 3D projection (or graphical projection) is a design technique used to display a three-dimensional (3D) object on a two-dimensional (2D) surface. These projections rely on visual perspective and aspect analysis to project a complex object for viewing capability on a simpler plane.
Part of a series on
Graphical projection
Planar
• Parallel projection
• Orthographic projection
• Isometric projection
• Oblique projection
• Perspective projection
• Curvilinear perspective
• Reverse perspective
Views
• Bird's-eye view
• Cross section
• Cutaway drawing
• Exploded view drawing
• Fisheye lens
• Multiviews
• Panorama
• Worm's-eye view
• Zoom lens
Topics
• 3D projection
• Anamorphosis
• Axonometry
• Computer graphics
• Computer-aided design
• Descriptive geometry
• Engineering drawing
• Map projection
• Picture plane
• Plans (drawings)
• Projection (linear algebra)
• Projection plane
• Projective geometry
• Stereoscopy
• Technical drawing
• True length
• Vanishing point
• Video game graphics
• Viewing frustum
3D projections use the primary qualities of an object's basic shape to create a map of points, that are then connected to one another to create a visual element. The result is a graphic that contains conceptual properties to interpret the figure or image as not actually flat (2D), but rather, as a solid object (3D) being viewed on a 2D display.
3D objects are largely displayed on two-dimensional mediums (such as paper and computer monitors). As such, graphical projections are a commonly used design element; notably, in engineering drawing, drafting, and computer graphics. Projections can be calculated through employment of mathematical analysis and formulae, or by using various geometric and optical techniques.
Overview
Projection is achieved by the use of imaginary "projectors"; the projected, mental image becomes the technician's vision of the desired, finished picture. Methods provide a uniform imaging procedure among people trained in technical graphics (mechanical drawing, computer aided design, etc.). By following a method, the technician may produce the envisioned picture on a planar surface such as drawing paper.
There are two graphical projection categories, each with its own method:
• parallel projection
• perspective projection
• Multiview projection (elevation)
• Isometric projection
• Military projection
• Cabinet projection
• One-point perspective
• Two-point perspective
• Three-point perspective
Parallel projection
Main article: Parallel projection
In parallel projection, the lines of sight from the object to the projection plane are parallel to each other. Thus, lines that are parallel in three-dimensional space remain parallel in the two-dimensional projected image. Parallel projection also corresponds to a perspective projection with an infinite focal length (the distance from a camera's lens and focal point), or "zoom".
Images drawn in parallel projection rely upon the technique of axonometry ("to measure along axes"), as described in Pohlke's theorem. In general, the resulting image is oblique (the rays are not perpendicular to the image plane); but in special cases the result is orthographic (the rays are perpendicular to the image plane). Axonometry should not be confused with axonometric projection, as in English literature the latter usually refers only to a specific class of pictorials (see below).
Orthographic projection
See also: Geometric transformation
The orthographic projection is derived from the principles of descriptive geometry and is a two-dimensional representation of a three-dimensional object. It is a parallel projection (the lines of projection are parallel both in reality and in the projection plane). It is the projection type of choice for working drawings.
If the normal of the viewing plane (the camera direction) is parallel to one of the primary axes (which is the x, y, or z axis), the mathematical transformation is as follows; To project the 3D point $a_{x}$, $a_{y}$, $a_{z}$ onto the 2D point $b_{x}$, $b_{y}$ using an orthographic projection parallel to the y axis (where positive y represents forward direction - profile view), the following equations can be used:
$b_{x}=s_{x}a_{x}+c_{x}$
$b_{y}=s_{z}a_{z}+c_{z}$
where the vector s is an arbitrary scale factor, and c is an arbitrary offset. These constants are optional, and can be used to properly align the viewport. Using matrix multiplication, the equations become:
${\begin{bmatrix}b_{x}\\b_{y}\end{bmatrix}}={\begin{bmatrix}s_{x}&0&0\\0&0&s_{z}\end{bmatrix}}{\begin{bmatrix}a_{x}\\a_{y}\\a_{z}\end{bmatrix}}+{\begin{bmatrix}c_{x}\\c_{z}\end{bmatrix}}.$
While orthographically projected images represent the three dimensional nature of the object projected, they do not represent the object as it would be recorded photographically or perceived by a viewer observing it directly. In particular, parallel lengths at all points in an orthographically projected image are of the same scale regardless of whether they are far away or near to the virtual viewer. As a result, lengths are not foreshortened as they would be in a perspective projection.
Multiview projection
With multiview projections, up to six pictures (called primary views) of an object are produced, with each projection plane parallel to one of the coordinate axes of the object. The views are positioned relative to each other according to either of two schemes: first-angle or third-angle projection. In each, the appearances of views may be thought of as being projected onto planes that form a 6-sided box around the object. Although six different sides can be drawn, usually three views of a drawing give enough information to make a 3D object. These views are known as front view, top view, and end view. The terms elevation, plan and section are also used.
Oblique projection
Potting bench drawn in cabinet projection with an angle of 45° and a ratio of 2/3
Stone arch drawn in military perspective
In oblique projections the parallel projection rays are not perpendicular to the viewing plane as with orthographic projection, but strike the projection plane at an angle other than ninety degrees. In both orthographic and oblique projection, parallel lines in space appear parallel on the projected image. Because of its simplicity, oblique projection is used exclusively for pictorial purposes rather than for formal, working drawings. In an oblique pictorial drawing, the displayed angles among the axes as well as the foreshortening factors (scale) are arbitrary. The distortion created thereby is usually attenuated by aligning one plane of the imaged object to be parallel with the plane of projection thereby creating a true shape, full-size image of the chosen plane. Special types of oblique projections are:
Cavalier projection (45°)
In cavalier projection (sometimes cavalier perspective or high view point) a point of the object is represented by three coordinates, x, y and z. On the drawing, it is represented by only two coordinates, x″ and y″. On the flat drawing, two axes, x and z on the figure, are perpendicular and the length on these axes are drawn with a 1:1 scale; it is thus similar to the dimetric projections, although it is not an axonometric projection, as the third axis, here y, is drawn in diagonal, making an arbitrary angle with the x″ axis, usually 30 or 45°. The length of the third axis is not scaled.
Cabinet projection
The term cabinet projection (sometimes cabinet perspective) stems from its use in illustrations by the furniture industry. Like cavalier perspective, one face of the projected object is parallel to the viewing plane, and the third axis is projected as going off in an angle (typically 30° or 45° or arctan(2) = 63.4°). Unlike cavalier projection, where the third axis keeps its length, with cabinet projection the length of the receding lines is cut in half.
Military projection
A variant of oblique projection is called military projection. In this case, the horizontal sections are isometrically drawn so that the floor plans are not distorted and the verticals are drawn at an angle. The military projection is given by rotation in the xy-plane and a vertical translation an amount z.[1]
Axonometric projection
Axonometric projections show an image of an object as viewed from a skew direction in order to reveal all three directions (axes) of space in one picture.[2] Axonometric projections may be either orthographic or oblique. Axonometric instrument drawings are often used to approximate graphical perspective projections, but there is attendant distortion in the approximation. Because pictorial projections innately contain this distortion, in instrument drawings of pictorials great liberties may then be taken for economy of effort and best effect.
Axonometric projection is further subdivided into three categories: isometric projection, dimetric projection, and trimetric projection, depending on the exact angle at which the view deviates from the orthogonal.[3][4] A typical characteristic of orthographic pictorials is that one axis of space is usually displayed as vertical.
Isometric projection
In isometric pictorials (for methods, see Isometric projection), the direction of viewing is such that the three axes of space appear equally foreshortened, and there is a common angle of 120° between them. The distortion caused by foreshortening is uniform, therefore the proportionality of all sides and lengths are preserved, and the axes share a common scale. This enables measurements to be read or taken directly from the drawing.
Dimetric projection
In dimetric pictorials (for methods, see Dimetric projection), the direction of viewing is such that two of the three axes of space appear equally foreshortened, of which the attendant scale and angles of presentation are determined according to the angle of viewing; the scale of the third direction (vertical) is determined separately. Approximations are common in dimetric drawings.
Trimetric projection
In trimetric pictorials (for methods, see Trimetric projection), the direction of viewing is such that all of the three axes of space appear unequally foreshortened. The scale along each of the three axes and the angles among them are determined separately as dictated by the angle of viewing. Approximations in Trimetric drawings are common.
Limitations of parallel projection
See also: Impossible object
An example of the limitations of isometric projection. The height difference between the red and blue balls cannot be determined locally.
The Penrose stairs depicts a staircase which seems to ascend (anticlockwise) or descend (clockwise) yet forms a continuous loop.
Objects drawn with parallel projection do not appear larger or smaller as they extend closer to or away from the viewer. While advantageous for architectural drawings, where measurements must be taken directly from the image, the result is a perceived distortion, since unlike perspective projection, this is not how our eyes or photography normally work. It also can easily result in situations where depth and altitude are difficult to gauge, as is shown in the illustration to the right.
In this isometric drawing, the blue sphere is two units higher than the red one. However, this difference in elevation is not apparent if one covers the right half of the picture, as the boxes (which serve as clues suggesting height) are then obscured.
This visual ambiguity has been exploited in op art, as well as "impossible object" drawings. M. C. Escher's Waterfall (1961), while not strictly utilizing parallel projection, is a well-known example, in which a channel of water seems to travel unaided along a downward path, only to then paradoxically fall once again as it returns to its source. The water thus appears to disobey the law of conservation of energy. An extreme example is depicted in the film Inception, where by a forced perspective trick an immobile stairway changes its connectivity. The video game Fez uses tricks of perspective to determine where a player can and cannot move in a puzzle-like fashion.
Perspective projection
See also: Perspective (graphical), Transformation matrix, and Camera matrix
Perspective projection or perspective transformation is a nonlinear projection where three dimensional objects are projected on a picture plane. This has the effect that distant objects appear smaller than nearer objects.
It also means that lines which are parallel in nature (that is, meet at the point at infinity) appear to intersect in the projected image. For example, if railways are pictured with perspective projection, they appear to converge towards a single point, called the vanishing point. Photographic lenses and the human eye work in the same way, therefore perspective projection looks most realistic.[5] Perspective projection is usually categorized into one-point, two-point and three-point perspective, depending on the orientation of the projection plane towards the axes of the depicted object.[6]
Graphical projection methods rely on the duality between lines and points, whereby two straight lines determine a point while two points determine a straight line. The orthogonal projection of the eye point onto the picture plane is called the principal vanishing point (P.P. in the scheme on the right, from the Italian term punto principale, coined during the renaissance).[7]
Two relevant points of a line are:
• its intersection with the picture plane, and
• its vanishing point, found at the intersection between the parallel line from the eye point and the picture plane.
The principal vanishing point is the vanishing point of all horizontal lines perpendicular to the picture plane. The vanishing points of all horizontal lines lie on the horizon line. If, as is often the case, the picture plane is vertical, all vertical lines are drawn vertically, and have no finite vanishing point on the picture plane. Various graphical methods can be easily envisaged for projecting geometrical scenes. For example, lines traced from the eye point at 45° to the picture plane intersect the latter along a circle whose radius is the distance of the eye point from the plane, thus tracing that circle aids the construction of all the vanishing points of 45° lines; in particular, the intersection of that circle with the horizon line consists of two distance points. They are useful for drawing chessboard floors which, in turn, serve for locating the base of objects on the scene. In the perspective of a geometric solid on the right, after choosing the principal vanishing point —which determines the horizon line— the 45° vanishing point on the left side of the drawing completes the characterization of the (equally distant) point of view. Two lines are drawn from the orthogonal projection of each vertex, one at 45° and one at 90° to the picture plane. After intersecting the ground line, those lines go toward the distance point (for 45°) or the principal point (for 90°). Their new intersection locates the projection of the map. Natural heights are measured above the ground line and then projected in the same way until they meet the vertical from the map.
While orthographic projection ignores perspective to allow accurate measurements, perspective projection shows distant objects as smaller to provide additional realism.
Mathematical formula
The perspective projection requires a more involved definition as compared to orthographic projections. A conceptual aid to understanding the mechanics of this projection is to imagine the 2D projection as though the object(s) are being viewed through a camera viewfinder. The camera's position, orientation, and field of view control the behavior of the projection transformation. The following variables are defined to describe this transformation:
• $\mathbf {a} _{x,y,z}$ – the 3D position of a point A that is to be projected.
• $\mathbf {c} _{x,y,z}$ – the 3D position of a point C representing the camera.
• $\mathbf {\theta } _{x,y,z}$ – The orientation of the camera (represented by Tait–Bryan angles).
• $\mathbf {e} _{x,y,z}$ – the display surface's position relative to the camera pinhole C.[8]
Most conventions use positive z values (the plane being in front of the pinhole), however negative z values are physically more correct, but the image will be inverted both horizontally and vertically. Which results in:
• $\mathbf {b} _{x,y}$ – the 2D projection of $\mathbf {a} .$
When $\mathbf {c} _{x,y,z}=\langle 0,0,0\rangle ,$ and $\mathbf {\theta } _{x,y,z}=\langle 0,0,0\rangle ,$ the 3D vector $\langle 1,2,0\rangle $ is projected to the 2D vector $\langle 1,2\rangle $.
Otherwise, to compute $\mathbf {b} _{x,y}$ we first define a vector $\mathbf {d} _{x,y,z}$ as the position of point A with respect to a coordinate system defined by the camera, with origin in C and rotated by $\mathbf {\theta } $ with respect to the initial coordinate system. This is achieved by subtracting $\mathbf {c} $ from $\mathbf {a} $ and then applying a rotation by $-\mathbf {\theta } $ to the result. This transformation is often called a camera transform, and can be expressed as follows, expressing the rotation in terms of rotations about the x, y, and z axes (these calculations assume that the axes are ordered as a left-handed system of axes): [9] [10]
${\begin{bmatrix}\mathbf {d} _{x}\\\mathbf {d} _{y}\\\mathbf {d} _{z}\end{bmatrix}}={\begin{bmatrix}1&0&0\\0&\cos(\mathbf {\theta } _{x})&\sin(\mathbf {\theta } _{x})\\0&-\sin(\mathbf {\theta } _{x})&\cos(\mathbf {\theta } _{x})\end{bmatrix}}{\begin{bmatrix}\cos(\mathbf {\theta } _{y})&0&-\sin(\mathbf {\theta } _{y})\\0&1&0\\\sin(\mathbf {\theta } _{y})&0&\cos(\mathbf {\theta } _{y})\end{bmatrix}}{\begin{bmatrix}\cos(\mathbf {\theta } _{z})&\sin(\mathbf {\theta } _{z})&0\\-\sin(\mathbf {\theta } _{z})&\cos(\mathbf {\theta } _{z})&0\\0&0&1\end{bmatrix}}\left({{\begin{bmatrix}\mathbf {a} _{x}\\\mathbf {a} _{y}\\\mathbf {a} _{z}\\\end{bmatrix}}-{\begin{bmatrix}\mathbf {c} _{x}\\\mathbf {c} _{y}\\\mathbf {c} _{z}\\\end{bmatrix}}}\right)$
This representation corresponds to rotating by three Euler angles (more properly, Tait–Bryan angles), using the xyz convention, which can be interpreted either as "rotate about the extrinsic axes (axes of the scene) in the order z, y, x (reading right-to-left)" or "rotate about the intrinsic axes (axes of the camera) in the order x, y, z (reading left-to-right)". If the camera is not rotated ($\mathbf {\theta } _{x,y,z}=\langle 0,0,0\rangle $), then the matrices drop out (as identities), and this reduces to simply a shift: $\mathbf {d} =\mathbf {a} -\mathbf {c} .$
Alternatively, without using matrices (let us replace $a_{x}-c_{x}$ with $\mathbf {x} $ and so on, and abbreviate $\cos \left(\theta _{\alpha }\right)$ to $c_{\alpha }$ and $\sin \left(\theta _{\alpha }\right)$ to $s_{\alpha }$):
${\begin{aligned}\mathbf {d} _{x}&=c_{y}(s_{z}\mathbf {y} +c_{z}\mathbf {x} )-s_{y}\mathbf {z} \\\mathbf {d} _{y}&=s_{x}(c_{y}\mathbf {z} +s_{y}(s_{z}\mathbf {y} +c_{z}\mathbf {x} ))+c_{x}(c_{z}\mathbf {y} -s_{z}\mathbf {x} )\\\mathbf {d} _{z}&=c_{x}(c_{y}\mathbf {z} +s_{y}(s_{z}\mathbf {y} +c_{z}\mathbf {x} ))-s_{x}(c_{z}\mathbf {y} -s_{z}\mathbf {x} )\end{aligned}}$
This transformed point can then be projected onto the 2D plane using the formula (here, x/y is used as the projection plane; literature also may use x/z):[11]
${\begin{aligned}\mathbf {b} _{x}&={\frac {\mathbf {e} _{z}}{\mathbf {d} _{z}}}\mathbf {d} _{x}+\mathbf {e} _{x},\\[5pt]\mathbf {b} _{y}&={\frac {\mathbf {e} _{z}}{\mathbf {d} _{z}}}\mathbf {d} _{y}+\mathbf {e} _{y}.\end{aligned}}$
Or, in matrix form using homogeneous coordinates, the system
${\begin{bmatrix}\mathbf {f} _{x}\\\mathbf {f} _{y}\\\mathbf {f} _{w}\end{bmatrix}}={\begin{bmatrix}1&0&{\frac {\mathbf {e} _{x}}{\mathbf {e} _{z}}}\\0&1&{\frac {\mathbf {e} _{y}}{\mathbf {e} _{z}}}\\0&0&{\frac {1}{\mathbf {e} _{z}}}\end{bmatrix}}{\begin{bmatrix}\mathbf {d} _{x}\\\mathbf {d} _{y}\\\mathbf {d} _{z}\end{bmatrix}}$
in conjunction with an argument using similar triangles, leads to division by the homogeneous coordinate, giving
${\begin{aligned}\mathbf {b} _{x}&=\mathbf {f} _{x}/\mathbf {f} _{w}\\\mathbf {b} _{y}&=\mathbf {f} _{y}/\mathbf {f} _{w}\end{aligned}}$
The distance of the viewer from the display surface, $\mathbf {e} _{z}$, directly relates to the field of view, where $\alpha =2\cdot \arctan(1/\mathbf {e} _{z})$ is the viewed angle. (Note: This assumes that you map the points (-1,-1) and (1,1) to the corners of your viewing surface)
The above equations can also be rewritten as:
${\begin{aligned}\mathbf {b} _{x}&=(\mathbf {d} _{x}\mathbf {s} _{x})/(\mathbf {d} _{z}\mathbf {r} _{x})\mathbf {r} _{z},\\\mathbf {b} _{y}&=(\mathbf {d} _{y}\mathbf {s} _{y})/(\mathbf {d} _{z}\mathbf {r} _{y})\mathbf {r} _{z}.\end{aligned}}$
In which $\mathbf {s} _{x,y}$ is the display size, $\mathbf {r} _{x,y}$ is the recording surface size (CCD or Photographic film), $\mathbf {r} _{z}$ is the distance from the recording surface to the entrance pupil (camera center), and $\mathbf {d} _{z}$ is the distance, from the 3D point being projected, to the entrance pupil.
Subsequent clipping and scaling operations may be necessary to map the 2D plane onto any particular display media.
Weak perspective projection
A "weak" perspective projection uses the same principles of an orthographic projection, but requires the scaling factor to be specified, thus ensuring that closer objects appear bigger in the projection, and vice versa. It can be seen as a hybrid between an orthographic and a perspective projection, and described either as a perspective projection with individual point depths $Z_{i}$ replaced by an average constant depth $Z_{\text{ave}}$,[12] or simply as an orthographic projection plus a scaling.[13]
The weak-perspective model thus approximates perspective projection while using a simpler model, similar to the pure (unscaled) orthographic perspective. It is a reasonable approximation when the depth of the object along the line of sight is small compared to the distance from the camera, and the field of view is small. With these conditions, it can be assumed that all points on a 3D object are at the same distance $Z_{\text{ave}}$ from the camera without significant errors in the projection (compared to the full perspective model).
Equation
${\begin{aligned}&P_{x}={\frac {X}{Z_{\text{ave}}}}\\[5pt]&P_{y}={\frac {Y}{Z_{\text{ave}}}}\end{aligned}}$
assuming focal length $ f=1$.
Diagram
To determine which screen x-coordinate corresponds to a point at $A_{x},A_{z}$ multiply the point coordinates by:
$B_{x}=A_{x}{\frac {B_{z}}{A_{z}}}$
where
$B_{x}$ is the screen x coordinate
$A_{x}$ is the model x coordinate
$B_{z}$ is the focal length—the axial distance from the camera center to the image plane
$A_{z}$ is the subject distance.
Because the camera is in 3D, the same works for the screen y-coordinate, substituting y for x in the above diagram and equation.
Alternatively, one could use clipping techniques, replacing the variables with values of the point that's are out of the FOV-angle and the point inside Camera Matrix.
This technique, also known as "Inverse Camera", is a Perspective Projection Calculus with known values to calculate the last point on visible angle, projecting from the invisible point, after all needed transformations finished.
See also
• 3D computer graphics
• Camera matrix
• Computer graphics
• Cross section (geometry)
• Cross-sectional view
• Curvilinear perspective
• Cutaway drawing
• Descriptive geometry
• Engineering drawing
• Exploded-view drawing
• Homogeneous coordinates
• Homography
• Map projection (including Cylindrical projection)
• Multiview projection
• Perspective (graphical)
• Plan (drawing)
• Technical drawing
• Tesseract
• Texture mapping
• Transform, clipping, and lighting
• Video card
• Viewing frustum
• Virtual globe
References
1. "Axonometric projections - a technical overview". Retrieved 24 April 2015.
2. Mitchell, William; Malcolm McCullough (1994). Digital design media. John Wiley and Sons. p. 169. ISBN 978-0-471-28666-0.
3. Maynard, Patric (2005). Drawing distinctions: the varieties of graphic expression. Cornell University Press. p. 22. ISBN 978-0-8014-7280-0.
4. McReynolds, Tom; David Blythe (2005). Advanced graphics programming using openGL. Elsevier. p. 502. ISBN 978-1-55860-659-3.
5. D. Hearn, & M. Baker (1997). Computer Graphics, C Version. Englewood Cliffs: Prentice Hall], chapter 9
6. James Foley (1997). Computer Graphics. Boston: Addison-Wesley. ISBN 0-201-84840-6], chapter 6
7. Kirsti Andersen (2007), The geometry of an art, Springer, p. xxix, ISBN 9780387259611
8. Ingrid Carlbom, Joseph Paciorek (1978). "Planar Geometric Projections and Viewing Transformations" (PDF). ACM Computing Surveys. 10 (4): 465–502. CiteSeerX 10.1.1.532.4774. doi:10.1145/356744.356750. S2CID 708008.
9. Riley, K F (2006). Mathematical Methods for Physics and Engineering. Cambridge University Press. pp. 931, 942. ISBN 978-0-521-67971-8.
10. Goldstein, Herbert (1980). Classical Mechanics (2nd ed.). Reading, Mass.: Addison-Wesley Pub. Co. pp. 146–148. ISBN 978-0-201-02918-5.
11. Sonka, M; Hlavac, V; Boyle, R (1995). Image Processing, Analysis & Machine Vision (2nd ed.). Chapman and Hall. p. 14. ISBN 978-0-412-45570-4.
12. Subhashis Banerjee (2002-02-18). "The Weak-Perspective Camera".
13. Alter, T. D. (July 1992). 3D Pose from 3 Corresponding Points under Weak-Perspective Projection (PDF) (Technical report). MIT AI Lab.
Further reading
• Kenneth C. Finney (2004). 3D Game Programming All in One. Thomson Course. p. 93. ISBN 978-1-59200-136-1. 3D projection.
• Koehler; Ralph (December 2000). 2D/3D Graphics and Splines with Source Code. ISBN 978-0759611870.
External links
Wikimedia Commons has media related to 3D projection.
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| Wikipedia |
3D rotation group
In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space $\mathbb {R} ^{3}$ under the operation of composition.[1]
By definition, a rotation about the origin is a transformation that preserves the origin, Euclidean distance (so it is an isometry), and orientation (i.e., handedness of space). Composing two rotations results in another rotation, every rotation has a unique inverse rotation, and the identity map satisfies the definition of a rotation. Owing to the above properties (along composite rotations' associative property), the set of all rotations is a group under composition.
Every non-trivial rotation is determined by its axis of rotation (a line through the origin) and its angle of rotation. Rotations are not commutative (for example, rotating R 90° in the x-y plane followed by S 90° in the y-z plane is not the same as S followed by R), making the 3D rotation group a nonabelian group. Moreover, the rotation group has a natural structure as a manifold for which the group operations are smoothly differentiable, so it is in fact a Lie group. It is compact and has dimension 3.
Rotations are linear transformations of $\mathbb {R} ^{3}$ and can therefore be represented by matrices once a basis of $\mathbb {R} ^{3}$ has been chosen. Specifically, if we choose an orthonormal basis of $\mathbb {R} ^{3}$, every rotation is described by an orthogonal 3 × 3 matrix (i.e., a 3 × 3 matrix with real entries which, when multiplied by its transpose, results in the identity matrix) with determinant 1. The group SO(3) can therefore be identified with the group of these matrices under matrix multiplication. These matrices are known as "special orthogonal matrices", explaining the notation SO(3).
The group SO(3) is used to describe the possible rotational symmetries of an object, as well as the possible orientations of an object in space. Its representations are important in physics, where they give rise to the elementary particles of integer spin.
Length and angle
Besides just preserving length, rotations also preserve the angles between vectors. This follows from the fact that the standard dot product between two vectors u and v can be written purely in terms of length:
$\mathbf {u} \cdot \mathbf {v} ={\frac {1}{2}}\left(\|\mathbf {u} +\mathbf {v} \|^{2}-\|\mathbf {u} \|^{2}-\|\mathbf {v} \|^{2}\right).$
It follows that every length-preserving linear transformation in $\mathbb {R} ^{3}$ preserves the dot product, and thus the angle between vectors. Rotations are often defined as linear transformations that preserve the inner product on $\mathbb {R} ^{3}$, which is equivalent to requiring them to preserve length. See classical group for a treatment of this more general approach, where SO(3) appears as a special case.
Orthogonal and rotation matrices
Main articles: Orthogonal matrix and Rotation matrix
Every rotation maps an orthonormal basis of $\mathbb {R} ^{3}$ to another orthonormal basis. Like any linear transformation of finite-dimensional vector spaces, a rotation can always be represented by a matrix. Let R be a given rotation. With respect to the standard basis e1, e2, e3 of $\mathbb {R} ^{3}$ the columns of R are given by (Re1, Re2, Re3). Since the standard basis is orthonormal, and since R preserves angles and length, the columns of R form another orthonormal basis. This orthonormality condition can be expressed in the form
$R^{\mathsf {T}}R=RR^{\mathsf {T}}=I,$
where RT denotes the transpose of R and I is the 3 × 3 identity matrix. Matrices for which this property holds are called orthogonal matrices. The group of all 3 × 3 orthogonal matrices is denoted O(3), and consists of all proper and improper rotations.
In addition to preserving length, proper rotations must also preserve orientation. A matrix will preserve or reverse orientation according to whether the determinant of the matrix is positive or negative. For an orthogonal matrix R, note that det RT = det R implies (det R)2 = 1, so that det R = ±1. The subgroup of orthogonal matrices with determinant +1 is called the special orthogonal group, denoted SO(3).
Thus every rotation can be represented uniquely by an orthogonal matrix with unit determinant. Moreover, since composition of rotations corresponds to matrix multiplication, the rotation group is isomorphic to the special orthogonal group SO(3).
Improper rotations correspond to orthogonal matrices with determinant −1, and they do not form a group because the product of two improper rotations is a proper rotation.
Group structure
The rotation group is a group under function composition (or equivalently the product of linear transformations). It is a subgroup of the general linear group consisting of all invertible linear transformations of the real 3-space $\mathbb {R} ^{3}$.[2]
Furthermore, the rotation group is nonabelian. That is, the order in which rotations are composed makes a difference. For example, a quarter turn around the positive x-axis followed by a quarter turn around the positive y-axis is a different rotation than the one obtained by first rotating around y and then x.
The orthogonal group, consisting of all proper and improper rotations, is generated by reflections. Every proper rotation is the composition of two reflections, a special case of the Cartan–Dieudonné theorem.
Complete classification of finite subgroups
The finite subgroups of $\mathrm {SO} (3)$ are completely classified.[3]
Every finite subgroup is isomorphic to either an element of one of two countably infinite families of planar isometries: the cyclic groups $C_{n}$ or the dihedral groups $D_{2n}$, or to one of three other groups: the tetrahedral group $\cong A_{4}$, the octahedral group $\cong S_{4}$, or the icosahedral group $\cong A_{5}$.
Axis of rotation
Main article: Axis–angle representation
Every nontrivial proper rotation in 3 dimensions fixes a unique 1-dimensional linear subspace of $\mathbb {R} ^{3}$ which is called the axis of rotation (this is Euler's rotation theorem). Each such rotation acts as an ordinary 2-dimensional rotation in the plane orthogonal to this axis. Since every 2-dimensional rotation can be represented by an angle φ, an arbitrary 3-dimensional rotation can be specified by an axis of rotation together with an angle of rotation about this axis. (Technically, one needs to specify an orientation for the axis and whether the rotation is taken to be clockwise or counterclockwise with respect to this orientation).
For example, counterclockwise rotation about the positive z-axis by angle φ is given by
$R_{z}(\phi )={\begin{bmatrix}\cos \phi &-\sin \phi &0\\\sin \phi &\cos \phi &0\\0&0&1\end{bmatrix}}.$
Given a unit vector n in $\mathbb {R} ^{3}$ and an angle φ, let R(φ, n) represent a counterclockwise rotation about the axis through n (with orientation determined by n). Then
• R(0, n) is the identity transformation for any n
• R(φ, n) = R(−φ, −n)
• R(π + φ, n) = R(π − φ, −n).
Using these properties one can show that any rotation can be represented by a unique angle φ in the range 0 ≤ φ ≤ π and a unit vector n such that
• n is arbitrary if φ = 0
• n is unique if 0 < φ < π
• n is unique up to a sign if φ = π (that is, the rotations R(π, ±n) are identical).
In the next section, this representation of rotations is used to identify SO(3) topologically with three-dimensional real projective space.
Topology
Main article: Hypersphere of rotations
The Lie group SO(3) is diffeomorphic to the real projective space $\mathbb {P} ^{3}(\mathbb {R} ).$[4]
Consider the solid ball in $\mathbb {R} ^{3}$ of radius π (that is, all points of $\mathbb {R} ^{3}$ of distance π or less from the origin). Given the above, for every point in this ball there is a rotation, with axis through the point and the origin, and rotation angle equal to the distance of the point from the origin. The identity rotation corresponds to the point at the center of the ball. Rotation through angles between 0 and −π correspond to the point on the same axis and distance from the origin but on the opposite side of the origin. The one remaining issue is that the two rotations through π and through −π are the same. So we identify (or "glue together") antipodal points on the surface of the ball. After this identification, we arrive at a topological space homeomorphic to the rotation group.
Indeed, the ball with antipodal surface points identified is a smooth manifold, and this manifold is diffeomorphic to the rotation group. It is also diffeomorphic to the real 3-dimensional projective space $\mathbb {P} ^{3}(\mathbb {R} ),$ so the latter can also serve as a topological model for the rotation group.
These identifications illustrate that SO(3) is connected but not simply connected. As to the latter, in the ball with antipodal surface points identified, consider the path running from the "north pole" straight through the interior down to the south pole. This is a closed loop, since the north pole and the south pole are identified. This loop cannot be shrunk to a point, since no matter how you deform the loop, the start and end point have to remain antipodal, or else the loop will "break open". In terms of rotations, this loop represents a continuous sequence of rotations about the z-axis starting (by example) at identity (center of ball), through south pole, jump to north pole and ending again at the identity rotation (i.e., a series of rotation through an angle φ where φ runs from 0 to 2π).
Surprisingly, if you run through the path twice, i.e., run from north pole down to south pole, jump back to the north pole (using the fact that north and south poles are identified), and then again run from north pole down to south pole, so that φ runs from 0 to 4π, you get a closed loop which can be shrunk to a single point: first move the paths continuously to the ball's surface, still connecting north pole to south pole twice. The second path can then be mirrored over to the antipodal side without changing the path at all. Now we have an ordinary closed loop on the surface of the ball, connecting the north pole to itself along a great circle. This circle can be shrunk to the north pole without problems. The plate trick and similar tricks demonstrate this practically.
The same argument can be performed in general, and it shows that the fundamental group of SO(3) is the cyclic group of order 2 (a fundamental group with two elements). In physics applications, the non-triviality (more than one element) of the fundamental group allows for the existence of objects known as spinors, and is an important tool in the development of the spin–statistics theorem.
The universal cover of SO(3) is a Lie group called Spin(3). The group Spin(3) is isomorphic to the special unitary group SU(2); it is also diffeomorphic to the unit 3-sphere S3 and can be understood as the group of versors (quaternions with absolute value 1). The connection between quaternions and rotations, commonly exploited in computer graphics, is explained in quaternions and spatial rotations. The map from S3 onto SO(3) that identifies antipodal points of S3 is a surjective homomorphism of Lie groups, with kernel {±1}. Topologically, this map is a two-to-one covering map. (See the plate trick.)
Connection between SO(3) and SU(2)
In this section, we give two different constructions of a two-to-one and surjective homomorphism of SU(2) onto SO(3).
Using quaternions of unit norm
Main article: Quaternions and spatial rotation
The group SU(2) is isomorphic to the quaternions of unit norm via a map given by[5]
$q=a\mathbf {1} +b\mathbf {i} +c\mathbf {j} +d\mathbf {k} =\alpha +\beta \mathbf {j} \leftrightarrow {\begin{bmatrix}\alpha &-{\overline {\beta }}\\\beta &{\overline {\alpha }}\end{bmatrix}}=U$
restricted to $ a^{2}+b^{2}+c^{2}+d^{2}=|\alpha |^{2}+|\beta |^{2}=1$ where $ q\in \mathbb {H} $, $ a,b,c,d\in \mathbb {R} $, $ U\in \operatorname {SU} (2)$, and $\alpha =a+bi\in \mathbb {C} $, $\beta =c+di\in \mathbb {C} $.
Let us now identify $\mathbb {R} ^{3}$ with the span of $\mathbf {i} ,\mathbf {j} ,\mathbf {k} $. One can then verify that if $v$ is in $\mathbb {R} ^{3}$ and $q$ is a unit quaternion, then
$qvq^{-1}\in \mathbb {R} ^{3}.$
Furthermore, the map $v\mapsto qvq^{-1}$ is a rotation of $\mathbb {R} ^{3}.$ Moreover, $(-q)v(-q)^{-1}$ is the same as $qvq^{-1}$. This means that there is a 2:1 homomorphism from quaternions of unit norm to the 3D rotation group SO(3).
One can work this homomorphism out explicitly: the unit quaternion, q, with
${\begin{aligned}q&=w+x\mathbf {i} +y\mathbf {j} +z\mathbf {k} ,\\1&=w^{2}+x^{2}+y^{2}+z^{2},\end{aligned}}$
is mapped to the rotation matrix
$Q={\begin{bmatrix}1-2y^{2}-2z^{2}&2xy-2zw&2xz+2yw\\2xy+2zw&1-2x^{2}-2z^{2}&2yz-2xw\\2xz-2yw&2yz+2xw&1-2x^{2}-2y^{2}\end{bmatrix}}.$
This is a rotation around the vector (x, y, z) by an angle 2θ, where cos θ = w and |sin θ| = ‖(x, y, z)‖. The proper sign for sin θ is implied, once the signs of the axis components are fixed. The 2:1-nature is apparent since both q and −q map to the same Q.
Using Möbius transformations
The general reference for this section is Gelfand, Minlos & Shapiro (1963). The points P on the sphere
$\mathbf {S} =\left\{(x,y,z)\in \mathbb {R} ^{3}:x^{2}+y^{2}+z^{2}={\frac {1}{4}}\right\}$
can, barring the north pole N, be put into one-to-one bijection with points S(P) = P' on the plane M defined by z = −1/2, see figure. The map S is called stereographic projection.
Let the coordinates on M be (ξ, η). The line L passing through N and P can be parametrized as
$L(t)=N+t(N-P)=\left(0,0,{\frac {1}{2}}\right)+t\left(\left(0,0,{\frac {1}{2}}\right)-(x,y,z)\right),\quad t\in \mathbb {R} .$
Demanding that the z-coordinate of $L(t_{0})$ equals −1/2, one finds
$t_{0}={\frac {1}{z-{\frac {1}{2}}}}.$
We have $L(t_{0})=(\xi ,\eta ,-1/2).$ Hence the map
${\begin{cases}S:\mathbf {S} \to M\\P=(x,y,z)\longmapsto P'=(\xi ,\eta )=\left({\frac {x}{{\frac {1}{2}}-z}},{\frac {y}{{\frac {1}{2}}-z}}\right)\equiv \zeta =\xi +i\eta \end{cases}}$
where, for later convenience, the plane M is identified with the complex plane $\mathbb {C} .$
For the inverse, write L as
$L=N+s(P'-N)=\left(0,0,{\frac {1}{2}}\right)+s\left(\left(\xi ,\eta ,-{\frac {1}{2}}\right)-\left(0,0,{\frac {1}{2}}\right)\right),$
and demand x2 + y2 + z2 = 1/4 to find s = 1/1 + ξ2 + η2 and thus
${\begin{cases}S^{-1}:M\to \mathbf {S} \\P'=(\xi ,\eta )\longmapsto P=(x,y,z)=\left({\frac {\xi }{1+\xi ^{2}+\eta ^{2}}},{\frac {\eta }{1+\xi ^{2}+\eta ^{2}}},{\frac {-1+\xi ^{2}+\eta ^{2}}{2+2\xi ^{2}+2\eta ^{2}}}\right)\end{cases}}$
If g ∈ SO(3) is a rotation, then it will take points on S to points on S by its standard action Πs(g) on the embedding space $\mathbb {R} ^{3}.$ By composing this action with S one obtains a transformation S ∘ Πs(g) ∘ S−1 of M,
$\zeta =P'\longmapsto P\longmapsto \Pi _{s}(g)P=gP\longmapsto S(gP)\equiv \Pi _{u}(g)\zeta =\zeta '.$
Thus Πu(g) is a transformation of $\mathbb {C} $ associated to the transformation Πs(g) of $\mathbb {R} ^{3}$.
It turns out that g ∈ SO(3) represented in this way by Πu(g) can be expressed as a matrix Πu(g) ∈ SU(2) (where the notation is recycled to use the same name for the matrix as for the transformation of $\mathbb {C} $ it represents). To identify this matrix, consider first a rotation gφ about the z-axis through an angle φ,
${\begin{aligned}x'&=x\cos \phi -y\sin \phi ,\\y'&=x\sin \phi +y\cos \phi ,\\z'&=z.\end{aligned}}$
Hence
$\zeta '={\frac {x'+iy'}{{\frac {1}{2}}-z'}}={\frac {e^{i\phi }(x+iy)}{{\frac {1}{2}}-z}}=e^{i\phi }\zeta ={\frac {e^{\frac {i\phi }{2}}\zeta +0}{0\zeta +e^{-{\frac {i\phi }{2}}}}},$
which, unsurprisingly, is a rotation in the complex plane. In an analogous way, if gθ is a rotation about the x-axis through an angle θ, then
$w'=e^{i\theta }w,\quad w={\frac {y+iz}{{\frac {1}{2}}-x}},$
which, after a little algebra, becomes
$\zeta '={\frac {\cos {\frac {\theta }{2}}\zeta +i\sin {\frac {\theta }{2}}}{i\sin {\frac {\theta }{2}}\zeta +\cos {\frac {\theta }{2}}}}.$
These two rotations, $g_{\phi },g_{\theta },$ thus correspond to bilinear transforms of R2 ≃ C ≃ M, namely, they are examples of Möbius transformations.
A general Möbius transformation is given by
$\zeta '={\frac {\alpha \zeta +\beta }{\gamma \zeta +\delta }},\quad \alpha \delta -\beta \gamma \neq 0.$
The rotations, $g_{\phi },g_{\theta }$ generate all of SO(3) and the composition rules of the Möbius transformations show that any composition of $g_{\phi },g_{\theta }$ translates to the corresponding composition of Möbius transformations. The Möbius transformations can be represented by matrices
${\begin{pmatrix}\alpha &\beta \\\gamma &\delta \end{pmatrix}},\qquad \alpha \delta -\beta \gamma =1,$
since a common factor of α, β, γ, δ cancels.
For the same reason, the matrix is not uniquely defined since multiplication by −I has no effect on either the determinant or the Möbius transformation. The composition law of Möbius transformations follow that of the corresponding matrices. The conclusion is that each Möbius transformation corresponds to two matrices g, −g ∈ SL(2, C).
Using this correspondence one may write
${\begin{aligned}\Pi _{u}(g_{\phi })&=\Pi _{u}\left[{\begin{pmatrix}\cos \phi &-\sin \phi &0\\\sin \phi &\cos \phi &0\\0&0&1\end{pmatrix}}\right]=\pm {\begin{pmatrix}e^{i{\frac {\phi }{2}}}&0\\0&e^{-i{\frac {\phi }{2}}}\end{pmatrix}},\\\Pi _{u}(g_{\theta })&=\Pi _{u}\left[{\begin{pmatrix}1&0&0\\0&\cos \theta &-\sin \theta \\0&\sin \theta &\cos \theta \end{pmatrix}}\right]=\pm {\begin{pmatrix}\cos {\frac {\theta }{2}}&i\sin {\frac {\theta }{2}}\\i\sin {\frac {\theta }{2}}&\cos {\frac {\theta }{2}}\end{pmatrix}}.\end{aligned}}$
These matrices are unitary and thus Πu(SO(3)) ⊂ SU(2) ⊂ SL(2, C). In terms of Euler angles[nb 1] one finds for a general rotation
${\begin{aligned}g(\phi ,\theta ,\psi )=g_{\phi }g_{\theta }g_{\psi }&={\begin{pmatrix}\cos \phi &-\sin \phi &0\\\sin \phi &\cos \phi &0\\0&0&1\end{pmatrix}}{\begin{pmatrix}1&0&0\\0&\cos \theta &-\sin \theta \\0&\sin \theta &\cos \theta \end{pmatrix}}{\begin{pmatrix}\cos \psi &-\sin \psi &0\\\sin \psi &\cos \psi &0\\0&0&1\end{pmatrix}}\\&={\begin{pmatrix}\cos \phi \cos \psi -\cos \theta \sin \phi \sin \psi &-\cos \phi \sin \psi -\cos \theta \sin \phi \cos \psi &\sin \phi \sin \theta \\\sin \phi \cos \psi +\cos \theta \cos \phi \sin \psi &-\sin \phi \sin \psi +\cos \theta \cos \phi \cos \psi &-\cos \phi \sin \theta \\\sin \psi \sin \theta &\cos \psi \sin \theta &\cos \theta \end{pmatrix}},\end{aligned}}$
(1)
one has[6]
${\begin{aligned}\Pi _{u}(g(\phi ,\theta ,\psi ))&=\pm {\begin{pmatrix}e^{i{\frac {\phi }{2}}}&0\\0&e^{-i{\frac {\phi }{2}}}\end{pmatrix}}{\begin{pmatrix}\cos {\frac {\theta }{2}}&i\sin {\frac {\theta }{2}}\\i\sin {\frac {\theta }{2}}&\cos {\frac {\theta }{2}}\end{pmatrix}}{\begin{pmatrix}e^{i{\frac {\psi }{2}}}&0\\0&e^{-i{\frac {\psi }{2}}}\end{pmatrix}}\\&=\pm {\begin{pmatrix}\cos {\frac {\theta }{2}}e^{i{\frac {\phi +\psi }{2}}}&i\sin {\frac {\theta }{2}}e^{i{\frac {\phi -\psi }{2}}}\\i\sin {\frac {\theta }{2}}e^{-i{\frac {\phi -\psi }{2}}}&\cos {\frac {\theta }{2}}e^{-i{\frac {\phi +\psi }{2}}}\end{pmatrix}}.\end{aligned}}$
(2)
For the converse, consider a general matrix
$\pm \Pi _{u}(g_{\alpha ,\beta })=\pm {\begin{pmatrix}\alpha &\beta \\-{\overline {\beta }}&{\overline {\alpha }}\end{pmatrix}}\in \operatorname {SU} (2).$
Make the substitutions
${\begin{aligned}\cos {\frac {\theta }{2}}&=|\alpha |,&\sin {\frac {\theta }{2}}&=|\beta |,&(0\leq \theta \leq \pi ),\\{\frac {\phi +\psi }{2}}&=\arg \alpha ,&{\frac {\psi -\phi }{2}}&=\arg \beta .&\end{aligned}}$
With the substitutions, Π(gα, β) assumes the form of the right hand side (RHS) of (2), which corresponds under Πu to a matrix on the form of the RHS of (1) with the same φ, θ, ψ. In terms of the complex parameters α, β,
$g_{\alpha ,\beta }={\begin{pmatrix}{\frac {1}{2}}\left(\alpha ^{2}-\beta ^{2}+{\overline {\alpha ^{2}}}-{\overline {\beta ^{2}}}\right)&{\frac {i}{2}}\left(-\alpha ^{2}-\beta ^{2}+{\overline {\alpha ^{2}}}+{\overline {\beta ^{2}}}\right)&-\alpha \beta -{\overline {\alpha }}{\overline {\beta }}\\{\frac {i}{2}}\left(\alpha ^{2}-\beta ^{2}-{\overline {\alpha ^{2}}}+{\overline {\beta ^{2}}}\right)&{\frac {1}{2}}\left(\alpha ^{2}+\beta ^{2}+{\overline {\alpha ^{2}}}+{\overline {\beta ^{2}}}\right)&-i\left(+\alpha \beta -{\overline {\alpha }}{\overline {\beta }}\right)\\\alpha {\overline {\beta }}+{\overline {\alpha }}\beta &i\left(-\alpha {\overline {\beta }}+{\overline {\alpha }}\beta \right)&\alpha {\overline {\alpha }}-\beta {\overline {\beta }}\end{pmatrix}}.$
To verify this, substitute for α. β the elements of the matrix on the RHS of (2). After some manipulation, the matrix assumes the form of the RHS of (1).
It is clear from the explicit form in terms of Euler angles that the map
${\begin{cases}p:\operatorname {SU} (2)\to \operatorname {SO} (3)\\\Pi _{u}(\pm g_{\alpha \beta })\mapsto g_{\alpha \beta }\end{cases}}$
just described is a smooth, 2:1 and surjective group homomorphism. It is hence an explicit description of the universal covering space of SO(3) from the universal covering group SU(2).
Lie algebra
Associated with every Lie group is its Lie algebra, a linear space of the same dimension as the Lie group, closed under a bilinear alternating product called the Lie bracket. The Lie algebra of SO(3) is denoted by ${\mathfrak {so}}(3)$ and consists of all skew-symmetric 3 × 3 matrices.[7] This may be seen by differentiating the orthogonality condition, ATA = I, A ∈ SO(3).[nb 2] The Lie bracket of two elements of ${\mathfrak {so}}(3)$ is, as for the Lie algebra of every matrix group, given by the matrix commutator, [A1, A2] = A1A2 − A2A1, which is again a skew-symmetric matrix. The Lie algebra bracket captures the essence of the Lie group product in a sense made precise by the Baker–Campbell–Hausdorff formula.
The elements of ${\mathfrak {so}}(3)$ are the "infinitesimal generators" of rotations, i.e., they are the elements of the tangent space of the manifold SO(3) at the identity element. If $R(\phi ,{\boldsymbol {n}})$ denotes a counterclockwise rotation with angle φ about the axis specified by the unit vector ${\boldsymbol {n}},$ then
$\forall {\boldsymbol {u}}\in \mathbb {R} ^{3}:\qquad \left.{\frac {\operatorname {d} }{\operatorname {d} \phi }}\right|_{\phi =0}R(\phi ,{\boldsymbol {n}}){\boldsymbol {u}}={\boldsymbol {n}}\times {\boldsymbol {u}}.$
This can be used to show that the Lie algebra ${\mathfrak {so}}(3)$ (with commutator) is isomorphic to the Lie algebra $\mathbb {R} ^{3}$ (with cross product). Under this isomorphism, an Euler vector ${\boldsymbol {\omega }}\in \mathbb {R} ^{3}$ corresponds to the linear map ${\widetilde {\boldsymbol {\omega }}}$ defined by ${\widetilde {\boldsymbol {\omega }}}({\boldsymbol {u}})={\boldsymbol {\omega }}\times {\boldsymbol {u}}.$
In more detail, most often a suitable basis for ${\mathfrak {so}}(3)$ as a 3-dimensional vector space is
${\boldsymbol {L}}_{x}={\begin{bmatrix}0&0&0\\0&0&-1\\0&1&0\end{bmatrix}},\quad {\boldsymbol {L}}_{y}={\begin{bmatrix}0&0&1\\0&0&0\\-1&0&0\end{bmatrix}},\quad {\boldsymbol {L}}_{z}={\begin{bmatrix}0&-1&0\\1&0&0\\0&0&0\end{bmatrix}}.$
The commutation relations of these basis elements are,
$[{\boldsymbol {L}}_{x},{\boldsymbol {L}}_{y}]={\boldsymbol {L}}_{z},\quad [{\boldsymbol {L}}_{z},{\boldsymbol {L}}_{x}]={\boldsymbol {L}}_{y},\quad [{\boldsymbol {L}}_{y},{\boldsymbol {L}}_{z}]={\boldsymbol {L}}_{x}$
which agree with the relations of the three standard unit vectors of $\mathbb {R} ^{3}$ under the cross product.
As announced above, one can identify any matrix in this Lie algebra with an Euler vector ${\boldsymbol {\omega }}=(x,y,z)\in \mathbb {R} ^{3},$[8]
${\widehat {\boldsymbol {\omega }}}={\boldsymbol {\omega }}\cdot {\boldsymbol {L}}=x{\boldsymbol {L}}_{x}+y{\boldsymbol {L}}_{y}+z{\boldsymbol {L}}_{z}={\begin{bmatrix}0&-z&y\\z&0&-x\\-y&x&0\end{bmatrix}}\in {\mathfrak {so}}(3).$
This identification is sometimes called the hat-map.[9] Under this identification, the ${\mathfrak {so}}(3)$ bracket corresponds in $\mathbb {R} ^{3}$ to the cross product,
$\left[{\widehat {\boldsymbol {u}}},{\widehat {\boldsymbol {v}}}\right]={\widehat {{\boldsymbol {u}}\times {\boldsymbol {v}}}}.$
The matrix identified with a vector ${\boldsymbol {u}}$ has the property that
${\widehat {\boldsymbol {u}}}{\boldsymbol {v}}={\boldsymbol {u}}\times {\boldsymbol {v}},$
where the left-hand side we have ordinary matrix multiplication. This implies ${\boldsymbol {u}}$ is in the null space of the skew-symmetric matrix with which it is identified, because ${\boldsymbol {u}}\times {\boldsymbol {u}}={\boldsymbol {0}}.$
A note on Lie algebras
See also: Representation theory of SU(2) and Jordan map
In Lie algebra representations, the group SO(3) is compact and simple of rank 1, and so it has a single independent Casimir element, a quadratic invariant function of the three generators which commutes with all of them. The Killing form for the rotation group is just the Kronecker delta, and so this Casimir invariant is simply the sum of the squares of the generators, ${\boldsymbol {J}}_{x},{\boldsymbol {J}}_{y},{\boldsymbol {J}}_{z},$ of the algebra
$[{\boldsymbol {J}}_{x},{\boldsymbol {J}}_{y}]={\boldsymbol {J}}_{z},\quad [{\boldsymbol {J}}_{z},{\boldsymbol {J}}_{x}]={\boldsymbol {J}}_{y},\quad [{\boldsymbol {J}}_{y},{\boldsymbol {J}}_{z}]={\boldsymbol {J}}_{x}.$
That is, the Casimir invariant is given by
${\boldsymbol {J}}^{2}\equiv {\boldsymbol {J}}\cdot {\boldsymbol {J}}={\boldsymbol {J}}_{x}^{2}+{\boldsymbol {J}}_{y}^{2}+{\boldsymbol {J}}_{z}^{2}\propto {\boldsymbol {I}}.$
For unitary irreducible representations Dj, the eigenvalues of this invariant are real and discrete, and characterize each representation, which is finite dimensional, of dimensionality $2j+1$. That is, the eigenvalues of this Casimir operator are
${\boldsymbol {J}}^{2}=-j(j+1){\boldsymbol {I}}_{2j+1},$
where j is integer or half-integer, and referred to as the spin or angular momentum.
So, the 3 × 3 generators L displayed above act on the triplet (spin 1) representation, while the 2 × 2 generators below, t, act on the doublet (spin-1/2) representation. By taking Kronecker products of D1/2 with itself repeatedly, one may construct all higher irreducible representations Dj. That is, the resulting generators for higher spin systems in three spatial dimensions, for arbitrarily large j, can be calculated using these spin operators and ladder operators.
For every unitary irreducible representations Dj there is an equivalent one, D−j−1. All infinite-dimensional irreducible representations must be non-unitary, since the group is compact.
In quantum mechanics, the Casimir invariant is the "angular-momentum-squared" operator; integer values of spin j characterize bosonic representations, while half-integer values fermionic representations. The antihermitian matrices used above are utilized as spin operators, after they are multiplied by i, so they are now hermitian (like the Pauli matrices). Thus, in this language,
$[{\boldsymbol {J}}_{x},{\boldsymbol {J}}_{y}]=i{\boldsymbol {J}}_{z},\quad [{\boldsymbol {J}}_{z},{\boldsymbol {J}}_{x}]=i{\boldsymbol {J}}_{y},\quad [{\boldsymbol {J}}_{y},{\boldsymbol {J}}_{z}]=i{\boldsymbol {J}}_{x}.$
and hence
${\boldsymbol {J}}^{2}=j(j+1){\boldsymbol {I}}_{2j+1}.$
Explicit expressions for these Dj are,
${\begin{aligned}\left({\boldsymbol {J}}_{z}^{(j)}\right)_{ba}&=(j+1-a)\delta _{b,a}\\\left({\boldsymbol {J}}_{x}^{(j)}\right)_{ba}&={\frac {1}{2}}\left(\delta _{b,a+1}+\delta _{b+1,a}\right){\sqrt {(j+1)(a+b-1)-ab}}\\\left({\boldsymbol {J}}_{y}^{(j)}\right)_{ba}&={\frac {1}{2i}}\left(\delta _{b,a+1}-\delta _{b+1,a}\right){\sqrt {(j+1)(a+b-1)-ab}}\\\end{aligned}}$
where j is arbitrary and $1\leq a,b\leq 2j+1$.
For example, the resulting spin matrices for spin 1 ($j=1$) are
${\begin{aligned}{\boldsymbol {J}}_{x}&={\frac {1}{\sqrt {2}}}{\begin{pmatrix}0&1&0\\1&0&1\\0&1&0\end{pmatrix}}\\{\boldsymbol {J}}_{y}&={\frac {1}{\sqrt {2}}}{\begin{pmatrix}0&-i&0\\i&0&-i\\0&i&0\end{pmatrix}}\\{\boldsymbol {J}}_{z}&={\begin{pmatrix}1&0&0\\0&0&0\\0&0&-1\end{pmatrix}}\end{aligned}}$
Note, however, how these are in an equivalent, but different basis, the spherical basis, than the above iL in the Cartesian basis.[nb 3]
For higher spins, such as spin 3/2 ($j={\tfrac {3}{2}}$):
${\begin{aligned}{\boldsymbol {J}}_{x}&={\frac {1}{2}}{\begin{pmatrix}0&{\sqrt {3}}&0&0\\{\sqrt {3}}&0&2&0\\0&2&0&{\sqrt {3}}\\0&0&{\sqrt {3}}&0\end{pmatrix}}\\{\boldsymbol {J}}_{y}&={\frac {1}{2}}{\begin{pmatrix}0&-i{\sqrt {3}}&0&0\\i{\sqrt {3}}&0&-2i&0\\0&2i&0&-i{\sqrt {3}}\\0&0&i{\sqrt {3}}&0\end{pmatrix}}\\{\boldsymbol {J}}_{z}&={\frac {1}{2}}{\begin{pmatrix}3&0&0&0\\0&1&0&0\\0&0&-1&0\\0&0&0&-3\end{pmatrix}}.\end{aligned}}$
For spin 5/2 ($j={\tfrac {5}{2}}$),
${\begin{aligned}{\boldsymbol {J}}_{x}&={\frac {1}{2}}{\begin{pmatrix}0&{\sqrt {5}}&0&0&0&0\\{\sqrt {5}}&0&2{\sqrt {2}}&0&0&0\\0&2{\sqrt {2}}&0&3&0&0\\0&0&3&0&2{\sqrt {2}}&0\\0&0&0&2{\sqrt {2}}&0&{\sqrt {5}}\\0&0&0&0&{\sqrt {5}}&0\end{pmatrix}}\\{\boldsymbol {J}}_{y}&={\frac {1}{2}}{\begin{pmatrix}0&-i{\sqrt {5}}&0&0&0&0\\i{\sqrt {5}}&0&-2i{\sqrt {2}}&0&0&0\\0&2i{\sqrt {2}}&0&-3i&0&0\\0&0&3i&0&-2i{\sqrt {2}}&0\\0&0&0&2i{\sqrt {2}}&0&-i{\sqrt {5}}\\0&0&0&0&i{\sqrt {5}}&0\end{pmatrix}}\\{\boldsymbol {J}}_{z}&={\frac {1}{2}}{\begin{pmatrix}5&0&0&0&0&0\\0&3&0&0&0&0\\0&0&1&0&0&0\\0&0&0&-1&0&0\\0&0&0&0&-3&0\\0&0&0&0&0&-5\end{pmatrix}}.\end{aligned}}$
Isomorphism with 𝖘𝖚(2)
The Lie algebras ${\mathfrak {so}}(3)$ and ${\mathfrak {su}}(2)$ are isomorphic. One basis for ${\mathfrak {su}}(2)$ is given by[10]
${\boldsymbol {t}}_{1}={\frac {1}{2}}{\begin{bmatrix}0&-i\\-i&0\end{bmatrix}},\quad {\boldsymbol {t}}_{2}={\frac {1}{2}}{\begin{bmatrix}0&-1\\1&0\end{bmatrix}},\quad {\boldsymbol {t}}_{3}={\frac {1}{2}}{\begin{bmatrix}-i&0\\0&i\end{bmatrix}}.$
These are related to the Pauli matrices by
${\boldsymbol {t}}_{i}\longleftrightarrow {\frac {1}{2i}}\sigma _{i}.$
The Pauli matrices abide by the physicists' convention for Lie algebras. In that convention, Lie algebra elements are multiplied by i, the exponential map (below) is defined with an extra factor of i in the exponent and the structure constants remain the same, but the definition of them acquires a factor of i. Likewise, commutation relations acquire a factor of i. The commutation relations for the ${\boldsymbol {t}}_{i}$ are
$[{\boldsymbol {t}}_{i},{\boldsymbol {t}}_{j}]=\varepsilon _{ijk}{\boldsymbol {t}}_{k},$
where εijk is the totally anti-symmetric symbol with ε123 = 1. The isomorphism between ${\mathfrak {so}}(3)$ and ${\mathfrak {su}}(2)$ can be set up in several ways. For later convenience, ${\mathfrak {so}}(3)$ and ${\mathfrak {su}}(2)$ are identified by mapping
${\boldsymbol {L}}_{x}\longleftrightarrow {\boldsymbol {t}}_{1},\quad {\boldsymbol {L}}_{y}\longleftrightarrow {\boldsymbol {t}}_{2},\quad {\boldsymbol {L}}_{z}\longleftrightarrow {\boldsymbol {t}}_{3},$
and extending by linearity.
Exponential map
The exponential map for SO(3), is, since SO(3) is a matrix Lie group, defined using the standard matrix exponential series,
${\begin{cases}\exp :{\mathfrak {so}}(3)\to \operatorname {SO} (3)\\A\mapsto e^{A}=\sum _{k=0}^{\infty }{\frac {1}{k!}}A^{k}=I+A+{\tfrac {1}{2}}A^{2}+\cdots .\end{cases}}$ :{\mathfrak {so}}(3)\to \operatorname {SO} (3)\\A\mapsto e^{A}=\sum _{k=0}^{\infty }{\frac {1}{k!}}A^{k}=I+A+{\tfrac {1}{2}}A^{2}+\cdots .\end{cases}}}
For any skew-symmetric matrix A ∈ 𝖘𝖔(3), eA is always in SO(3). The proof uses the elementary properties of the matrix exponential
$\left(e^{A}\right)^{\textsf {T}}e^{A}=e^{A^{\textsf {T}}}e^{A}=e^{A^{\textsf {T}}+A}=e^{-A+A}=e^{A-A}=e^{A}\left(e^{A}\right)^{\textsf {T}}=e^{0}=I.$
since the matrices A and AT commute, this can be easily proven with the skew-symmetric matrix condition. This is not enough to show that 𝖘𝖔(3) is the corresponding Lie algebra for SO(3), and shall be proven separately.
The level of difficulty of proof depends on how a matrix group Lie algebra is defined. Hall (2003) defines the Lie algebra as the set of matrices
$\left\{A\in \operatorname {M} (n,\mathbb {R} )\left|e^{tA}\in \operatorname {SO} (3)\forall t\right.\right\},$
in which case it is trivial. Rossmann (2002) uses for a definition derivatives of smooth curve segments in SO(3) through the identity taken at the identity, in which case it is harder.[11]
For a fixed A ≠ 0, etA, −∞ < t < ∞ is a one-parameter subgroup along a geodesic in SO(3). That this gives a one-parameter subgroup follows directly from properties of the exponential map.[12]
The exponential map provides a diffeomorphism between a neighborhood of the origin in the 𝖘𝖔(3) and a neighborhood of the identity in the SO(3).[13] For a proof, see Closed subgroup theorem.
The exponential map is surjective. This follows from the fact that every R ∈ SO(3), since every rotation leaves an axis fixed (Euler's rotation theorem), and is conjugate to a block diagonal matrix of the form
$D={\begin{pmatrix}\cos \theta &-\sin \theta &0\\\sin \theta &\cos \theta &0\\0&0&1\end{pmatrix}}=e^{\theta L_{z}},$
such that A = BDB−1, and that
$Be^{\theta L_{z}}B^{-1}=e^{B\theta L_{z}B^{-1}},$
together with the fact that 𝖘𝖔(3) is closed under the adjoint action of SO(3), meaning that BθLzB−1 ∈ 𝖘𝖔(3).
Thus, e.g., it is easy to check the popular identity
$e^{-\pi L_{x}/2}e^{\theta L_{z}}e^{\pi L_{x}/2}=e^{\theta L_{y}}.$
As shown above, every element A ∈ 𝖘𝖔(3) is associated with a vector ω = θ u, where u = (x,y,z) is a unit magnitude vector. Since u is in the null space of A, if one now rotates to a new basis, through some other orthogonal matrix O, with u as the z axis, the final column and row of the rotation matrix in the new basis will be zero.
Thus, we know in advance from the formula for the exponential that exp(OAOT) must leave u fixed. It is mathematically impossible to supply a straightforward formula for such a basis as a function of u, because its existence would violate the hairy ball theorem; but direct exponentiation is possible, and yields
${\begin{aligned}\exp({\tilde {\boldsymbol {\omega }}})&=\exp(\theta ({\boldsymbol {u\cdot L}}))=\exp \left(\theta {\begin{bmatrix}0&-z&y\\z&0&-x\\-y&x&0\end{bmatrix}}\right)\\[4pt]&={\boldsymbol {I}}+2cs({\boldsymbol {u\cdot L}})+2s^{2}({\boldsymbol {u\cdot L}})^{2}\\[4pt]&={\begin{bmatrix}2\left(x^{2}-1\right)s^{2}+1&2xys^{2}-2zcs&2xzs^{2}+2ycs\\2xys^{2}+2zcs&2\left(y^{2}-1\right)s^{2}+1&2yzs^{2}-2xcs\\2xzs^{2}-2ycs&2yzs^{2}+2xcs&2\left(z^{2}-1\right)s^{2}+1\end{bmatrix}},\end{aligned}}$
where $ c=\cos {\frac {\theta }{2}}$ and $ s=\sin {\frac {\theta }{2}}$. This is recognized as a matrix for a rotation around axis u by the angle θ: cf. Rodrigues' rotation formula.
Logarithm map
Given R ∈ SO(3), let $A={\tfrac {1}{2}}\left(R-R^{\mathrm {T} }\right)$ denote the antisymmetric part and let $ \|A\|={\sqrt {-{\frac {1}{2}}\operatorname {Tr} \left(A^{2}\right)}}.$ Then, the logarithm of R is given by[9]
$\log R={\frac {\sin ^{-1}\|A\|}{\|A\|}}A.$
This is manifest by inspection of the mixed symmetry form of Rodrigues' formula,
$e^{X}=I+{\frac {\sin \theta }{\theta }}X+2{\frac {\sin ^{2}{\frac {\theta }{2}}}{\theta ^{2}}}X^{2},\quad \theta =\|X\|,$
where the first and last term on the right-hand side are symmetric.
Uniform random sampling
$SO(3)$ is doubly covered by the group of unit quaternions, which is isomorphic to the 3-sphere. Since the Haar measure on the unit quaternions is just the 3-area measure in 4 dimensions, the Haar measure on $SO(3)$ is just the pushforward of the 3-area measure.
Consequently, generating a uniformly random rotation in $\mathbb {R} ^{3}$ is equivalent to generating a uniformly random point on the 3-sphere. This can be accomplished by the following
$({\sqrt {1-u_{1}}}\sin(2\pi u_{2}),{\sqrt {1-u_{1}}}\cos(2\pi u_{2}),{\sqrt {u_{1}}}\sin(2\pi u_{3}),{\sqrt {u_{1}}}\cos(2\pi u_{3}))$
where $u_{1},u_{2},u_{3}$ are uniformly random samples of $[0,1]$.[14]
Baker–Campbell–Hausdorff formula
Main article: Baker–Campbell–Hausdorff formula
Suppose X and Y in the Lie algebra are given. Their exponentials, exp(X) and exp(Y), are rotation matrices, which can be multiplied. Since the exponential map is a surjection, for some Z in the Lie algebra, exp(Z) = exp(X) exp(Y), and one may tentatively write
$Z=C(X,Y),$
for C some expression in X and Y. When exp(X) and exp(Y) commute, then Z = X + Y, mimicking the behavior of complex exponentiation.
The general case is given by the more elaborate BCH formula, a series expansion of nested Lie brackets.[15] For matrices, the Lie bracket is the same operation as the commutator, which monitors lack of commutativity in multiplication. This general expansion unfolds as follows,[nb 4]
$Z=C(X,Y)=X+Y+{\frac {1}{2}}[X,Y]+{\tfrac {1}{12}}[X,[X,Y]]-{\frac {1}{12}}[Y,[X,Y]]+\cdots .$
The infinite expansion in the BCH formula for SO(3) reduces to a compact form,
$Z=\alpha X+\beta Y+\gamma [X,Y],$
for suitable trigonometric function coefficients (α, β, γ).
The trigonometric coefficients
The (α, β, γ) are given by
$\alpha =\phi \cot \left({\frac {\phi }{2}}\right)\gamma ,\qquad \beta =\theta \cot \left({\frac {\theta }{2}}\right)\gamma ,\qquad \gamma ={\frac {\sin ^{-1}d}{d}}{\frac {c}{\theta \phi }},$
where
${\begin{aligned}c&={\frac {1}{2}}\sin \theta \sin \phi -2\sin ^{2}{\frac {\theta }{2}}\sin ^{2}{\frac {\phi }{2}}\cos(\angle (u,v)),\quad a=c\cot \left({\frac {\phi }{2}}\right),\quad b=c\cot \left({\frac {\theta }{2}}\right),\\d&={\sqrt {a^{2}+b^{2}+2ab\cos(\angle (u,v))+c^{2}\sin ^{2}(\angle (u,v))}},\end{aligned}}$
for
$\theta ={\frac {1}{\sqrt {2}}}\|X\|,\quad \phi ={\frac {1}{\sqrt {2}}}\|Y\|,\quad \angle (u,v)=\cos ^{-1}{\frac {\langle X,Y\rangle }{\|X\|\|Y\|}}.$
The inner product is the Hilbert–Schmidt inner product and the norm is the associated norm. Under the hat-isomorphism,
$\langle u,v\rangle ={\frac {1}{2}}\operatorname {Tr} X^{\mathrm {T} }Y,$
which explains the factors for θ and φ. This drops out in the expression for the angle.
See also: Rotation formalisms in three dimensions § Rodrigues parameters and Gibbs representation
It is worthwhile to write this composite rotation generator as
$\alpha X+\beta Y+\gamma [X,Y]{\underset {{\mathfrak {so}}(3)}{=}}X+Y+{\frac {1}{2}}[X,Y]+{\frac {1}{12}}[X,[X,Y]]-{\frac {1}{12}}[Y,[X,Y]]+\cdots ,$
to emphasize that this is a Lie algebra identity.
The above identity holds for all faithful representations of 𝖘𝖔(3). The kernel of a Lie algebra homomorphism is an ideal, but 𝖘𝖔(3), being simple, has no nontrivial ideals and all nontrivial representations are hence faithful. It holds in particular in the doublet or spinor representation. The same explicit formula thus follows in a simpler way through Pauli matrices, cf. the 2×2 derivation for SU(2).
The SU(2) case
The Pauli vector version of the same BCH formula is the somewhat simpler group composition law of SU(2),
$e^{ia'\left({\hat {u}}\cdot {\vec {\sigma }}\right)}e^{ib'\left({\hat {v}}\cdot {\vec {\sigma }}\right)}=\exp \left({\frac {c'}{\sin c'}}\sin a'\sin b'\left(\left(i\cot b'{\hat {u}}+i\cot a'{\hat {v}}\right)\cdot {\vec {\sigma }}+{\frac {1}{2}}\left[i{\hat {u}}\cdot {\vec {\sigma }},i{\hat {v}}\cdot {\vec {\sigma }}\right]\right)\right),$
where
$\cos c'=\cos a'\cos b'-{\hat {u}}\cdot {\hat {v}}\sin a'\sin b',$
the spherical law of cosines. (Note a', b', c' are angles, not the a, b, c above.)
This is manifestly of the same format as above,
$Z=\alpha 'X+\beta 'Y+\gamma '[X,Y],$
with
$X=ia'{\hat {u}}\cdot \mathbf {\sigma } ,\quad Y=ib'{\hat {v}}\cdot \mathbf {\sigma } \in {\mathfrak {su}}(2),$
so that
${\begin{aligned}\alpha '&={\frac {c'}{\sin c'}}{\frac {\sin a'}{a'}}\cos b'\\\beta '&={\frac {c'}{\sin c'}}{\frac {\sin b'}{b'}}\cos a'\\\gamma '&={\frac {1}{2}}{\frac {c'}{\sin c'}}{\frac {\sin a'}{a'}}{\frac {\sin b'}{b'}}.\end{aligned}}$
For uniform normalization of the generators in the Lie algebra involved, express the Pauli matrices in terms of t-matrices, σ → 2i t, so that
$a'\mapsto -{\frac {\theta }{2}},\quad b'\mapsto -{\frac {\phi }{2}}.$
To verify then these are the same coefficients as above, compute the ratios of the coefficients,
${\begin{aligned}{\frac {\alpha '}{\gamma '}}&=\theta \cot {\frac {\theta }{2}}&={\frac {\alpha }{\gamma }}\\{\frac {\beta '}{\gamma '}}&=\phi \cot {\frac {\phi }{2}}&={\frac {\beta }{\gamma }}.\end{aligned}}$
Finally, γ = γ' given the identity d = sin 2c'.
For the general n × n case, one might use Ref.[16]
The quaternion case
The quaternion formulation of the composition of two rotations RB and RA also yields directly the rotation axis and angle of the composite rotation RC = RBRA.
Let the quaternion associated with a spatial rotation R is constructed from its rotation axis S and the rotation angle φ this axis. The associated quaternion is given by,
$S=\cos {\frac {\phi }{2}}+\sin {\frac {\phi }{2}}\mathbf {S} .$
Then the composition of the rotation RR with RA is the rotation RC = RBRA with rotation axis and angle defined by the product of the quaternions
$A=\cos {\frac {\alpha }{2}}+\sin {\frac {\alpha }{2}}\mathbf {A} \quad {\text{ and }}\quad B=\cos {\frac {\beta }{2}}+\sin {\frac {\beta }{2}}\mathbf {B} ,$
that is
$C=\cos {\frac {\gamma }{2}}+\sin {\frac {\gamma }{2}}\mathbf {C} =\left(\cos {\frac {\beta }{2}}+\sin {\frac {\beta }{2}}\mathbf {B} \right)\left(\cos {\frac {\alpha }{2}}+\sin {\frac {\alpha }{2}}\mathbf {A} \right).$
Expand this product to obtain
$\cos {\frac {\gamma }{2}}+\sin {\frac {\gamma }{2}}\mathbf {C} =\left(\cos {\frac {\beta }{2}}\cos {\frac {\alpha }{2}}-\sin {\frac {\beta }{2}}\sin {\frac {\alpha }{2}}\mathbf {B} \cdot \mathbf {A} \right)+\left(\sin {\frac {\beta }{2}}\cos {\frac {\alpha }{2}}\mathbf {B} +\sin {\frac {\alpha }{2}}\cos {\frac {\beta }{2}}\mathbf {A} +\sin {\frac {\beta }{2}}\sin {\frac {\alpha }{2}}\mathbf {B} \times \mathbf {A} \right).$
Divide both sides of this equation by the identity, which is the law of cosines on a sphere,
$\cos {\frac {\gamma }{2}}=\cos {\frac {\beta }{2}}\cos {\frac {\alpha }{2}}-\sin {\frac {\beta }{2}}\sin {\frac {\alpha }{2}}\mathbf {B} \cdot \mathbf {A} ,$
and compute
$\tan {\frac {\gamma }{2}}\mathbf {C} ={\frac {\tan {\frac {\beta }{2}}\mathbf {B} +\tan {\frac {\alpha }{2}}\mathbf {A} +\tan {\frac {\beta }{2}}\tan {\frac {\alpha }{2}}\mathbf {B} \times \mathbf {A} }{1-\tan {\frac {\beta }{2}}\tan {\frac {\alpha }{2}}\mathbf {B} \cdot \mathbf {A} }}.$
This is Rodrigues' formula for the axis of a composite rotation defined in terms of the axes of the two rotations. He derived this formula in 1840 (see page 408).[17]
The three rotation axes A, B, and C form a spherical triangle and the dihedral angles between the planes formed by the sides of this triangle are defined by the rotation angles.
Infinitesimal rotations
This section is an excerpt from Infinitesimal rotation matrix.[edit]
An infinitesimal rotation matrix or differential rotation matrix is a matrix representing an infinitely small rotation.
While a rotation matrix is an orthogonal matrix $R^{\mathsf {T}}=R^{-1}$ representing an element of $SO(n)$ (the special orthogonal group), the differential of a rotation is a skew-symmetric matrix $A^{\mathsf {T}}=-A$ in the tangent space ${\mathfrak {so}}(n)$ (the special orthogonal Lie algebra), which is not itself a rotation matrix.
An infinitesimal rotation matrix has the form
$I+d\theta \,A,$
where $I$ is the identity matrix, $d\theta $ is vanishingly small, and $A\in {\mathfrak {so}}(n).$
For example, if $A=L_{x},$ representing an infinitesimal three-dimensional rotation about the x-axis, a basis element of ${\mathfrak {so}}(3),$
$dL_{x}={\begin{bmatrix}1&0&0\\0&1&-d\theta \\0&d\theta &1\end{bmatrix}}.$
The computation rules for infinitesimal rotation matrices are as usual except that infinitesimals of second order are routinely dropped. With these rules, these matrices do not satisfy all the same properties as ordinary finite rotation matrices under the usual treatment of infinitesimals.[18] It turns out that the order in which infinitesimal rotations are applied is irrelevant.
Realizations of rotations
Main article: Rotation formalisms in three dimensions
See also: Charts on SO(3)
We have seen that there are a variety of ways to represent rotations:
• as orthogonal matrices with determinant 1,
• by axis and rotation angle
• in quaternion algebra with versors and the map 3-sphere S3 → SO(3) (see quaternions and spatial rotations)
• in geometric algebra as a rotor
• as a sequence of three rotations about three fixed axes; see Euler angles.
Spherical harmonics
Main article: Spherical harmonics
See also: Representations of SO(3)
The group SO(3) of three-dimensional Euclidean rotations has an infinite-dimensional representation on the Hilbert space
$L^{2}\left(\mathbf {S} ^{2}\right)=\operatorname {span} \left\{Y_{m}^{\ell },\ell \in \mathbb {N} ^{+},-\ell \leq m\leq \ell \right\},$
where $Y_{m}^{\ell }$ are spherical harmonics. Its elements are square integrable complex-valued functions[nb 5] on the sphere. The inner product on this space is given by
$\langle f,g\rangle =\int _{\mathbf {S} ^{2}}{\overline {f}}g\,d\Omega =\int _{0}^{2\pi }\int _{0}^{\pi }{\overline {f}}g\sin \theta \,d\theta \,d\phi .$
(H1)
If f is an arbitrary square integrable function defined on the unit sphere S2, then it can be expressed as[19]
$|f\rangle =\sum _{\ell =1}^{\infty }\sum _{m=-\ell }^{m=\ell }\left|Y_{m}^{\ell }\right\rangle \left\langle Y_{m}^{\ell }|f\right\rangle ,\qquad f(\theta ,\phi )=\sum _{\ell =1}^{\infty }\sum _{m=-\ell }^{m=\ell }f_{\ell m}Y_{m}^{\ell }(\theta ,\phi ),$
(H2)
where the expansion coefficients are given by
$f_{\ell m}=\left\langle Y_{m}^{\ell },f\right\rangle =\int _{\mathbf {S} ^{2}}{\overline {Y_{m}^{\ell }}}f\,d\Omega =\int _{0}^{2\pi }\int _{0}^{\pi }{\overline {Y_{m}^{\ell }}}(\theta ,\phi )f(\theta ,\phi )\sin \theta \,d\theta \,d\phi .$
(H3)
The Lorentz group action restricts to that of SO(3) and is expressed as
$(\Pi (R)f)(\theta (x),\phi (x))=\sum _{\ell =1}^{\infty }\sum _{m=-\ell }^{m=\ell }\sum _{m'=-\ell }^{m'=\ell }D_{mm'}^{(\ell )}(R)f_{\ell m'}Y_{m}^{\ell }\left(\theta \left(R^{-1}x\right),\phi \left(R^{-1}x\right)\right),\qquad R\in \operatorname {SO} (3),\quad x\in \mathbf {S} ^{2}.$
(H4)
This action is unitary, meaning that
$\langle \Pi (R)f,\Pi (R)g\rangle =\langle f,g\rangle \qquad \forall f,g\in \mathbf {S} ^{2},\quad \forall R\in \operatorname {SO} (3).$
(H5)
The D(ℓ) can be obtained from the D(m, n) of above using Clebsch–Gordan decomposition, but they are more easily directly expressed as an exponential of an odd-dimensional su(2)-representation (the 3-dimensional one is exactly 𝖘𝖔(3)).[20][21] In this case the space L2(S2) decomposes neatly into an infinite direct sum of irreducible odd finite-dimensional representations V2i + 1, i = 0, 1, ... according to[22]
$L^{2}\left(\mathbf {S} ^{2}\right)=\sum _{i=0}^{\infty }V_{2i+1}\equiv \bigoplus _{i=0}^{\infty }\operatorname {span} \left\{Y_{m}^{2i+1}\right\}.$
(H6)
This is characteristic of infinite-dimensional unitary representations of SO(3). If Π is an infinite-dimensional unitary representation on a separable[nb 6] Hilbert space, then it decomposes as a direct sum of finite-dimensional unitary representations.[19] Such a representation is thus never irreducible. All irreducible finite-dimensional representations (Π, V) can be made unitary by an appropriate choice of inner product,[19]
$\langle f,g\rangle _{U}\equiv \int _{\operatorname {SO} (3)}\langle \Pi (R)f,\Pi (R)g\rangle \,dg={\frac {1}{8\pi ^{2}}}\int _{0}^{2\pi }\int _{0}^{\pi }\int _{0}^{2\pi }\langle \Pi (R)f,\Pi (R)g\rangle \sin \theta \,d\phi \,d\theta \,d\psi ,\quad f,g\in V,$
where the integral is the unique invariant integral over SO(3) normalized to 1, here expressed using the Euler angles parametrization. The inner product inside the integral is any inner product on V.
Generalizations
The rotation group generalizes quite naturally to n-dimensional Euclidean space, $\mathbb {R} ^{n}$ with its standard Euclidean structure. The group of all proper and improper rotations in n dimensions is called the orthogonal group O(n), and the subgroup of proper rotations is called the special orthogonal group SO(n), which is a Lie group of dimension n(n − 1)/2.
In special relativity, one works in a 4-dimensional vector space, known as Minkowski space rather than 3-dimensional Euclidean space. Unlike Euclidean space, Minkowski space has an inner product with an indefinite signature. However, one can still define generalized rotations which preserve this inner product. Such generalized rotations are known as Lorentz transformations and the group of all such transformations is called the Lorentz group.
The rotation group SO(3) can be described as a subgroup of E+(3), the Euclidean group of direct isometries of Euclidean $\mathbb {R} ^{3}.$ This larger group is the group of all motions of a rigid body: each of these is a combination of a rotation about an arbitrary axis and a translation, or put differently, a combination of an element of SO(3) and an arbitrary translation.
In general, the rotation group of an object is the symmetry group within the group of direct isometries; in other words, the intersection of the full symmetry group and the group of direct isometries. For chiral objects it is the same as the full symmetry group.
See also
• Orthogonal group
• Angular momentum
• Coordinate rotations
• Charts on SO(3)
• Representations of SO(3)
• Euler angles
• Rodrigues' rotation formula
• Infinitesimal rotation
• Pin group
• Quaternions and spatial rotations
• Rigid body
• Spherical harmonics
• Plane of rotation
• Lie group
• Pauli matrix
• Plate trick
• Three-dimensional rotation operator
Footnotes
1. This is effected by first applying a rotation $g_{\theta }$ through φ about the z-axis to take the x-axis to the line L, the intersection between the planes xy and x'y', the latter being the rotated xy-plane. Then rotate with $g_{\theta }$ through θ about L to obtain the new z-axis from the old one, and finally rotate by $g_{\psi }$ through an angle ψ about the new z-axis, where ψ is the angle between L and the new x-axis. In the equation, $g_{\theta }$ and $g_{\psi }$ are expressed in a temporary rotated basis at each step, which is seen from their simple form. To transform these back to the original basis, observe that $\mathbf {g} _{\theta }=g_{\phi }g_{\theta }g_{\phi }^{-1}.$ Here boldface means that the rotation is expressed in the original basis. Likewise,
$\mathbf {g} _{\psi }=g_{\phi }g_{\theta }g_{\phi }^{-1}g_{\phi }g_{\psi }\left[g_{\phi }g_{\theta }g_{\phi }^{-1}g_{\phi }\right]^{-1}.$
Thus
$\mathbf {g} _{\psi }\mathbf {g} _{\theta }\mathbf {g} _{\phi }=g_{\phi }g_{\theta }g_{\phi }^{-1}g_{\phi }g_{\psi }\left[g_{\phi }g_{\theta }g_{\phi }^{-1}g_{\phi }\right]^{-1}*g_{\phi }g_{\theta }g_{\phi }^{-1}*g_{\phi }=g_{\phi }g_{\theta }g_{\psi }.$
2. For an alternative derivation of ${\mathfrak {so}}(3)$, see Classical group.
3. Specifically, ${\boldsymbol {U}}{\boldsymbol {J}}_{\alpha }{\boldsymbol {U}}^{\dagger }=i{\boldsymbol {L}}_{\alpha }$ for
${\boldsymbol {U}}={\frac {1}{\sqrt {2}}}{\begin{pmatrix}-1&0&1\\-i&0&-i\\0&{\sqrt {2}}&0\end{pmatrix}}.$
4. For a full proof, see Derivative of the exponential map. Issues of convergence of this series to the correct element of the Lie algebra are here swept under the carpet. Convergence is guaranteed when $\|X\|+\|Y\|<\log 2$ and $\|Z\|<\log 2.$ The series may still converge even if these conditions are not fulfilled. A solution always exists since exp is onto in the cases under consideration.
5. The elements of L2(S2) are actually equivalence classes of functions. two functions are declared equivalent if they differ merely on a set of measure zero. The integral is the Lebesgue integral in order to obtain a complete inner product space.
6. A Hilbert space is separable if and only if it has a countable basis. All separable Hilbert spaces are isomorphic.
References
1. Jacobson (2009), p. 34, Ex. 14.
2. n × n real matrices are identical to linear transformations of $\mathbb {R} ^{n}$ expressed in its standard basis.
3. Coxeter, H. S. M. (1973). Regular polytopes (Third ed.). New York. p. 53. ISBN 0-486-61480-8.{{cite book}}: CS1 maint: location missing publisher (link)
4. Hall 2015 Proposition 1.17
5. Rossmann 2002 p. 95.
6. These expressions were, in fact, seminal in the development of quantum mechanics in the 1930s, cf. Ch III, § 16, B.L. van der Waerden, 1932/1932
7. Hall 2015 Proposition 3.24
8. Rossmann 2002
9. Engø 2001
10. Hall 2015 Example 3.27
11. See Rossmann 2002, theorem 3, section 2.2.
12. Rossmann 2002 Section 1.1.
13. Hall 2003 Theorem 2.27.
14. Shoemake, Ken (1992-01-01), Kirk, DAVID (ed.), "III.6 - UNIFORM RANDOM ROTATIONS", Graphics Gems III (IBM Version), San Francisco: Morgan Kaufmann, pp. 124–132, ISBN 978-0-12-409673-8, retrieved 2022-07-29
15. Hall 2003, Ch. 3; Varadarajan 1984, §2.15
16. Curtright, Fairlie & Zachos 2014 Group elements of SU(2) are expressed in closed form as finite polynomials of the Lie algebra generators, for all definite spin representations of the rotation group.
17. Rodrigues, O. (1840), Des lois géométriques qui régissent les déplacements d'un système solide dans l'espace, et la variation des coordonnées provenant de ses déplacements con- sidérés indépendamment des causes qui peuvent les produire, Journal de Mathématiques Pures et Appliquées de Liouville 5, 380–440.
18. (Goldstein, Poole & Safko 2002, §4.8)
19. Gelfand, Minlos & Shapiro 1963
20. In Quantum Mechanics – non-relativistic theory by Landau and Lifshitz the lowest order D are calculated analytically.
21. Curtright, Fairlie & Zachos 2014 A formula for D(ℓ) valid for all ℓ is given.
22. Hall 2003 Section 4.3.5.
Bibliography
• Boas, Mary L. (2006), Mathematical Methods in the Physical Sciences (3rd ed.), John Wiley & sons, pp. 120, 127, 129, 155ff and 535, ISBN 978-0471198260
• Curtright, T. L.; Fairlie, D. B.; Zachos, C. K. (2014), "A compact formula for rotations as spin matrix polynomials", SIGMA, 10: 084, arXiv:1402.3541, Bibcode:2014SIGMA..10..084C, doi:10.3842/SIGMA.2014.084, S2CID 18776942
• Engø, Kenth (2001), "On the BCH-formula in 𝖘𝖔(3)", BIT Numerical Mathematics, 41 (3): 629–632, doi:10.1023/A:1021979515229, ISSN 0006-3835, S2CID 126053191
• Gelfand, I.M.; Minlos, R.A.; Shapiro, Z.Ya. (1963), Representations of the Rotation and Lorentz Groups and their Applications, New York: Pergamon Press
• Goldstein, Herbert; Poole, Charles P.; Safko, John L. (2002), Classical Mechanics (third ed.), Addison Wesley, ISBN 978-0-201-65702-9
• Hall, Brian C. (2015), Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, ISBN 978-3319134666
• Hall, Brian C. (2003). Lie groups, Lie algebras, and representations : an elementary introduction. Graduate Texts in Mathematics. Vol. 222. New York. ISBN 0-387-40122-9.{{cite book}}: CS1 maint: location missing publisher (link)
• Jacobson, Nathan (2009), Basic algebra, vol. 1 (2nd ed.), Dover Publications, ISBN 978-0-486-47189-1
• Joshi, A. W. (2007), Elements of Group Theory for Physicists, New Age International, pp. 111ff, ISBN 978-81-224-0975-8
• Rossmann, Wulf (2002), Lie Groups – An Introduction Through Linear Groups, Oxford Graduate Texts in Mathematics, Oxford Science Publications, ISBN 0-19-859683-9
• van der Waerden, B. L. (1952), Group Theory and Quantum Mechanics, Springer Publishing, ISBN 978-3642658624 (translation of the original 1932 edition, Die Gruppentheoretische Methode in Der Quantenmechanik).
• Varadarajan, V. S. (1984). Lie groups, Lie algebras, and their representations. New York: Springer-Verlag. ISBN 978-0-387-90969-1.
• Veltman, M.; 't Hooft, G.; de Wit, B. (2007). "Lie Groups in Physics (online lecture)" (PDF). Retrieved 2016-10-24..
| Wikipedia |
Order-6 pentagonal tiling
In geometry, the order-6 pentagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {5,6}.
Order-6 pentagonal tiling
Poincaré disk model of the hyperbolic plane
TypeHyperbolic regular tiling
Vertex configuration56
Schläfli symbol{5,6}
Wythoff symbol6 | 5 2
Coxeter diagram
Symmetry group[6,5], (*652)
DualOrder-5 hexagonal tiling
PropertiesVertex-transitive, edge-transitive, face-transitive
Uniform coloring
This regular tiling can also be constructed from [(5,5,3)] symmetry alternating two colors of pentagons, represented by t1(5,5,3).
Symmetry
This tiling represents a hyperbolic kaleidoscope of 6 mirrors defining a regular hexagon fundamental domain, and 5 mirrors meeting at a point. This symmetry by orbifold notation is called *33333 with 5 order-3 mirror intersections.
Related polyhedra and tiling
This tiling is topologically related as a part of sequence of regular tilings with order-6 vertices with Schläfli symbol {n,6}, and Coxeter diagram , progressing to infinity.
Regular tilings {n,6}
Spherical Euclidean Hyperbolic tilings
{2,6}
{3,6}
{4,6}
{5,6}
{6,6}
{7,6}
{8,6}
...
{∞,6}
Uniform hexagonal/pentagonal tilings
Symmetry: [6,5], (*652) [6,5]+, (652) [6,5+], (5*3) [1+,6,5], (*553)
{6,5} t{6,5} r{6,5} 2t{6,5}=t{5,6} 2r{6,5}={5,6} rr{6,5} tr{6,5} sr{6,5} s{5,6} h{6,5}
Uniform duals
V65 V5.12.12 V5.6.5.6 V6.10.10 V56 V4.5.4.6 V4.10.12 V3.3.5.3.6 V3.3.3.5.3.5 V(3.5)5
[(5,5,3)] reflective symmetry uniform tilings
References
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 19, The Hyperbolic Archimedean Tessellations)
• "Chapter 10: Regular honeycombs in hyperbolic space". The Beauty of Geometry: Twelve Essays. Dover Publications. 1999. ISBN 0-486-40919-8. LCCN 99035678.
See also
Wikimedia Commons has media related to Order-6 pentagonal tiling.
• Square tiling
• Uniform tilings in hyperbolic plane
• List of regular polytopes
External links
• Weisstein, Eric W. "Hyperbolic tiling". MathWorld.
• Weisstein, Eric W. "Poincaré hyperbolic disk". MathWorld.
• Hyperbolic and Spherical Tiling Gallery
• KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
• Hyperbolic Planar Tessellations, Don Hatch
Tessellation
Periodic
• Pythagorean
• Rhombille
• Schwarz triangle
• Rectangle
• Domino
• Uniform tiling and honeycomb
• Coloring
• Convex
• Kisrhombille
• Wallpaper group
• Wythoff
Aperiodic
• Ammann–Beenker
• Aperiodic set of prototiles
• List
• Einstein problem
• Socolar–Taylor
• Gilbert
• Penrose
• Pentagonal
• Pinwheel
• Quaquaversal
• Rep-tile and Self-tiling
• Sphinx
• Socolar
• Truchet
Other
• Anisohedral and Isohedral
• Architectonic and catoptric
• Circle Limit III
• Computer graphics
• Honeycomb
• Isotoxal
• List
• Packing
• Problems
• Domino
• Wang
• Heesch's
• Squaring
• Dividing a square into similar rectangles
• Prototile
• Conway criterion
• Girih
• Regular Division of the Plane
• Regular grid
• Substitution
• Voronoi
• Voderberg
By vertex type
Spherical
• 2n
• 33.n
• V33.n
• 42.n
• V42.n
Regular
• 2∞
• 36
• 44
• 63
Semi-
regular
• 32.4.3.4
• V32.4.3.4
• 33.42
• 33.∞
• 34.6
• V34.6
• 3.4.6.4
• (3.6)2
• 3.122
• 42.∞
• 4.6.12
• 4.82
Hyper-
bolic
• 32.4.3.5
• 32.4.3.6
• 32.4.3.7
• 32.4.3.8
• 32.4.3.∞
• 32.5.3.5
• 32.5.3.6
• 32.6.3.6
• 32.6.3.8
• 32.7.3.7
• 32.8.3.8
• 33.4.3.4
• 32.∞.3.∞
• 34.7
• 34.8
• 34.∞
• 35.4
• 37
• 38
• 3∞
• (3.4)3
• (3.4)4
• 3.4.62.4
• 3.4.7.4
• 3.4.8.4
• 3.4.∞.4
• 3.6.4.6
• (3.7)2
• (3.8)2
• 3.142
• 3.162
• (3.∞)2
• 3.∞2
• 42.5.4
• 42.6.4
• 42.7.4
• 42.8.4
• 42.∞.4
• 45
• 46
• 47
• 48
• 4∞
• (4.5)2
• (4.6)2
• 4.6.12
• 4.6.14
• V4.6.14
• 4.6.16
• V4.6.16
• 4.6.∞
• (4.7)2
• (4.8)2
• 4.8.10
• V4.8.10
• 4.8.12
• 4.8.14
• 4.8.16
• 4.8.∞
• 4.102
• 4.10.12
• 4.122
• 4.12.16
• 4.142
• 4.162
• 4.∞2
• (4.∞)2
• 54
• 55
• 56
• 5∞
• 5.4.6.4
• (5.6)2
• 5.82
• 5.102
• 5.122
• (5.∞)2
• 64
• 65
• 66
• 68
• 6.4.8.4
• (6.8)2
• 6.82
• 6.102
• 6.122
• 6.162
• 73
• 74
• 77
• 7.62
• 7.82
• 7.142
• 83
• 84
• 86
• 88
• 8.62
• 8.122
• 8.162
• ∞3
• ∞4
• ∞5
• ∞∞
• ∞.62
• ∞.82
| Wikipedia |
Order-6 hexagonal tiling
In geometry, the order-6 hexagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {6,6} and is self-dual.
Order-6 hexagonal tiling
Poincaré disk model of the hyperbolic plane
TypeHyperbolic regular tiling
Vertex configuration66
Schläfli symbol{6,6}
Wythoff symbol6 | 6 2
Coxeter diagram
Symmetry group[6,6], (*662)
Dualself dual
PropertiesVertex-transitive, edge-transitive, face-transitive
Symmetry
This tiling represents a hyperbolic kaleidoscope of 6 mirrors defining a regular hexagon fundamental domain. This symmetry by orbifold notation is called *333333 with 6 order-3 mirror intersections. In Coxeter notation can be represented as [6*,6], removing two of three mirrors (passing through the hexagon center) in the [6,6] symmetry.
The even/odd fundamental domains of this kaleidoscope can be seen in the alternating colorings of the tiling:
Related polyhedra and tiling
This tiling is topologically related as a part of sequence of regular tilings with order-6 vertices with Schläfli symbol {n,6}, and Coxeter diagram , progressing to infinity.
Regular tilings {n,6}
Spherical Euclidean Hyperbolic tilings
{2,6}
{3,6}
{4,6}
{5,6}
{6,6}
{7,6}
{8,6}
...
{∞,6}
This tiling is topologically related as a part of sequence of regular tilings with hexagonal faces, starting with the hexagonal tiling, with Schläfli symbol {6,n}, and Coxeter diagram , progressing to infinity.
*n62 symmetry mutation of regular tilings: {6,n}
Spherical Euclidean Hyperbolic tilings
{6,2}
{6,3}
{6,4}
{6,5}
{6,6}
{6,7}
{6,8}
...
{6,∞}
Uniform hexahexagonal tilings
Symmetry: [6,6], (*662)
=
=
=
=
=
=
=
=
=
=
=
=
=
=
{6,6}
= h{4,6}
t{6,6}
= h2{4,6}
r{6,6}
{6,4}
t{6,6}
= h2{4,6}
{6,6}
= h{4,6}
rr{6,6}
r{6,4}
tr{6,6}
t{6,4}
Uniform duals
V66 V6.12.12 V6.6.6.6 V6.12.12 V66 V4.6.4.6 V4.12.12
Alternations
[1+,6,6]
(*663)
[6+,6]
(6*3)
[6,1+,6]
(*3232)
[6,6+]
(6*3)
[6,6,1+]
(*663)
[(6,6,2+)]
(2*33)
[6,6]+
(662)
= = =
h{6,6} s{6,6} hr{6,6} s{6,6} h{6,6} hrr{6,6} sr{6,6}
Similar H2 tilings in *3232 symmetry
Coxeter
diagrams
Vertex
figure
66 (3.4.3.4)2 3.4.6.6.4 6.4.6.4
Image
Dual
References
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 19, The Hyperbolic Archimedean Tessellations)
• "Chapter 10: Regular honeycombs in hyperbolic space". The Beauty of Geometry: Twelve Essays. Dover Publications. 1999. ISBN 0-486-40919-8. LCCN 99035678.
See also
Wikimedia Commons has media related to Order-6 hexagonal tiling.
• Square tiling
• Tilings of regular polygons
• List of uniform planar tilings
• List of regular polytopes
External links
• Weisstein, Eric W. "Hyperbolic tiling". MathWorld.
• Weisstein, Eric W. "Poincaré hyperbolic disk". MathWorld.
• Hyperbolic and Spherical Tiling Gallery
• KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
• Hyperbolic Planar Tessellations, Don Hatch
Tessellation
Periodic
• Pythagorean
• Rhombille
• Schwarz triangle
• Rectangle
• Domino
• Uniform tiling and honeycomb
• Coloring
• Convex
• Kisrhombille
• Wallpaper group
• Wythoff
Aperiodic
• Ammann–Beenker
• Aperiodic set of prototiles
• List
• Einstein problem
• Socolar–Taylor
• Gilbert
• Penrose
• Pentagonal
• Pinwheel
• Quaquaversal
• Rep-tile and Self-tiling
• Sphinx
• Socolar
• Truchet
Other
• Anisohedral and Isohedral
• Architectonic and catoptric
• Circle Limit III
• Computer graphics
• Honeycomb
• Isotoxal
• List
• Packing
• Problems
• Domino
• Wang
• Heesch's
• Squaring
• Dividing a square into similar rectangles
• Prototile
• Conway criterion
• Girih
• Regular Division of the Plane
• Regular grid
• Substitution
• Voronoi
• Voderberg
By vertex type
Spherical
• 2n
• 33.n
• V33.n
• 42.n
• V42.n
Regular
• 2∞
• 36
• 44
• 63
Semi-
regular
• 32.4.3.4
• V32.4.3.4
• 33.42
• 33.∞
• 34.6
• V34.6
• 3.4.6.4
• (3.6)2
• 3.122
• 42.∞
• 4.6.12
• 4.82
Hyper-
bolic
• 32.4.3.5
• 32.4.3.6
• 32.4.3.7
• 32.4.3.8
• 32.4.3.∞
• 32.5.3.5
• 32.5.3.6
• 32.6.3.6
• 32.6.3.8
• 32.7.3.7
• 32.8.3.8
• 33.4.3.4
• 32.∞.3.∞
• 34.7
• 34.8
• 34.∞
• 35.4
• 37
• 38
• 3∞
• (3.4)3
• (3.4)4
• 3.4.62.4
• 3.4.7.4
• 3.4.8.4
• 3.4.∞.4
• 3.6.4.6
• (3.7)2
• (3.8)2
• 3.142
• 3.162
• (3.∞)2
• 3.∞2
• 42.5.4
• 42.6.4
• 42.7.4
• 42.8.4
• 42.∞.4
• 45
• 46
• 47
• 48
• 4∞
• (4.5)2
• (4.6)2
• 4.6.12
• 4.6.14
• V4.6.14
• 4.6.16
• V4.6.16
• 4.6.∞
• (4.7)2
• (4.8)2
• 4.8.10
• V4.8.10
• 4.8.12
• 4.8.14
• 4.8.16
• 4.8.∞
• 4.102
• 4.10.12
• 4.122
• 4.12.16
• 4.142
• 4.162
• 4.∞2
• (4.∞)2
• 54
• 55
• 56
• 5∞
• 5.4.6.4
• (5.6)2
• 5.82
• 5.102
• 5.122
• (5.∞)2
• 64
• 65
• 66
• 68
• 6.4.8.4
• (6.8)2
• 6.82
• 6.102
• 6.122
• 6.162
• 73
• 74
• 77
• 7.62
• 7.82
• 7.142
• 83
• 84
• 86
• 88
• 8.62
• 8.122
• 8.162
• ∞3
• ∞4
• ∞5
• ∞∞
• ∞.62
• ∞.82
| Wikipedia |
Order-6 octagonal tiling
In geometry, the order-6 octagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {8,6}.
Order-6 octagonal tiling
Poincaré disk model of the hyperbolic plane
TypeHyperbolic regular tiling
Vertex configuration86
Schläfli symbol{8,6}
Wythoff symbol6 | 8 2
Coxeter diagram
Symmetry group[8,6], (*862)
DualOrder-8 hexagonal tiling
PropertiesVertex-transitive, edge-transitive, face-transitive
Symmetry
This tiling represents a hyperbolic kaleidoscope of 8 mirrors meeting at a point and bounding regular octagon fundamental domains. This symmetry by orbifold notation is called *33333333 with 8 order-3 mirror intersections. In Coxeter notation can be represented as [8*,6], removing two of three mirrors (passing through the octagon center) in the [8,6] symmetry.
Uniform constructions
There are four uniform constructions of this tiling, three of them as constructed by mirror removal from the [8,6] kaleidoscope. Removing the mirror between the order 2 and 6 points, [8,6,1+], gives [(8,8,3)], (*883). Removing two mirrors as [8,6*], leaves remaining mirrors (*444444).
Four uniform constructions of 8.8.8.8
Uniform
Coloring
Symmetry [8,6]
(*862)
[8,6,1+] = [(8,8,3)]
(*883)
=
[8,1+,6]
(*4232)
=
[8,6*]
(*444444)
Symbol {8,6} {8,6}1⁄2 r(8,6,8)
Coxeter
diagram
= =
Related polyhedra and tiling
This tiling is topologically related as a part of sequence of regular tilings with octagonal faces, starting with the octagonal tiling, with Schläfli symbol {8,n}, and Coxeter diagram , progressing to infinity.
n82 symmetry mutations of regular tilings: 8n
Space Spherical Compact hyperbolic Paracompact
Tiling
Config. 8.8 83 84 85 86 87 88 ...8∞
Regular tilings {n,6}
Spherical Euclidean Hyperbolic tilings
{2,6}
{3,6}
{4,6}
{5,6}
{6,6}
{7,6}
{8,6}
...
{∞,6}
Uniform octagonal/hexagonal tilings
Symmetry: [8,6], (*862)
{8,6} t{8,6}
r{8,6} 2t{8,6}=t{6,8} 2r{8,6}={6,8} rr{8,6} tr{8,6}
Uniform duals
V86 V6.16.16 V(6.8)2 V8.12.12 V68 V4.6.4.8 V4.12.16
Alternations
[1+,8,6]
(*466)
[8+,6]
(8*3)
[8,1+,6]
(*4232)
[8,6+]
(6*4)
[8,6,1+]
(*883)
[(8,6,2+)]
(2*43)
[8,6]+
(862)
h{8,6} s{8,6} hr{8,6} s{6,8} h{6,8} hrr{8,6} sr{8,6}
Alternation duals
V(4.6)6 V3.3.8.3.8.3 V(3.4.4.4)2 V3.4.3.4.3.6 V(3.8)8 V3.45 V3.3.6.3.8
See also
Wikimedia Commons has media related to Order-6 octagonal tiling.
• Square tiling
• Tilings of regular polygons
• List of uniform planar tilings
• List of regular polytopes
References
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 19, The Hyperbolic Archimedean Tessellations)
• "Chapter 10: Regular honeycombs in hyperbolic space". The Beauty of Geometry: Twelve Essays. Dover Publications. 1999. ISBN 0-486-40919-8. LCCN 99035678.
External links
• Weisstein, Eric W. "Hyperbolic tiling". MathWorld.
• Weisstein, Eric W. "Poincaré hyperbolic disk". MathWorld.
• Hyperbolic and Spherical Tiling Gallery
• KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
• Hyperbolic Planar Tessellations, Don Hatch
Tessellation
Periodic
• Pythagorean
• Rhombille
• Schwarz triangle
• Rectangle
• Domino
• Uniform tiling and honeycomb
• Coloring
• Convex
• Kisrhombille
• Wallpaper group
• Wythoff
Aperiodic
• Ammann–Beenker
• Aperiodic set of prototiles
• List
• Einstein problem
• Socolar–Taylor
• Gilbert
• Penrose
• Pentagonal
• Pinwheel
• Quaquaversal
• Rep-tile and Self-tiling
• Sphinx
• Socolar
• Truchet
Other
• Anisohedral and Isohedral
• Architectonic and catoptric
• Circle Limit III
• Computer graphics
• Honeycomb
• Isotoxal
• List
• Packing
• Problems
• Domino
• Wang
• Heesch's
• Squaring
• Dividing a square into similar rectangles
• Prototile
• Conway criterion
• Girih
• Regular Division of the Plane
• Regular grid
• Substitution
• Voronoi
• Voderberg
By vertex type
Spherical
• 2n
• 33.n
• V33.n
• 42.n
• V42.n
Regular
• 2∞
• 36
• 44
• 63
Semi-
regular
• 32.4.3.4
• V32.4.3.4
• 33.42
• 33.∞
• 34.6
• V34.6
• 3.4.6.4
• (3.6)2
• 3.122
• 42.∞
• 4.6.12
• 4.82
Hyper-
bolic
• 32.4.3.5
• 32.4.3.6
• 32.4.3.7
• 32.4.3.8
• 32.4.3.∞
• 32.5.3.5
• 32.5.3.6
• 32.6.3.6
• 32.6.3.8
• 32.7.3.7
• 32.8.3.8
• 33.4.3.4
• 32.∞.3.∞
• 34.7
• 34.8
• 34.∞
• 35.4
• 37
• 38
• 3∞
• (3.4)3
• (3.4)4
• 3.4.62.4
• 3.4.7.4
• 3.4.8.4
• 3.4.∞.4
• 3.6.4.6
• (3.7)2
• (3.8)2
• 3.142
• 3.162
• (3.∞)2
• 3.∞2
• 42.5.4
• 42.6.4
• 42.7.4
• 42.8.4
• 42.∞.4
• 45
• 46
• 47
• 48
• 4∞
• (4.5)2
• (4.6)2
• 4.6.12
• 4.6.14
• V4.6.14
• 4.6.16
• V4.6.16
• 4.6.∞
• (4.7)2
• (4.8)2
• 4.8.10
• V4.8.10
• 4.8.12
• 4.8.14
• 4.8.16
• 4.8.∞
• 4.102
• 4.10.12
• 4.122
• 4.12.16
• 4.142
• 4.162
• 4.∞2
• (4.∞)2
• 54
• 55
• 56
• 5∞
• 5.4.6.4
• (5.6)2
• 5.82
• 5.102
• 5.122
• (5.∞)2
• 64
• 65
• 66
• 68
• 6.4.8.4
• (6.8)2
• 6.82
• 6.102
• 6.122
• 6.162
• 73
• 74
• 77
• 7.62
• 7.82
• 7.142
• 83
• 84
• 86
• 88
• 8.62
• 8.122
• 8.162
• ∞3
• ∞4
• ∞5
• ∞∞
• ∞.62
• ∞.82
| Wikipedia |
Water pouring puzzle
Water pouring puzzles (also called water jug problems, decanting problems,[1][2] measuring puzzles, or Die Hard with a Vengeance puzzles) are a class of puzzle involving a finite collection of water jugs of known integer capacities (in terms of a liquid measure such as liters or gallons). Initially each jug contains a known integer volume of liquid, not necessarily equal to its capacity.
Puzzles of this type ask how many steps of pouring water from one jug to another (until either one jug becomes empty or the other becomes full) are needed to reach a goal state, specified in terms of the volume of liquid that must be present in some jug or jugs.[3]
By Bézout's identity, such puzzles have solution if and only if the desired volume is a multiple of the greatest common divisor of all the integer volume capacities of jugs.
Rules
It is a common assumption, stated as part of these puzzles, that the jugs in the puzzle are irregularly shaped and unmarked, so that it is impossible to accurately measure any quantity of water that does not completely fill a jug. Other assumptions of these problems may include that no water can be spilled, and that each step pouring water from a source jug to a destination jug stops when either the source jug is empty or the destination jug is full, whichever happens first.
Standard example
The standard puzzle of this kind works with three jugs of capacity 8, 5 and 3 liters. These are initially filled with 8, 0 and 0 liters. In the goal state they should be filled with 4, 4 and 0 liters. The puzzle may be solved in seven steps, passing through the following sequence of states (denoted as a bracketed triple of the three volumes of water in the three jugs):
[8,0,0] → [3,5,0] → [3,2,3] → [6,2,0] → [6,0,2] → [1,5,2] → [1,4,3] → [4,4,0].
Cowley (1926) writes that this particular puzzle "dates back to mediaeval times" and notes its occurrence in Bachet's 17th-century mathematics textbook.
Reversibility of actions
Since the rules only allows stopping/turning on the boundaries of the Cartesian grid (i.e. at the full capacities of each jug), the only reversible actions (reversible in one step) are:
• Transferring water from a full jug to any jug
• Transferring water from any jug to an empty jug
The only irreversible actions that can't be reversed in one step are:
• Transferring water from a partially full jug to another partially full jug
By restricting ourselves to reversible actions only, we can construct the solution to the problem from the desired result. From the point [4,4,0], there are only two reversible action: Transferring 3 liters from the 8 liter jug to the empty 3 liter jug [1,4,3], or transferring 3 liters from the 5 liter jug to the empty 3 liter jug [4,1,3]. Therefore, there are only two solutions to this problem:
[4,4,0] ↔ [1,4,3] ↔ [1,5,2] ↔ [6,0,2] ↔ [6,2,0] ↔ [3,2,3] ↔ [3,5,0] ↔ [8,0,0]
[4,4,0] ↔ [4,1,3] ↔ [7,1,0] ↔ [7,0,1] ↔ [2,5,1] ↔ [2,3,3] ↔ [5,3,0] ↔ [5,0,3] ↔ [8,0,0]
Variant with taps and sinks
The rules are sometimes formulated by adding a tap (a source "jug" with infinite water) and a sink (a drain "jug" that accepts any amount of water without limit). Filling a jug to the rim from the tap or pouring the entire contents of jug into the drain each count as one step while solving the problem. This version of the puzzle was featured in a scene of the 1995 movie Die Hard with a Vengeance.[4] This variant has an optimal solution that can be obtained using a billiard-shape barycentric plot (or a mathematical billiard).[5]
The graph shows two ways to obtain 4 liters using 3-liter and 5-liter jugs, and a water source and sink on a Cartesian grid with diagonal lines of slope −1 (such that $x+y=const.$ on these diagonal lines, which represent pouring water from one jug to the other jug). The x and y axes represent the amounts in the 5 and 3 L jugs, respectively. Starting from (0, 0), we traverse the grid along the line segments, turning only on its boundaries, until we reach the black line denoting 4 L in the 5 L jug. Solid lines denote pouring between jugs, dashed lines denote filling a jug and dotted lines denote emptying a jug.
Concatenating either solution, traversal of the 4 L line and the reverse of the other solution returns to (0, 0), yielding a cycle graph. If and only if the jugs' volumes are co-prime, every boundary point is visited, giving an algorithm to measure any integer amount up to the sum of the volumes.
As shown in the previous section, we can construct the solution to the problem from the desired result by using reversible actions only (emptying a full jug into the sink and filling an empty jug from the tap are both reversible). To obtain 4 liters using 3-liter and 5-liter jugs, we want to reach the point (4, 0). From the point (4, 0), there are only two reversible actions: filling the empty 3-liter jug to full from the tap (4,3), or transferring 1 liter of water from the 5-liter jug to the 3-liter jug (1,3). Therefore, there are only two solutions to the problem:
(4, 0) ↔ (4, 3) ↔ (5, 2) ↔ (0, 2) ↔ (2, 0) ↔ (2, 3) ↔ (5, 0) ↔ (0, 0)
(4, 0) ↔ (1, 3) ↔ (1, 0) ↔ (0, 1) ↔ (5, 1) ↔ (3, 3) ↔ (3, 0) ↔ (0, 3) ↔ (0, 0)
The cycle graph can be represented by the ordered pairs connected by reversible actions:
(0, 0) ↔ (5, 0) ↔ (2, 3) ↔ (2, 0) ↔ (0, 2) ↔ (5, 2) ↔ (4, 3) ↔ (4, 0) ↔ (1, 3) ↔ (1, 0) ↔ (0, 1) ↔ (5, 1) ↔ (3, 3) ↔ (3, 0) ↔ (0, 3) ↔ (0, 0)
which contains all the possible states reachable with a 3-liter jug and a 5-liter jug. The state (1, 2), for example, is impossible to reach from an initial state of (0, 0), since (1, 2) has both jugs partially full, and no reversible action is possible from this state.
Jug with initial water
Another variant[6] is when one of the jugs has a known volume of water to begin with; In that case, the achievable volumes are either a multiple of the greatest common divisor between the two containers away from the existing known volume, or from zero. For example, if one jug that holds 8 liters is empty and the other jug that hold 12 liters has 9 liters of water in it to begin with, then with a source (tap) and a drain (sink), these two jugs can measure volumes of 9 liters, 5 liters, 1 liter, as well as 12 liters, 8 liters, 4 liters and 0 liters. The simplest solution for 5 liters is (9,0) → (9,8) → (12,5); The simplest solution for 4 liters is (9,0) → (12,0) → (4,8). These solutions can be visualized by red and blue arrows in a Cartesian grid with diagonal lines (of slope -1 such that $x+y=const.$ on these diagonal lines) spaced 4 liters apart, both horizontally and vertically.
Again, if we restrict ourselves to reversible actions only, from the desired point (5,0), there are only two reversible actions: transferring 5 liter of water from the 12-liter jug to the 8-liter jug (0,5), or filling the empty 8 liter jug to full from the tap (5,8). Therefore, there are only two solutions to the problem:
(5, 0) ↔ (0, 5) ↔ (12, 5) ↔ (9, 8) ↔ (9, 0)
(5, 0) ↔ (5, 8) ↔ (12, 1) ↔ (0, 1) ↔ (1, 0) ↔ (1, 8) ↔ (9, 0)
For the 4 liter question, since $4\equiv 0\!\mod \!4$, one irreversible action is necessary at the start of the solution; It could be simply pouring the whole 9 liters of water from the 12-liter jug to the sink (0,0), or fully fill it to 12 liters from the tap (12,0). Then, we can construct our solutions backwards as before:
(4, 0) ↔ (4, 8) ↔ (12, 0) ← (9, 0)
(4, 0) ↔ (0, 4) ↔ (12, 4) ↔ (8, 8) ↔ (8, 0) ↔ (0, 8) ↔ (0, 0) ← (9, 0)
Solution for three jugs using a barycentric plot
If the number of jugs is three, the filling status after each step can be described in a diagram of barycentric coordinates, because the sum of all three integers stays the same throughout all steps.[7] In consequence the steps can be visualized as billiard moves in the (clipped) coordinate system on a triangular lattice.
The barycentric plot on the right gives two solutions to the 8, 5 and 3 L puzzle. The yellow area denotes combinations achievable with the jugs. Starting at the square, solid red and dashed blue paths show pourable transitions. When a vertex lands on the dotted black triangle, 4 L has been measured. Another pour to the diamond yields 4 L in each 8 and 5 L jugs.
The blue path is one step shorter than the path for the two-jug puzzle with tap and drain, as we can accumulate 4 L in the 8 L jug, absent in the two-jug variant.
See also
• Rope-burning puzzle, another class of puzzles involving the combination of measurements
• Einstellung effect
Literature
• Cowley, Elizabeth B. (1926). "Note on a Linear Diophantine Equation". Questions and Discussions. American Mathematical Monthly. 33 (7): 379–381. doi:10.2307/2298647. JSTOR 2298647. MR 1520987.
• Tweedie, M. C. K. (1939). "A graphical method of solving Tartaglian measuring puzzles". The Mathematical Gazette. Vol. 23, no. 255. pp. 278–282. JSTOR 3606420.
• Saksena, J. P. (1968). "Stochastic optimal routing". Unternehmensforschung. Vol. 12, no. 1. pp. 173–177. doi:10.1007/BF01918326.
• Atwood, Michael E.; Polson, Peter G. (1976). "A process model for water jug problems". Cognitive Psychology. Vol. 8. pp. 191–216. doi:10.1016/0010-0285(76)90023-2.
• Rem, Martin; Choo, Young il (1982). "A fixed-space program of linear output complexity for the problem of the three vessels". Science of Computer Programming. Vol. 2, no. 2. pp. 133–141. doi:10.1016/0167-6423(82)90011-9.
• Thomas, Glanffrwd P. (1995). "The water jugs problem: solutions from artificial intelligence and mathematical viewpoints". Mathematics in School. Vol. 24, no. 2. pp. 34–37. JSTOR 30215221.
• Murray-Lasso, M. A. (2003). "Math puzzles, powerful ideas, algorithms and computers in teaching problem-solving". Journal of Applied Research and Technology. Vol. 1, no. 3. pp. 215–234.
• Lalchev, Zdravko Voutov; Varbanova, Margarita Genova; Voutova, Irirna Zdravkova (2009). "Perlman's geometric method of solving liquid pouring problems".
• Goetschalckx, Marc (2011). "Single flow routing through a network". International Series in Operations Research & Management Science. Vol. 161. pp. 155–180. doi:10.1007/978-1-4419-6512-7_6.
References
1. Weisstein, Eric W. "Three Jug Problem". mathworld.wolfram.com. Retrieved 2020-01-21.
2. "Solving Decanting Problems by Graph Theory". Wolfram Alpha.{{cite web}}: CS1 maint: url-status (link)
3. "Decanting Problems and Dijkstra's Algorithm". Francisco Blanco-Silva. 2016-07-29. Retrieved 2020-05-25.
4. Hint to Riddle #22: The 3 & 5 Litre Die Hard Water Puzzle. Puzzles.nigelcoldwell.co.uk. Retrieved on 2017-07-09.
5. How not to Die Hard with Math, retrieved 2020-05-25
6. "Choose Your Volume". brilliant.org. Retrieved 2020-09-22.
7. Weisstein, Eric W. "Three Jug Problem". mathworld.wolfram.com. Retrieved 27 August 2019.
| Wikipedia |
3-fold
In algebraic geometry, a 3-fold or threefold is a 3-dimensional algebraic variety.
The Mori program showed that 3-folds have minimal models.
References
• Matsuki, Kenji (2002), Introduction to the Mori program, Universitext, Berlin, New York: Springer-Verlag, ISBN 978-0-387-98465-0, MR 1875410
• Reid, Miles (1980), "Canonical 3-folds", Journées de Géometrie Algébrique d'Angers, Juillet 1979/Algebraic Geometry, Angers, 1979, Alphen aan den Rijn: Sijthoff & Noordhoff, pp. 273–310, MR 0605348
• Roth, L. (1955), Algebraic threefolds, with special regard to problems of rationality, Ergebnisse der Mathematik und ihrer Grenzgebiete (N.F.), Heft 6, Berlin, New York: Springer-Verlag, MR 0076426
Authority control: National
• Israel
• United States
| Wikipedia |
3-step group
In mathematics, a 3-step group is a special sort of group of Fitting length at most 3, that is used in the classification of CN groups and in the Feit–Thompson theorem. The definition of a 3-step group in these two cases is slightly different.
CN groups
In the theory of CN groups, a 3-step group (for some prime p) is a group such that:
• G = Op,p',p(G)
• Op,p′(G) is a Frobenius group with kernel Op(G)
• G/Op(G) is a Frobenius group with kernel Op,p′(G)/Op(G)
Any 3-step group is a solvable CN-group, and conversely any solvable CN-group is either nilpotent, or a Frobenius group, or a 3-step group.
Example: the symmetric group S4 is a 3-step group for the prime p = 2.
Odd order groups
Feit & Thompson (1963, p.780) defined a three-step group to be a group G satisfying the following conditions:
• The derived group of G is a Hall subgroup with a cyclic complement Q.
• If H is the maximal normal nilpotent Hall subgroup of G, then G′′⊆HCG(H)⊆G′ and HCG is nilpotent and H is noncyclic.
• For q∈Q nontrivial, CG(q) is cyclic and non-trivial and independent of q.
References
• Feit, Walter; Thompson, John G. (1963), "Solvability of groups of odd order", Pacific Journal of Mathematics, 13: 775–1029, doi:10.2140/pjm.1963.13.775, ISSN 0030-8730, MR 0166261
• Feit, Walter; Thompson, John G.; Hall, Marshall, Jr. (1960), "Finite groups in which the centralizer of any non-identity element is nilpotent", Mathematische Zeitschrift, 74: 1–17, doi:10.1007/BF01180468, ISSN 0025-5874, MR 0114856{{citation}}: CS1 maint: multiple names: authors list (link)
• Gorenstein, D. (1980), Finite Groups, New York: Chelsea, ISBN 978-0-8284-0301-6, MR 0569209
| Wikipedia |
3D Life
3D Life is a cellular automaton. It is a three-dimensional extension of Game of Life, investigated by Carter Bays. A number of different semitotalistic rules for the 3D rectangular Moore neighborhood were investigated.
It was popularized by A. K. Dewdney in his "Computer Recreations" column in Scientific American magazine.
References
• Bays, Carter (1987), "Candidates for the Game of Life in Three Dimensions", Complex Systems, 3 (1): 373–400.
• Bays, Carter (2006), "A Note About the Discovery of Many New Rules for the Game of Three-Dimensional Life", Complex Systems, 16 (4): 381–386.
• Dewdney, A. K. (February 1987), "The game of life acquires some successors in three dimensions", Scientific American: 8–13.
External links
• Kaleidoscope of 3D Life
| Wikipedia |
Collatz conjecture
The Collatz conjecture is one of the most famous unsolved problems in mathematics. The conjecture asks whether repeating two simple arithmetic operations will eventually transform every positive integer into 1. It concerns sequences of integers in which each term is obtained from the previous term as follows: if the previous term is even, the next term is one half of the previous term. If the previous term is odd, the next term is 3 times the previous term plus 1. The conjecture is that these sequences always reach 1, no matter which positive integer is chosen to start the sequence.
Unsolved problem in mathematics:
Does the Collatz sequence eventually reach 1 for all positive integer initial values?
(more unsolved problems in mathematics)
It is named after the mathematician Lothar Collatz, who introduced the idea in 1937, two years after receiving his doctorate.[1] It is also known as the 3n + 1 problem (or conjecture), the 3x + 1 problem (or conjecture), the Ulam conjecture (after Stanisław Ulam), Kakutani's problem (after Shizuo Kakutani), the Thwaites conjecture (after Sir Bryan Thwaites), Hasse's algorithm (after Helmut Hasse), or the Syracuse problem.[2][4] The sequence of numbers involved is sometimes referred to as the hailstone sequence, hailstone numbers or hailstone numerals (because the values are usually subject to multiple descents and ascents like hailstones in a cloud),[5] or as wondrous numbers.[6]
Paul Erdős said about the Collatz conjecture: "Mathematics may not be ready for such problems."[7] Jeffrey Lagarias stated in 2010 that the Collatz conjecture "is an extraordinarily difficult problem, completely out of reach of present day mathematics".[8]
Statement of the problem
Consider the following operation on an arbitrary positive integer:
• If the number is even, divide it by two.
• If the number is odd, triple it and add one.
In modular arithmetic notation, define the function f as follows:
$f(n)={\begin{cases}n/2&{\text{if }}n\equiv 0{\pmod {2}},\\[4px]3n+1&{\text{if }}n\equiv 1{\pmod {2}}.\end{cases}}$
Now form a sequence by performing this operation repeatedly, beginning with any positive integer, and taking the result at each step as the input at the next.
In notation:
$a_{i}={\begin{cases}n&{\text{for }}i=0,\\f(a_{i-1})&{\text{for }}i>0\end{cases}}$
(that is: ai is the value of f applied to n recursively i times; ai = f i(n)).
The Collatz conjecture is: This process will eventually reach the number 1, regardless of which positive integer is chosen initially.
If the conjecture is false, it can only be because there is some starting number which gives rise to a sequence that does not contain 1. Such a sequence would either enter a repeating cycle that excludes 1, or increase without bound. No such sequence has been found.
The smallest i such that ai < a0 is called the stopping time of n. Similarly, the smallest k such that ak = 1 is called the total stopping time of n.[3] If one of the indexes i or k doesn't exist, we say that the stopping time or the total stopping time, respectively, is infinite.
The Collatz conjecture asserts that the total stopping time of every n is finite. It is also equivalent to saying that every n ≥ 2 has a finite stopping time.
Since 3n + 1 is even whenever n is odd, one may instead use the "shortcut" form of the Collatz function:
$f(n)={\begin{cases}{\frac {n}{2}}&{\text{if }}n\equiv 0{\pmod {2}},\\[4px]{\frac {3n+1}{2}}&{\text{if }}n\equiv 1{\pmod {2}}.\end{cases}}$
This definition yields smaller values for the stopping time and total stopping time without changing the overall dynamics of the process.
Empirical data
For instance, starting with n = 12 and applying the function f without "shortcut", one gets the sequence 12, 6, 3, 10, 5, 16, 8, 4, 2, 1.
The number n = 19 takes longer to reach 1: 19, 58, 29, 88, 44, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1.
The sequence for n = 27, listed and graphed below, takes 111 steps (41 steps through odd numbers, in bold), climbing as high as 9232 before descending to 1.
27, 82, 41, 124, 62, 31, 94, 47, 142, 71, 214, 107, 322, 161, 484, 242, 121, 364, 182, 91, 274, 137, 412, 206, 103, 310, 155, 466, 233, 700, 350, 175, 526, 263, 790, 395, 1186, 593, 1780, 890, 445, 1336, 668, 334, 167, 502, 251, 754, 377, 1132, 566, 283, 850, 425, 1276, 638, 319, 958, 479, 1438, 719, 2158, 1079, 3238, 1619, 4858, 2429, 7288, 3644, 1822, 911, 2734, 1367, 4102, 2051, 6154, 3077, 9232, 4616, 2308, 1154, 577, 1732, 866, 433, 1300, 650, 325, 976, 488, 244, 122, 61, 184, 92, 46, 23, 70, 35, 106, 53, 160, 80, 40, 20, 10, 5, 16, 8, 4, 2, 1 (sequence A008884 in the OEIS)
Numbers with a total stopping time longer than that of any smaller starting value form a sequence beginning with:
1, 2, 3, 6, 7, 9, 18, 25, 27, 54, 73, 97, 129, 171, 231, 313, 327, 649, 703, 871, 1161, 2223, 2463, 2919, 3711, 6171, ... (sequence A006877 in the OEIS).
The starting values whose maximum trajectory point is greater than that of any smaller starting value are as follows:
1, 2, 3, 7, 15, 27, 255, 447, 639, 703, 1819, 4255, 4591, 9663, 20895, 26623, 31911, 60975, 77671, 113383, 138367, 159487, 270271, 665215, 704511, ... (sequence A006884 in the OEIS)
Number of steps for n to reach 1 are
0, 1, 7, 2, 5, 8, 16, 3, 19, 6, 14, 9, 9, 17, 17, 4, 12, 20, 20, 7, 7, 15, 15, 10, 23, 10, 111, 18, 18, 18, 106, 5, 26, 13, 13, 21, 21, 21, 34, 8, 109, 8, 29, 16, 16, 16, 104, 11, 24, 24, ... (sequence A006577 in the OEIS)
The starting value having the largest total stopping time while being
less than 10 is 9, which has 19 steps,
less than 100 is 97, which has 118 steps,
less than 1000 is 871, which has 178 steps,
less than 104 is 6171, which has 261 steps,
less than 105 is 77031, which has 350 steps,
less than 106 is 837799, which has 524 steps,
less than 107 is 8400511, which has 685 steps,
less than 108 is 63728127, which has 949 steps,
less than 109 is 670617279, which has 986 steps,
less than 1010 is 9780657630, which has 1132 steps,[9]
less than 1011 is 75128138247, which has 1228 steps,
less than 1012 is 989345275647, which has 1348 steps.[10] (sequence A284668 in the OEIS)
These numbers are the lowest ones with the indicated step count, but not necessarily the only ones below the given limit. As an example, 9780657631 has 1132 steps, as does 9780657630.
The starting values having the smallest total stopping time with respect to their number of digits (in base 2) are the powers of two since 2n is halved n times to reach 1, and is never increased.
Visualizations
• Directed graph showing the orbits of the first 1000 numbers.
• The x axis represents starting number, the y axis represents the highest number reached during the chain to 1. This plot shows a restricted y axis: some x values produce intermediates as high as 2.7×107 (for x = 9663)
• The same plot on the left but on log scale, so all y values are shown. The first thick line towards the middle of the plot corresponds to the tip at 27, which reaches a maximum at 9232.
• The tree of all the numbers having fewer than 20 steps.
• The number of iterations it takes to get to one for the first 100 million numbers.
Supporting arguments
Although the conjecture has not been proven, most mathematicians who have looked into the problem think the conjecture is true because experimental evidence and heuristic arguments support it.
Experimental evidence
As of 2020, the conjecture has been checked by computer for all starting values up to 268 ≈ 2.95×1020. All initial values tested so far eventually end in the repeating cycle (4; 2; 1) of period 3.[11]
This computer evidence is still not rigorous proof that the conjecture is true for all starting values, as counterexamples may be found when considering very large (or possibly immense) positive integers, as in the case of the disproven Pólya conjecture.
However, such verifications may have other implications. For example, one can derive additional constraints on the period and structural form of a non-trivial cycle.[12][13][14]
A probabilistic heuristic
If one considers only the odd numbers in the sequence generated by the Collatz process, then each odd number is on average 3/4 of the previous one.[15] (More precisely, the geometric mean of the ratios of outcomes is 3/4.) This yields a heuristic argument that every Hailstone sequence should decrease in the long run, although this is not evidence against other cycles, only against divergence. The argument is not a proof because it assumes that Hailstone sequences are assembled from uncorrelated probabilistic events. (It does rigorously establish that the 2-adic extension of the Collatz process has two division steps for every multiplication step for almost all 2-adic starting values.)
Stopping times
As proven by Riho Terras, almost every positive integer has a finite stopping time.[16] In other words, almost every Collatz sequence reaches a point that is strictly below its initial value. The proof is based on the distribution of parity vectors and uses the central limit theorem.
In 2019, Terence Tao improved this result by showing, using logarithmic density, that almost all (in the sense of logarithmic density) Collatz orbits are descending below any given function of the starting point, provided that this function diverges to infinity, no matter how slowly. Responding to this work, Quanta Magazine wrote that Tao "came away with one of the most significant results on the Collatz conjecture in decades".[17][18]
Lower bounds
In a computer-aided proof, Krasikov and Lagarias showed that the number of integers in the interval [1,x] that eventually reach 1 is at least equal to x0.84 for all sufficiently large x.[19]
Cycles
In this part, consider the shortcut form of the Collatz function
$f(n)={\begin{cases}{\frac {n}{2}}&{\text{if }}n\equiv 0{\pmod {2}},\\[4px]{\frac {3n+1}{2}}&{\text{if }}n\equiv 1{\pmod {2}}.\end{cases}}$
A cycle is a sequence (a0, a1, ..., aq) of distinct positive integers where f(a0) = a1, f(a1) = a2, ..., and f(aq) = a0.
The only known cycle is (1,2) of period 2, called the trivial cycle.
Cycle length
The length of a non-trivial cycle is known to be at least 186265759595. If it can be shown that for all positive integers less than $3\times 2^{69}$ the Collatz sequences reach 1, then this bound would raise to 355504839929.[20][13] In fact, Eliahou (1993) proved that the period p of any non-trivial cycle is of the form
$p=301994a+17087915b+85137581c$
where a, b and c are non-negative integers, b ≥ 1 and ac = 0. This result is based on the continued fraction expansion of ln 3/ln 2.[13]
k-cycles
A k-cycle is a cycle that can be partitioned into k contiguous subsequences, each consisting of an increasing sequence of odd numbers, followed by a decreasing sequence of even numbers.[14] For instance, if the cycle consists of a single increasing sequence of odd numbers followed by a decreasing sequence of even numbers, it is called a 1-cycle.
Steiner (1977) proved that there is no 1-cycle other than the trivial (1; 2).[21] Simons (2005) used Steiner's method to prove that there is no 2-cycle.[22] Simons & de Weger (2005) extended this proof up to 68-cycles; there is no k-cycle up to k = 68.[14] Hercher extended the method further and proved that there exists no k-cycle with k≤91.[20] As exhaustive computer searches continue, larger k values may be ruled out. To state the argument more intuitively; we do not have to search for cycles that have less than 92 subsequences, where each subsequence consists of consecutive ups followed by consecutive downs.
Other formulations of the conjecture
In reverse
There is another approach to prove the conjecture, which considers the bottom-up method of growing the so-called Collatz graph. The Collatz graph is a graph defined by the inverse relation
$R(n)={\begin{cases}\{2n\}&{\text{if }}n\equiv 0,1,2,3,5\\[4px]\left\{2n,{\frac {n-1}{3}}\right\}&{\text{if }}n\equiv 4\end{cases}}{\pmod {6}}.$
So, instead of proving that all positive integers eventually lead to 1, we can try to prove that 1 leads backwards to all positive integers. For any integer n, n ≡ 1 (mod 2) if and only if 3n + 1 ≡ 4 (mod 6). Equivalently, n − 1/3 ≡ 1 (mod 2) if and only if n ≡ 4 (mod 6). Conjecturally, this inverse relation forms a tree except for the 1–2–4 loop (the inverse of the 4–2–1 loop of the unaltered function f defined in the Statement of the problem section of this article).
When the relation 3n + 1 of the function f is replaced by the common substitute "shortcut" relation 3n + 1/2, the Collatz graph is defined by the inverse relation,
$R(n)={\begin{cases}\{2n\}&{\text{if }}n\equiv 0,1\\[4px]\left\{2n,{\frac {2n-1}{3}}\right\}&{\text{if }}n\equiv 2\end{cases}}{\pmod {3}}.$
For any integer n, n ≡ 1 (mod 2) if and only if 3n + 1/2 ≡ 2 (mod 3). Equivalently, 2n − 1/3 ≡ 1 (mod 2) if and only if n ≡ 2 (mod 3). Conjecturally, this inverse relation forms a tree except for a 1–2 loop (the inverse of the 1–2 loop of the function f(n) revised as indicated above).
Alternatively, replace the 3n + 1 with n′/H(n′) where n′ = 3n + 1 and H(n′) is the highest power of 2 that divides n′ (with no remainder). The resulting function f maps from odd numbers to odd numbers. Now suppose that for some odd number n, applying this operation k times yields the number 1 (that is, fk(n) = 1). Then in binary, the number n can be written as the concatenation of strings wk wk−1 ... w1 where each wh is a finite and contiguous extract from the representation of 1/3h.[23] The representation of n therefore holds the repetends of 1/3h, where each repetend is optionally rotated and then replicated up to a finite number of bits. It is only in binary that this occurs.[24] Conjecturally, every binary string s that ends with a '1' can be reached by a representation of this form (where we may add or delete leading '0's to s).
As an abstract machine that computes in base two
Repeated applications of the Collatz function can be represented as an abstract machine that handles strings of bits. The machine will perform the following three steps on any odd number until only one 1 remains:
1. Append 1 to the (right) end of the number in binary (giving 2n + 1);
2. Add this to the original number by binary addition (giving 2n + 1 + n = 3n + 1);
3. Remove all trailing 0s (that is, repeatedly divide by 2 until the result is odd).
Example
The starting number 7 is written in base two as 111. The resulting Collatz sequence is:
111
1111
10110
10111
100010
100011
110100
11011
101000
1011
10000
As a parity sequence
For this section, consider the Collatz function in the slightly modified form
$f(n)={\begin{cases}{\frac {n}{2}}&{\text{if }}n\equiv 0\\[4px]{\frac {3n+1}{2}}&{\text{if }}n\equiv 1\end{cases}}{\pmod {2}}.$
This can be done because when n is odd, 3n + 1 is always even.
If P(...) is the parity of a number, that is P(2n) = 0 and P(2n + 1) = 1, then we can define the Collatz parity sequence (or parity vector) for a number n as pi = P(ai), where a0 = n, and ai+1 = f(ai).
Which operation is performed, 3n + 1/2 or n/2, depends on the parity. The parity sequence is the same as the sequence of operations.
Using this form for f(n), it can be shown that the parity sequences for two numbers m and n will agree in the first k terms if and only if m and n are equivalent modulo 2k. This implies that every number is uniquely identified by its parity sequence, and moreover that if there are multiple Hailstone cycles, then their corresponding parity cycles must be different.[3][16]
Applying the f function k times to the number n = 2ka + b will give the result 3ca + d, where d is the result of applying the f function k times to b, and c is how many increases were encountered during that sequence. For example, for 25a + 1 there are 3 increases as 1 iterates to 2, 1, 2, 1, and finally to 2 so the result is 33a + 2; for 22a + 1 there is only 1 increase as 1 rises to 2 and falls to 1 so the result is 3a + 1. When b is 2k − 1 then there will be k rises and the result will be 3ka + 3k − 1. The power of 3 multiplying a is independent of the value of a; it depends only on the behavior of b. This allows one to predict that certain forms of numbers will always lead to a smaller number after a certain number of iterations: for example, 4a + 1 becomes 3a + 1 after two applications of f and 16a + 3 becomes 9a + 2 after four applications of f. Whether those smaller numbers continue to 1, however, depends on the value of a.
As a tag system
For the Collatz function in the form
$f(n)={\begin{cases}{\frac {n}{2}}&{\text{if }}n\equiv 0\\[4px]{\frac {3n+1}{2}}&{\text{if }}n\equiv 1.\end{cases}}{\pmod {2}}$
Hailstone sequences can be computed by the 2-tag system with production rules
a → bc, b → a, c → aaa.
In this system, the positive integer n is represented by a string of n copies of a, and iteration of the tag operation halts on any word of length less than 2. (Adapted from De Mol.)
The Collatz conjecture equivalently states that this tag system, with an arbitrary finite string of a as the initial word, eventually halts (see Tag system for a worked example).
Extensions to larger domains
Iterating on all integers
An extension to the Collatz conjecture is to include all integers, not just positive integers. Leaving aside the cycle 0 → 0 which cannot be entered from outside, there are a total of four known cycles, which all nonzero integers seem to eventually fall into under iteration of f. These cycles are listed here, starting with the well-known cycle for positive n:
Odd values are listed in large bold. Each cycle is listed with its member of least absolute value (which is always odd) first.
CycleOdd-value cycle lengthFull cycle length
1 → 4 → 2 → 1 ...13
−1 → −2 → −1 ...12
−5 → −14 → −7 → −20 → −10 → −5 ...25
−17 → −50 → −25 → −74 → −37 → −110 → −55 → −164 → −82 → −41 → −122 → −61 → −182 → −91 → −272 → −136 → −68 → −34 → −17 ...718
The generalized Collatz conjecture is the assertion that every integer, under iteration by f, eventually falls into one of the four cycles above or the cycle 0 → 0. (The 0 → 0 cycle is only included for the sake of completeness.)
Iterating on rationals with odd denominators
The Collatz map can be extended to (positive or negative) rational numbers which have odd denominators when written in lowest terms. The number is taken to be 'odd' or 'even' according to whether its numerator is odd or even. Then the formula for the map is exactly the same as when the domain is the integers: an 'even' such rational is divided by 2; an 'odd' such rational is multiplied by 3 and then 1 is added. A closely related fact is that the Collatz map extends to the ring of 2-adic integers, which contains the ring of rationals with odd denominators as a subring.
When using the "shortcut" definition of the Collatz map, it is known that any periodic parity sequence is generated by exactly one rational.[25] Conversely, it is conjectured that every rational with an odd denominator has an eventually cyclic parity sequence (Periodicity Conjecture[3]).
If a parity cycle has length n and includes odd numbers exactly m times at indices k0 < ⋯ < km−1, then the unique rational which generates immediately and periodically this parity cycle is
${\frac {3^{m-1}2^{k_{0}}+\cdots +3^{0}2^{k_{m-1}}}{2^{n}-3^{m}}}.$
(1)
For example, the parity cycle (1 0 1 1 0 0 1) has length 7 and four odd terms at indices 0, 2, 3, and 6. It is repeatedly generated by the fraction
${\frac {3^{3}2^{0}+3^{2}2^{2}+3^{1}2^{3}+3^{0}2^{6}}{2^{7}-3^{4}}}={\frac {151}{47}}$
as the latter leads to the rational cycle
${\frac {151}{47}}\rightarrow {\frac {250}{47}}\rightarrow {\frac {125}{47}}\rightarrow {\frac {211}{47}}\rightarrow {\frac {340}{47}}\rightarrow {\frac {170}{47}}\rightarrow {\frac {85}{47}}\rightarrow {\frac {151}{47}}.$
Any cyclic permutation of (1 0 1 1 0 0 1) is associated to one of the above fractions. For instance, the cycle (0 1 1 0 0 1 1) is produced by the fraction
${\frac {3^{3}2^{1}+3^{2}2^{2}+3^{1}2^{5}+3^{0}2^{6}}{2^{7}-3^{4}}}={\frac {250}{47}}.$
For a one-to-one correspondence, a parity cycle should be irreducible, that is, not partitionable into identical sub-cycles. As an illustration of this, the parity cycle (1 1 0 0 1 1 0 0) and its sub-cycle (1 1 0 0) are associated to the same fraction 5/7 when reduced to lowest terms.
In this context, assuming the validity of the Collatz conjecture implies that (1 0) and (0 1) are the only parity cycles generated by positive whole numbers (1 and 2, respectively).
If the odd denominator d of a rational is not a multiple of 3, then all the iterates have the same denominator and the sequence of numerators can be obtained by applying the "3n + d " generalization[26] of the Collatz function
$T_{d}(x)={\begin{cases}{\frac {x}{2}}&{\text{if }}x\equiv 0{\pmod {2}},\\[4px]{\frac {3x+d}{2}}&{\text{if }}x\equiv 1{\pmod {2}}.\end{cases}}$
2-adic extension
The function
$T(x)={\begin{cases}{\frac {x}{2}}&{\text{if }}x\equiv 0{\pmod {2}}\\[4px]{\frac {3x+1}{2}}&{\text{if }}x\equiv 1{\pmod {2}}\end{cases}}$
is well-defined on the ring $\mathbb {Z} _{2}$ of 2-adic integers, where it is continuous and measure-preserving with respect to the 2-adic measure. Moreover, its dynamics is known to be ergodic.[3]
Define the parity vector function Q acting on $\mathbb {Z} _{2}$ as
$Q(x)=\sum _{k=0}^{\infty }\left(T^{k}(x)\mod 2\right)2^{k}.$
The function Q is a 2-adic isometry.[27] Consequently, every infinite parity sequence occurs for exactly one 2-adic integer, so that almost all trajectories are acyclic in $\mathbb {Z} _{2}$.
An equivalent formulation of the Collatz conjecture is that
$Q\left(\mathbb {Z} ^{+}\right)\subset {\tfrac {1}{3}}\mathbb {Z} .$
Iterating on real or complex numbers
The Collatz map (with shortcut) can be viewed as the restriction to the integers of the smooth map
$f(z)={\frac {1}{2}}z\cos ^{2}\left({\frac {\pi }{2}}z\right)+{\frac {3z+1}{2}}\sin ^{2}\left({\frac {\pi }{2}}z\right).$
The iterations of this map on the real line lead to a dynamical system, further investigated by Chamberland.[28] He showed that the conjecture does not hold for positive real numbers since there are infinitely many fixed points, as well as orbits escaping monotonically to infinity. The function f has two attracting cycles of period 2, (1; 2) and (1.1925...; 2.1386...). Moreover, the set of unbounded orbits is conjectured to be of measure 0.
Letherman, Schleicher, and Wood extended the study to the complex plane, where most of the points have orbits that diverge to infinity (colored region on the illustration).[29] The boundary between the colored region and the black components, namely the Julia set of f, is a fractal pattern, sometimes called the "Collatz fractal".
Optimizations
Time–space tradeoff
The section As a parity sequence above gives a way to speed up simulation of the sequence. To jump ahead k steps on each iteration (using the f function from that section), break up the current number into two parts, b (the k least significant bits, interpreted as an integer), and a (the rest of the bits as an integer). The result of jumping ahead k is given by
fk(2ka + b) = 3c(b, k)a + d(b, k).
The values of c (or better 3c) and d can be precalculated for all possible k-bit numbers b, where d(b, k) is the result of applying the f function k times to b, and c(b, k) is the number of odd numbers encountered on the way.[30] For example, if k = 5, one can jump ahead 5 steps on each iteration by separating out the 5 least significant bits of a number and using
c(0...31, 5) = { 0, 3, 2, 2, 2, 2, 2, 4, 1, 4, 1, 3, 2, 2, 3, 4, 1, 2, 3, 3, 1, 1, 3, 3, 2, 3, 2, 4, 3, 3, 4, 5 },
d(0...31, 5) = { 0, 2, 1, 1, 2, 2, 2, 20, 1, 26, 1, 10, 4, 4, 13, 40, 2, 5, 17, 17, 2, 2, 20, 20, 8, 22, 8, 71, 26, 26, 80, 242 }.
This requires 2k precomputation and storage to speed up the resulting calculation by a factor of k, a space–time tradeoff.
Modular restrictions
For the special purpose of searching for a counterexample to the Collatz conjecture, this precomputation leads to an even more important acceleration, used by Tomás Oliveira e Silva in his computational confirmations of the Collatz conjecture up to large values of n. If, for some given b and k, the inequality
fk(2ka + b) = 3c(b)a + d(b) < 2ka + b
holds for all a, then the first counterexample, if it exists, cannot be b modulo 2k.[12] For instance, the first counterexample must be odd because f(2n) = n, smaller than 2n; and it must be 3 mod 4 because f2(4n + 1) = 3n + 1, smaller than 4n + 1. For each starting value a which is not a counterexample to the Collatz conjecture, there is a k for which such an inequality holds, so checking the Collatz conjecture for one starting value is as good as checking an entire congruence class. As k increases, the search only needs to check those residues b that are not eliminated by lower values of k. Only an exponentially small fraction of the residues survive.[31] For example, the only surviving residues mod 32 are 7, 15, 27, and 31.
Syracuse function
If k is an odd integer, then 3k + 1 is even, so 3k + 1 = 2ak′ with k′ odd and a ≥ 1. The Syracuse function is the function f from the set I of odd integers into itself, for which f(k) = k′ (sequence A075677 in the OEIS).
Some properties of the Syracuse function are:
• For all k ∈ I, f(4k + 1) = f(k). (Because 3(4k + 1) + 1 = 12k + 4 = 4(3k + 1).)
• In more generality: For all p ≥ 1 and odd h, fp − 1(2ph − 1) = 2 × 3p − 1h − 1. (Here fp − 1 is function iteration notation.)
• For all odd h, f(2h − 1) ≤ 3h − 1/2
The Collatz conjecture is equivalent to the statement that, for all k in I, there exists an integer n ≥ 1 such that fn(k) = 1.
Undecidable generalizations
In 1972, John Horton Conway proved that a natural generalization of the Collatz problem is algorithmically undecidable.[32]
Specifically, he considered functions of the form
${g(n)=a_{i}n+b_{i}}{\text{ when }}{n\equiv i{\pmod {P}}},$
where a0, b0, ..., aP − 1, bP − 1 are rational numbers which are so chosen that g(n) is always an integer. The standard Collatz function is given by P = 2, a0 = 1/2, b0 = 0, a1 = 3, b1 = 1. Conway proved that the problem
Given g and n, does the sequence of iterates gk(n) reach 1?
is undecidable, by representing the halting problem in this way.
Closer to the Collatz problem is the following universally quantified problem:
Given g, does the sequence of iterates gk(n) reach 1, for all n > 0?
Modifying the condition in this way can make a problem either harder or easier to solve (intuitively, it is harder to justify a positive answer but might be easier to justify a negative one). Kurtz and Simon[33] proved that the universally quantified problem is, in fact, undecidable and even higher in the arithmetical hierarchy; specifically, it is Π0
2
-complete. This hardness result holds even if one restricts the class of functions g by fixing the modulus P to 6480.[34]
Iterations of in a simplified version of this form, with all $b_{i}$ equal to zero, are formalized in an esoteric programming language called FRACTRAN.
In popular culture
In the movie Incendies, a graduate student in pure mathematics explains the Collatz conjecture to a group of undergraduates. She puts her studies on hold for a time to address some unresolved questions about her family's past. Late in the movie, the Collatz conjecture turns out to have foreshadowed a disturbing and difficult discovery that she makes about her family.[35][36]
See also
Wikimedia Commons has media related to Collatz conjecture.
• 3x + 1 semigroup
• Arithmetic dynamics
• Modular arithmetic
• Residue-class-wise affine group
Further reading
• The Ultimate Challenge: The 3x + 1 Problem,[8] published in 2010 by the American Mathematical Society and edited by Jeffrey Lagarias, is a compendium of information on the Collatz conjecture, methods of approaching it, and generalizations. It includes two survey papers by the editor and five by other authors concerning the history of the problem, generalizations, statistical approaches, and results from the theory of computation. It also includes reprints of early papers on the subject, including the paper by Lothar Collatz.
References
1. O'Connor, J.J.; Robertson, E.F. (2006). "Lothar Collatz". St Andrews University School of Mathematics and Statistics, Scotland.
2. Maddux, Cleborne D.; Johnson, D. Lamont (1997). Logo: A Retrospective. New York: Haworth Press. p. 160. ISBN 0-7890-0374-0. The problem is also known by several other names, including: Ulam's conjecture, the Hailstone problem, the Syracuse problem, Kakutani's problem, Hasse's algorithm, and the Collatz problem.
3. Lagarias, Jeffrey C. (1985). "The 3x + 1 problem and its generalizations". The American Mathematical Monthly. 92 (1): 3–23. doi:10.1080/00029890.1985.11971528. JSTOR 2322189.
4. According to Lagarias (1985),[3] p. 4, the name "Syracuse problem" was proposed by Hasse in the 1950s, during a visit to Syracuse University.
5. Pickover, Clifford A. (2001). Wonders of Numbers. Oxford: Oxford University Press. pp. 116–118. ISBN 0-19-513342-0.
6. Hofstadter, Douglas R. (1979). Gödel, Escher, Bach. New York: Basic Books. pp. 400–2. ISBN 0-465-02685-0.
7. Guy, Richard K. (2004). ""E16: The 3x+1 problem"". Unsolved Problems in Number Theory (3rd ed.). Springer-Verlag. pp. 330–6. ISBN 0-387-20860-7. Zbl 1058.11001.
8. Lagarias, Jeffrey C., ed. (2010). The Ultimate Challenge: The 3x + 1 Problem. American Mathematical Society. ISBN 978-0-8218-4940-8. Zbl 1253.11003.
9. Leavens, Gary T.; Vermeulen, Mike (December 1992). "3x + 1 search programs". Computers & Mathematics with Applications. 24 (11): 79–99. doi:10.1016/0898-1221(92)90034-F.
10. Roosendaal, Eric. "3x+1 delay records". Retrieved 14 March 2020. (Note: "Delay records" are total stopping time records.)
11. Barina, David (2020). "Convergence verification of the Collatz problem". The Journal of Supercomputing. 77 (3): 2681–2688. doi:10.1007/s11227-020-03368-x. S2CID 220294340.
12. Garner, Lynn E. (1981). "On the Collatz 3n + 1 algorithm". Proceedings of the American Mathematical Society. 82 (1): 19–22. doi:10.1090/S0002-9939-1981-0603593-2. JSTOR 2044308.
13. Eliahou, Shalom (1993). "The 3x + 1 problem: new lower bounds on nontrivial cycle lengths". Discrete Mathematics. 118 (1): 45–56. doi:10.1016/0012-365X(93)90052-U.
14. Simons, J.; de Weger, B. (2005). "Theoretical and computational bounds for m-cycles of the 3n + 1 problem" (PDF). Acta Arithmetica. 117 (1): 51–70. Bibcode:2005AcAri.117...51S. doi:10.4064/aa117-1-3.
15. Lagarias (1985),[3] section "A heuristic argument".
16. Terras, Riho (1976). "A stopping time problem on the positive integers" (PDF). Acta Arithmetica. 30 (3): 241–252. doi:10.4064/aa-30-3-241-252. MR 0568274.
17. Tao, Terence (2022). "Almost all orbits of the Collatz map attain almost bounded values". Forum of Mathematics, Pi. 10: e12. arXiv:1909.03562. doi:10.1017/fmp.2022.8. ISSN 2050-5086.
18. Hartnett, Kevin (December 11, 2019). "Mathematician Proves Huge Result on 'Dangerous' Problem". Quanta Magazine.
19. Krasikov, Ilia; Lagarias, Jeffrey C. (2003). "Bounds for the 3x + 1 problem using difference inequalities". Acta Arithmetica. 109 (3): 237–258. arXiv:math/0205002. Bibcode:2003AcAri.109..237K. doi:10.4064/aa109-3-4. MR 1980260. S2CID 18467460.
20. Hercher, C. (2023). "There are no Collatz m-cycles with m <= 91" (PDF). Journal of Integer Sequences. 26 (3): Article 23.3.5.
21. Steiner, R. P. (1977). "A theorem on the syracuse problem". Proceedings of the 7th Manitoba Conference on Numerical Mathematics. pp. 553–9. MR 0535032.
22. Simons, John L. (2005). "On the nonexistence of 2-cycles for the 3x + 1 problem". Math. Comp. 74: 1565–72. Bibcode:2005MaCom..74.1565S. doi:10.1090/s0025-5718-04-01728-4. MR 2137019.
23. Colussi, Livio (9 September 2011). "The convergence classes of Collatz function". Theoretical Computer Science. 412 (39): 5409–5419. doi:10.1016/j.tcs.2011.05.056.
24. Hew, Patrick Chisan (7 March 2016). "Working in binary protects the repetends of 1/3h: Comment on Colussi's 'The convergence classes of Collatz function'". Theoretical Computer Science. 618: 135–141. doi:10.1016/j.tcs.2015.12.033.
25. Lagarias, Jeffrey (1990). "The set of rational cycles for the 3x+1 problem". Acta Arithmetica. 56 (1): 33–53. doi:10.4064/aa-56-1-33-53. ISSN 0065-1036.
26. Belaga, Edward G.; Mignotte, Maurice (1998). "Embedding the 3x+1 Conjecture in a 3x+d Context". Experimental Mathematics. 7 (2): 145–151. doi:10.1080/10586458.1998.10504364. S2CID 17925995.
27. Bernstein, Daniel J.; Lagarias, Jeffrey C. (1996). "The 3x + 1 conjugacy map". Canadian Journal of Mathematics. 48 (6): 1154–1169. doi:10.4153/CJM-1996-060-x. ISSN 0008-414X.
28. Chamberland, Marc (1996). "A continuous extension of the 3x + 1 problem to the real line". Dynam. Contin. Discrete Impuls Systems. 2 (4): 495–509.
29. Letherman, Simon; Schleicher, Dierk; Wood, Reg (1999). "The (3n + 1)-problem and holomorphic dynamics". Experimental Mathematics. 8 (3): 241–252. doi:10.1080/10586458.1999.10504402.
30. Scollo, Giuseppe (2007). "Looking for class records in the 3x + 1 problem by means of the COMETA grid infrastructure" (PDF). Grid Open Days at the University of Palermo.
31. Lagarias (1985),[3] Theorem D.
32. Conway, John H. (1972). "Unpredictable iterations". Proc. 1972 Number Theory Conf., Univ. Colorado, Boulder. pp. 49–52.
33. Kurtz, Stuart A.; Simon, Janos (2007). "The undecidability of the generalized Collatz problem". In Cai, J.-Y.; Cooper, S. B.; Zhu, H. (eds.). Proceedings of the 4th International Conference on Theory and Applications of Models of Computation, TAMC 2007, held in Shanghai, China in May 2007. pp. 542–553. doi:10.1007/978-3-540-72504-6_49. ISBN 978-3-540-72503-9. As PDF
34. Ben-Amram, Amir M. (2015). "Mortality of iterated piecewise affine functions over the integers: Decidability and complexity". Computability. 1 (1): 19–56. doi:10.3233/COM-150032.
35. Emmer, Michele (2012). Imagine Math: Between Culture and Mathematics. Springer Publishing. pp. 260–264. ISBN 978-8-847-02426-7.
36. Mazmanian, Adam (19 May 2011). "MOVIE REVIEW: 'Incendies'". The Washington Times. Retrieved 7 December 2019.
External links
• Matthews, Keith. "3 x + 1 page".
• An ongoing volunteer computing project Archived 2021-08-30 at the Wayback Machine by David Bařina verifies Convergence of the Collatz conjecture for large values. (furthest progress so far)
• (BOINC) volunteer computing project that verifies the Collatz conjecture for larger values.
• An ongoing volunteer computing project by Eric Roosendaal verifies the Collatz conjecture for larger and larger values.
• Another ongoing volunteer computing project by Tomás Oliveira e Silva continues to verify the Collatz conjecture (with fewer statistics than Eric Roosendaal's page but with further progress made).
• Weisstein, Eric W. "Collatz Problem". MathWorld.
• Collatz Problem at PlanetMath..
• Nochella, Jesse. "Collatz Paths". Wolfram Demonstrations Project.
• Eisenbud, D. (8 August 2016). Uncrackable? The Collatz conjecture (short video). Numberphile. Archived from the original on 2021-12-11 – via YouTube.
• Eisenbud, D. (August 9, 2016). Uncrackable? Collatz conjecture (extra footage). Numberphile. Archived from the original on 2021-12-11 – via YouTube.
• Alex Kontorovich (featuring) (30 July 2021). The simplest math problem no one can solve (short video). Veritasium – via YouTube.
• Are computers ready to solve this notoriously unwieldy math problem?
| Wikipedia |
Three-dimensional space
In geometry, a three-dimensional space (3D space, 3-space or, rarely, tri-dimensional space) is a mathematical space in which three values (coordinates) are required to determine the position of a point. Most commonly, it is the three-dimensional Euclidean space, the Euclidean n-space of dimension n=3 that models physical space. More general three-dimensional spaces are called 3-manifolds.
Technically, a tuple of n numbers can be understood as the Cartesian coordinates of a location in a n-dimensional Euclidean space. The set of these n-tuples is commonly denoted $\mathbb {R} ^{n},$ and can be identified to the pair formed by a n-dimensional Euclidean space and a Cartesian coordinate system. When n = 3, this space is called the three-dimensional Euclidean space (or simply "Euclidean space" when the context is clear).[1] It serves as a model of the physical universe (when relativity theory is not considered), in which all known matter exists. While this space remains the most compelling and useful way to model the world as it is experienced,[2] it is only one example of a large variety of spaces in three dimensions called 3-manifolds. In this classical example, when the three values refer to measurements in different directions (coordinates), any three directions can be chosen, provided that vectors in these directions do not all lie in the same 2-space (plane). Furthermore, in this case, these three values can be labeled by any combination of three chosen from the terms width/breadth, height/depth, and length.
History
Books XI to XIII of Euclid's Elements dealt with three-dimensional geometry. Book XI develops notions of orthogonality and parallelism of lines and planes, and defines solids including parallelpipeds, pyramids, prisms, spheres, octahedra, icosahedra and dodecahedra. Book XII develops notions of similarity of solids. Book XIII describes the construction of the five regular Platonic solids in a sphere.
In the 17th century, three-dimensional space was described with Cartesian coordinates, with the advent of analytic geometry developed by René Descartes in his work La Géométrie and Pierre de Fermat in the manuscript Ad locos planos et solidos isagoge (Introduction to Plane and Solid Loci), which was unpublished during Fermat's lifetime. However, only Fermat's work dealt with three-dimensional space.
In the 19th century, developments of the geometry of three-dimensional space came with William Rowan Hamilton's development of the quaternions. In fact, it was Hamilton who coined the terms scalar and vector, and they were first defined within his geometric framework for quaternions. Three dimensional space could then be described by quaternions $q=a+ui+vj+wk$ which had vanishing scalar component, that is, $a=0$. While not explicitly studied by Hamilton, this indirectly introduced notions of basis, here given by the quaternion elements $i,j,k$, as well as the dot product and cross product, which correspond to (the negative of) the scalar part and the vector part of the product of two vector quaternions.
It was not until Josiah Willard Gibbs that these two products were identified in their own right, and the modern notation for the dot and cross product were introduced in his classroom teaching notes, found also in the 1901 textbook Vector Analysis written by Edwin Bidwell Wilson based on Gibbs' lectures.
Also during the 19th century came developments in the abstract formalism of vector spaces, with the work of Hermann Grassmann and Giuseppe Peano, the latter of whom first gave the modern definition of vector spaces as an algebraic structure.
In Euclidean geometry
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In mathematics, analytic geometry (also called Cartesian geometry) describes every point in three-dimensional space by means of three coordinates. Three coordinate axes are given, each perpendicular to the other two at the origin, the point at which they cross. They are usually labeled x, y, and z. Relative to these axes, the position of any point in three-dimensional space is given by an ordered triple of real numbers, each number giving the distance of that point from the origin measured along the given axis, which is equal to the distance of that point from the plane determined by the other two axes.[3]
Other popular methods of describing the location of a point in three-dimensional space include cylindrical coordinates and spherical coordinates, though there are an infinite number of possible methods. For more, see Euclidean space.
Below are images of the above-mentioned systems.
• Cartesian coordinate system
• Cylindrical coordinate system
• Spherical coordinate system
Lines and planes
Two distinct points always determine a (straight) line. Three distinct points are either collinear or determine a unique plane. On the other hand, four distinct points can either be collinear, coplanar, or determine the entire space.
Two distinct lines can either intersect, be parallel or be skew. Two parallel lines, or two intersecting lines, lie in a unique plane, so skew lines are lines that do not meet and do not lie in a common plane.
Two distinct planes can either meet in a common line or are parallel (i.e., do not meet). Three distinct planes, no pair of which are parallel, can either meet in a common line, meet in a unique common point, or have no point in common. In the last case, the three lines of intersection of each pair of planes are mutually parallel.
A line can lie in a given plane, intersect that plane in a unique point, or be parallel to the plane. In the last case, there will be lines in the plane that are parallel to the given line.
A hyperplane is a subspace of one dimension less than the dimension of the full space. The hyperplanes of a three-dimensional space are the two-dimensional subspaces, that is, the planes. In terms of Cartesian coordinates, the points of a hyperplane satisfy a single linear equation, so planes in this 3-space are described by linear equations. A line can be described by a pair of independent linear equations—each representing a plane having this line as a common intersection.
Varignon's theorem states that the midpoints of any quadrilateral in ℝ3 form a parallelogram, and hence are coplanar.
Spheres and balls
Main article: Sphere
A sphere in 3-space (also called a 2-sphere because it is a 2-dimensional object) consists of the set of all points in 3-space at a fixed distance r from a central point P. The solid enclosed by the sphere is called a ball (or, more precisely a 3-ball).
The volume of the ball is given by
$V={\frac {4}{3}}\pi r^{3},$
and the surface area of the sphere is
$A=4\pi r^{2}.$
Another type of sphere arises from a 4-ball, whose three-dimensional surface is the 3-sphere: points equidistant to the origin of the euclidean space ℝ4. If a point has coordinates, P(x, y, z, w), then x2 + y2 + z2 + w2 = 1 characterizes those points on the unit 3-sphere centered at the origin.
This 3-sphere is an example of a 3-manifold: a space which is 'looks locally' like 3D space. In precise topological terms, each point of the 3-sphere has a neighborhood which is homeomorphic to an open subset of 3D space.
Polytopes
Main article: Polyhedron
In three dimensions, there are nine regular polytopes: the five convex Platonic solids and the four nonconvex Kepler-Poinsot polyhedra.
Regular polytopes in three dimensions
Class Platonic solids Kepler-Poinsot polyhedra
Symmetry Td Oh Ih
Coxeter group A3, [3,3] B3, [4,3] H3, [5,3]
Order 24 48 120
Regular
polyhedron
{3,3}
{4,3}
{3,4}
{5,3}
{3,5}
{5/2,5}
{5,5/2}
{5/2,3}
{3,5/2}
Surfaces of revolution
Main article: Surface of revolution
A surface generated by revolving a plane curve about a fixed line in its plane as an axis is called a surface of revolution. The plane curve is called the generatrix of the surface. A section of the surface, made by intersecting the surface with a plane that is perpendicular (orthogonal) to the axis, is a circle.
Simple examples occur when the generatrix is a line. If the generatrix line intersects the axis line, the surface of revolution is a right circular cone with vertex (apex) the point of intersection. However, if the generatrix and axis are parallel, then the surface of revolution is a circular cylinder.
Quadric surfaces
Main article: Quadric surface
In analogy with the conic sections, the set of points whose Cartesian coordinates satisfy the general equation of the second degree, namely,
$Ax^{2}+By^{2}+Cz^{2}+Fxy+Gyz+Hxz+Jx+Ky+Lz+M=0,$
where A, B, C, F, G, H, J, K, L and M are real numbers and not all of A, B, C, F, G and H are zero, is called a quadric surface.[4]
There are six types of non-degenerate quadric surfaces:
1. Ellipsoid
2. Hyperboloid of one sheet
3. Hyperboloid of two sheets
4. Elliptic cone
5. Elliptic paraboloid
6. Hyperbolic paraboloid
The degenerate quadric surfaces are the empty set, a single point, a single line, a single plane, a pair of planes or a quadratic cylinder (a surface consisting of a non-degenerate conic section in a plane π and all the lines of ℝ3 through that conic that are normal to π).[4] Elliptic cones are sometimes considered to be degenerate quadric surfaces as well.
Both the hyperboloid of one sheet and the hyperbolic paraboloid are ruled surfaces, meaning that they can be made up from a family of straight lines. In fact, each has two families of generating lines, the members of each family are disjoint and each member one family intersects, with just one exception, every member of the other family.[5] Each family is called a regulus.
In linear algebra
Another way of viewing three-dimensional space is found in linear algebra, where the idea of independence is crucial. Space has three dimensions because the length of a box is independent of its width or breadth. In the technical language of linear algebra, space is three-dimensional because every point in space can be described by a linear combination of three independent vectors.
Dot product, angle, and length
Main article: Dot product
A vector can be pictured as an arrow. The vector's magnitude is its length, and its direction is the direction the arrow points. A vector in ℝ3 can be represented by an ordered triple of real numbers. These numbers are called the components of the vector.
The dot product of two vectors A = [A1, A2, A3] and B = [B1, B2, B3] is defined as:[6]
$\mathbf {A} \cdot \mathbf {B} =A_{1}B_{1}+A_{2}B_{2}+A_{3}B_{3}=\sum _{i=1}^{3}A_{i}B_{i}.$
The magnitude of a vector A is denoted by ||A||. The dot product of a vector A = [A1, A2, A3] with itself is
$\mathbf {A} \cdot \mathbf {A} =\|\mathbf {A} \|^{2}=A_{1}^{2}+A_{2}^{2}+A_{3}^{2},$
which gives
$\|\mathbf {A} \|={\sqrt {\mathbf {A} \cdot \mathbf {A} }}={\sqrt {A_{1}^{2}+A_{2}^{2}+A_{3}^{2}}},$
the formula for the Euclidean length of the vector.
Without reference to the components of the vectors, the dot product of two non-zero Euclidean vectors A and B is given by[7]
$\mathbf {A} \cdot \mathbf {B} =\|\mathbf {A} \|\,\|\mathbf {B} \|\cos \theta ,$
where θ is the angle between A and B.
Cross product
Main article: Cross product
The cross product or vector product is a binary operation on two vectors in three-dimensional space and is denoted by the symbol ×. The cross product A × B of the vectors A and B is a vector that is perpendicular to both and therefore normal to the plane containing them. It has many applications in mathematics, physics, and engineering.
In function language, the cross product is a function $\times :\mathbb {R} ^{3}\times \mathbb {R} ^{3}\rightarrow \mathbb {R} ^{3}$ :\mathbb {R} ^{3}\times \mathbb {R} ^{3}\rightarrow \mathbb {R} ^{3}} .
The components of the cross product are $\mathbf {A} \times \mathbf {B} =[A_{2}B_{3}-B_{2}A_{3},A_{3}B_{1}-B_{3}A_{1},A_{1}B_{2}-B_{1}A_{2}]$, and can also be written in components, using Einstein summation convention as $(\mathbf {A} \times \mathbf {B} )_{i}=\epsilon _{ijk}A_{j}B_{k}$ where $\epsilon _{ijk}$ is the Levi-Civita symbol. It has the property that $\mathbf {A} \times \mathbf {B} =-\mathbf {B} \times \mathbf {A} $.
Its magnitude is related to the angle $\theta $ between $\mathbf {A} $ and $\mathbf {B} $ by the identity
$||\mathbf {A} \times \mathbf {B} ||=||\mathbf {A} ||\cdot ||\mathbf {B} ||\cdot |\sin \theta |.$
The space and product form an algebra over a field, which is not commutative nor associative, but is a Lie algebra with the cross product being the Lie bracket. Specifically, the space together with the product, $(\mathbb {R} ^{3},\times )$ is isomorphic to the Lie algebra of three-dimensional rotations, denoted ${\mathfrak {so}}(3)$. In order to satisfy the axioms of a Lie algebra, instead of associativity the cross product satisfies the Jacobi identity. For any three vectors $\mathbf {A} ,\mathbf {B} $ and $\mathbf {C} $
$\mathbf {A} \times (\mathbf {B} \times \mathbf {C} )+\mathbf {B} \times (\mathbf {C} \times \mathbf {A} )+\mathbf {C} \times (\mathbf {A} \times \mathbf {B} )=0$
One can in n dimensions take the product of n − 1 vectors to produce a vector perpendicular to all of them. But if the product is limited to non-trivial binary products with vector results, it exists only in three and seven dimensions.[8]
Abstract description
See also: vector space
It can be useful to describe three-dimensional space as a three-dimensional vector space $V$ over the real numbers. This differs from $\mathbb {R} ^{3}$ in a subtle way. By definition, there exists a basis ${\mathcal {B}}=\{e_{1},e_{2},e_{3}\}$ for $V$. This corresponds to an isomorphism between $V$ and $\mathbb {R} ^{3}$: the construction for the isomorphism is found here. However, there is no 'preferred' or 'canonical basis' for $V$.
On the other hand, there is a preferred basis for $\mathbb {R} ^{3}$, which is due to its description as a Cartesian product of copies of $\mathbb {R} $, that is, $\mathbb {R} ^{3}=\mathbb {R} \times \mathbb {R} \times \mathbb {R} $. This allows the definition of canonical projections, $\pi _{i}:\mathbb {R} ^{3}\rightarrow \mathbb {R} $, where $1\leq i\leq 3$. For example, $\pi _{1}(x_{1},x_{2},x_{3})=x$. This then allows the definition of the standard basis ${\mathcal {B}}_{\text{Standard}}=\{E_{1},E_{2},E_{3}\}$ defined by
$\pi _{i}(E_{j})=\delta _{ij}$
where $\delta _{ij}$ is the Kronecker delta. Written out in full, the standard basis is
$E_{1}={\begin{pmatrix}1\\0\\0\end{pmatrix}},E_{2}={\begin{pmatrix}0\\1\\0\end{pmatrix}},E_{3}={\begin{pmatrix}0\\0\\1\end{pmatrix}}.$
Therefore $\mathbb {R} ^{3}$ can be viewed as the abstract vector space, together with the additional structure of a choice of basis. Conversely, $V$ can be obtained by starting with $\mathbb {R} ^{3}$ and 'forgetting' the Cartesian product structure, or equivalently the standard choice of basis.
As opposed to a general vector space $V$, the space $\mathbb {R} ^{3}$ is sometimes referred to as a coordinate space.[9]
Physically, it is conceptually desirable to use the abstract formalism in order to assume as little structure as possible if it is not given by the parameters of a particular problem. For example, in a problem with rotational symmetry, working with the more concrete description of three-dimensional space $\mathbb {R} ^{3}$ assumes a choice of basis, corresponding to a set of axes. But in rotational symmetry, there is no reason why one set of axes is preferred to say, the same set of axes which has been rotated arbitrarily. Stated another way, a preferred choice of axes breaks the rotational symmetry of physical space.
Computationally, it is necessary to work with the more concrete description $\mathbb {R} ^{3}$ in order to do concrete computations.
Affine description
See also: affine space and Euclidean space
A more abstract description still is to model physical space as a three-dimensional affine space $E(3)$ over the real numbers. This is unique up to affine isomorphism. It is sometimes referred to as three-dimensional Euclidean space. Just as the vector space description came from 'forgetting the preferred basis' of $\mathbb {R} ^{3}$, the affine space description comes from 'forgetting the origin' of the vector space. Euclidean spaces are sometimes called Euclidean affine spaces for distinguishing them from Euclidean vector spaces.[10]
This is physically appealing as it makes the translation invariance of physical space manifest. A preferred origin breaks the translational invariance.
Inner product space
See also: inner product space
The above discussion does not involve the dot product. The dot product is an example of an inner product. Physical space can be modelled as a vector space which additionally has the structure of an inner product. The inner product defines notions of length and angle (and therefore in particular the notion of orthogonality). For any inner product, there exist bases under which the inner product agrees with the dot product, but again, there are many different possible bases, none of which are preferred. They differ from one another by a rotation, an element of the group of rotations SO(3).
In calculus
Main article: vector calculus
Gradient, divergence and curl
In a rectangular coordinate system, the gradient of a (differentiable) function $f:\mathbb {R} ^{3}\rightarrow \mathbb {R} $ is given by
$\nabla f={\frac {\partial f}{\partial x}}\mathbf {i} +{\frac {\partial f}{\partial y}}\mathbf {j} +{\frac {\partial f}{\partial z}}\mathbf {k} $
and in index notation is written
$(\nabla f)_{i}=\partial _{i}f.$
The divergence of a (differentiable) vector field F = U i + V j + W k, that is, a function $\mathbf {F} :\mathbb {R} ^{3}\rightarrow \mathbb {R} ^{3}$ :\mathbb {R} ^{3}\rightarrow \mathbb {R} ^{3}} , is equal to the scalar-valued function:
$\operatorname {div} \,\mathbf {F} =\nabla \cdot \mathbf {F} ={\frac {\partial U}{\partial x}}+{\frac {\partial V}{\partial y}}+{\frac {\partial W}{\partial z}}.$
In index notation, with Einstein summation convention this is
$\nabla \cdot \mathbf {F} =\partial _{i}F_{i}.$
Expanded in Cartesian coordinates (see Del in cylindrical and spherical coordinates for spherical and cylindrical coordinate representations), the curl ∇ × F is, for F composed of [Fx, Fy, Fz]:
${\begin{vmatrix}\mathbf {i} &\mathbf {j} &\mathbf {k} \\\\{\frac {\partial }{\partial x}}&{\frac {\partial }{\partial y}}&{\frac {\partial }{\partial z}}\\\\F_{x}&F_{y}&F_{z}\end{vmatrix}}$
where i, j, and k are the unit vectors for the x-, y-, and z-axes, respectively. This expands as follows:[11]
$\left({\frac {\partial F_{z}}{\partial y}}-{\frac {\partial F_{y}}{\partial z}}\right)\mathbf {i} +\left({\frac {\partial F_{x}}{\partial z}}-{\frac {\partial F_{z}}{\partial x}}\right)\mathbf {j} +\left({\frac {\partial F_{y}}{\partial x}}-{\frac {\partial F_{x}}{\partial y}}\right)\mathbf {k} .$
In index notation, with Einstein summation convention this is
$(\nabla \times \mathbf {F} )_{i}=\epsilon _{ijk}\partial _{j}F_{k},$
where $\epsilon _{ijk}$ is the totally antisymmetric symbol, the Levi-Civita symbol.
Line integrals, surface integrals, and volume integrals
For some scalar field f : U ⊆ Rn → R, the line integral along a piecewise smooth curve C ⊂ U is defined as
$\int \limits _{C}f\,ds=\int _{a}^{b}f(\mathbf {r} (t))|\mathbf {r} '(t)|\,dt.$
where r: [a, b] → C is an arbitrary bijective parametrization of the curve C such that r(a) and r(b) give the endpoints of C and $a<b$.
For a vector field F : U ⊆ Rn → Rn, the line integral along a piecewise smooth curve C ⊂ U, in the direction of r, is defined as
$\int \limits _{C}\mathbf {F} (\mathbf {r} )\cdot \,d\mathbf {r} =\int _{a}^{b}\mathbf {F} (\mathbf {r} (t))\cdot \mathbf {r} '(t)\,dt.$
where · is the dot product and r: [a, b] → C is a bijective parametrization of the curve C such that r(a) and r(b) give the endpoints of C.
A surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analog of the line integral. To find an explicit formula for the surface integral, we need to parameterize the surface of interest, S, by considering a system of curvilinear coordinates on S, like the latitude and longitude on a sphere. Let such a parameterization be x(s, t), where (s, t) varies in some region T in the plane. Then, the surface integral is given by
$\iint _{S}f\,\mathrm {d} S=\iint _{T}f(\mathbf {x} (s,t))\left\|{\partial \mathbf {x} \over \partial s}\times {\partial \mathbf {x} \over \partial t}\right\|\mathrm {d} s\,\mathrm {d} t$
where the expression between bars on the right-hand side is the magnitude of the cross product of the partial derivatives of x(s, t), and is known as the surface element. Given a vector field v on S, that is a function that assigns to each x in S a vector v(x), the surface integral can be defined component-wise according to the definition of the surface integral of a scalar field; the result is a vector.
A volume integral refers to an integral over a 3-dimensional domain.
It can also mean a triple integral within a region D in R3 of a function $f(x,y,z),$ and is usually written as:
$\iiint \limits _{D}f(x,y,z)\,dx\,dy\,dz.$
Fundamental theorem of line integrals
Main article: Fundamental theorem of line integrals
The fundamental theorem of line integrals, says that a line integral through a gradient field can be evaluated by evaluating the original scalar field at the endpoints of the curve.
Let $\varphi :U\subseteq \mathbb {R} ^{n}\to \mathbb {R} $. Then
$\varphi \left(\mathbf {q} \right)-\varphi \left(\mathbf {p} \right)=\int _{\gamma [\mathbf {p} ,\,\mathbf {q} ]}\nabla \varphi (\mathbf {r} )\cdot d\mathbf {r} .$
Stokes' theorem
Main article: Stokes' theorem
Stokes' theorem relates the surface integral of the curl of a vector field F over a surface Σ in Euclidean three-space to the line integral of the vector field over its boundary ∂Σ:
$\iint _{\Sigma }\nabla \times \mathbf {F} \cdot \mathrm {d} \mathbf {\Sigma } =\oint _{\partial \Sigma }\mathbf {F} \cdot \mathrm {d} \mathbf {r} .$
Divergence theorem
Main article: Divergence theorem
Suppose V is a subset of $\mathbb {R} ^{n}$ (in the case of n = 3, V represents a volume in 3D space) which is compact and has a piecewise smooth boundary S (also indicated with ∂V = S ). If F is a continuously differentiable vector field defined on a neighborhood of V, then the divergence theorem says:[12]
$\iiint _{V}\left(\mathbf {\nabla } \cdot \mathbf {F} \right)\,dV=$ $\scriptstyle S$ $(\mathbf {F} \cdot \mathbf {n} )\,dS.$
The left side is a volume integral over the volume V, the right side is the surface integral over the boundary of the volume V. The closed manifold ∂V is quite generally the boundary of V oriented by outward-pointing normals, and n is the outward pointing unit normal field of the boundary ∂V. (dS may be used as a shorthand for ndS.)
In topology
Three-dimensional space has a number of topological properties that distinguish it from spaces of other dimension numbers. For example, at least three dimensions are required to tie a knot in a piece of string.[13]
In differential geometry the generic three-dimensional spaces are 3-manifolds, which locally resemble ${\mathbb {R} }^{3}$.
In finite geometry
Many ideas of dimension can be tested with finite geometry. The simplest instance is PG(3,2), which has Fano planes as its 2-dimensional subspaces. It is an instance of Galois geometry, a study of projective geometry using finite fields. Thus, for any Galois field GF(q), there is a projective space PG(3,q) of three dimensions. For example, any three skew lines in PG(3,q) are contained in exactly one regulus.[14]
See also
• 3D rotation
• Rotation formalisms in three dimensions
• Dimensional analysis
• Distance from a point to a plane
• Four-dimensional space
• Skew lines § Distance
• Three-dimensional graph
• Solid geometry
• Two-dimensional space
Notes
1. "Euclidean space - Encyclopedia of Mathematics". encyclopediaofmath.org. Retrieved 2020-08-12.
2. "Euclidean space | geometry". Encyclopedia Britannica. Retrieved 2020-08-12.
3. Hughes-Hallett, Deborah; McCallum, William G.; Gleason, Andrew M. (2013). Calculus : Single and Multivariable (6 ed.). John wiley. ISBN 978-0470-88861-2.
4. Brannan, Esplen & Gray 1999, pp. 34–5
5. Brannan, Esplen & Gray 1999, pp. 41–2
6. Anton 1994, p. 133
7. Anton 1994, p. 131
8. Massey, WS (1983). "Cross products of vectors in higher dimensional Euclidean spaces". The American Mathematical Monthly. 90 (10): 697–701. doi:10.2307/2323537. JSTOR 2323537. If one requires only three basic properties of the cross product ... it turns out that a cross product of vectors exists only in 3-dimensional and 7-dimensional Euclidean space.
9. Lang 1987, ch. I.1
10. Berger 1987, Chapter 9. sfn error: no target: CITEREFBerger1987 (help)
11. Arfken, p. 43.
12. M. R. Spiegel; S. Lipschutz; D. Spellman (2009). Vector Analysis. Schaum's Outlines (2nd ed.). US: McGraw Hill. ISBN 978-0-07-161545-7.
13. Rolfsen, Dale (1976). Knots and Links. Berkeley, California: Publish or Perish. ISBN 0-914098-16-0.
14. Albrecht Beutelspacher & Ute Rosenbaum (1998) Projective Geometry, page 72, Cambridge University Press ISBN 0-521-48277-1
References
• Anton, Howard (1994), Elementary Linear Algebra (7th ed.), John Wiley & Sons, ISBN 978-0-471-58742-2
• Arfken, George B. and Hans J. Weber. Mathematical Methods For Physicists, Academic Press; 6 edition (June 21, 2005). ISBN 978-0-12-059876-2.
• Brannan, David A.; Esplen, Matthew F.; Gray, Jeremy J. (1999), Geometry, Cambridge University Press, ISBN 978-0-521-59787-6
External links
Wikiquote has quotations related to Three-dimensional space.
Wikimedia Commons has media related to 3D.
• The dictionary definition of three-dimensional at Wiktionary
• Weisstein, Eric W. "Four-Dimensional Geometry". MathWorld.
• Elementary Linear Algebra - Chapter 8: Three-dimensional Geometry Keith Matthews from University of Queensland, 1991
Dimension
Dimensional spaces
• Vector space
• Euclidean space
• Affine space
• Projective space
• Free module
• Manifold
• Algebraic variety
• Spacetime
Other dimensions
• Krull
• Lebesgue covering
• Inductive
• Hausdorff
• Minkowski
• Fractal
• Degrees of freedom
Polytopes and shapes
• Hyperplane
• Hypersurface
• Hypercube
• Hyperrectangle
• Demihypercube
• Hypersphere
• Cross-polytope
• Simplex
• Hyperpyramid
Dimensions by number
• Zero
• One
• Two
• Three
• Four
• Five
• Six
• Seven
• Eight
• n-dimensions
See also
• Hyperspace
• Codimension
Category
| Wikipedia |
3SUM
In computational complexity theory, the 3SUM problem asks if a given set of $n$ real numbers contains three elements that sum to zero. A generalized version, k-SUM, asks the same question on k numbers. 3SUM can be easily solved in $O(n^{2})$ time, and matching $\Omega (n^{\lceil k/2\rceil })$ lower bounds are known in some specialized models of computation (Erickson 1999).
Unsolved problem in computer science:
Is there an algorithm to solve the 3SUM problem in time $O(n^{2-\epsilon })$, for some $\epsilon >0$?
(more unsolved problems in computer science)
It was conjectured that any deterministic algorithm for the 3SUM requires $\Omega (n^{2})$ time. In 2014, the original 3SUM conjecture was refuted by Allan Grønlund and Seth Pettie who gave a deterministic algorithm that solves 3SUM in $O(n^{2}/({\log n}/{\log \log n})^{2/3})$ time.[1] Additionally, Grønlund and Pettie showed that the 4-linear decision tree complexity of 3SUM is $O(n^{3/2}{\sqrt {\log n}})$. These bounds were subsequently improved.[2][3][4] The current best known algorithm for 3SUM runs in $O(n^{2}(\log \log n)^{O(1)}/{\log ^{2}n})$ time.[4] Kane, Lovett, and Moran showed that the 6-linear decision tree complexity of 3SUM is $O(n{\log ^{2}n})$.[5] The latter bound is tight (up to a logarithmic factor). It is still conjectured that 3SUM is unsolvable in $O(n^{2-\Omega (1)})$ expected time.[6]
When the elements are integers in the range $[-N,\dots ,N]$, 3SUM can be solved in $O(n+N\log N)$ time by representing the input set $S$ as a bit vector, computing the set $S+S$ of all pairwise sums as a discrete convolution using the fast Fourier transform, and finally comparing this set to $S$.[7]
Quadratic algorithm
Suppose the input array is $S[0..n-1]$. In integer (word RAM) models of computing, 3SUM can be solved in $O(n^{2})$ time on average by inserting each number $S[i]$ into a hash table, and then, for each index $i$ and $j$, checking whether the hash table contains the integer $-(S[i]+S[j])$.
It is also possible to solve the problem in the same time in a comparison-based model of computing or real RAM, for which hashing is not allowed. The algorithm below first sorts the input array and then tests all possible pairs in a careful order that avoids the need to binary search for the pairs in the sorted list, achieving worst-case $O(n^{2})$ time, as follows.[8]
sort(S);
for i = 0 to n - 2 do
a = S[i];
start = i + 1;
end = n - 1;
while (start < end) do
b = S[start]
c = S[end];
if (a + b + c == 0) then
output a, b, c;
// Continue search for all triplet combinations summing to zero.
// We need to update both end and start together since the array values are distinct.
start = start + 1;
end = end - 1;
else if (a + b + c > 0) then
end = end - 1;
else
start = start + 1;
end
end
The following example shows this algorithm's execution on a small sorted array. Current values of a are shown in red, values of b and c are shown in magenta.
-25 -10 -7 -3 2 4 8 10 (a+b+c==-25)
-25 -10 -7 -3 2 4 8 10 (a+b+c==-22)
. . .
-25 -10 -7 -3 2 4 8 10 (a+b+c==-7)
-25 -10 -7 -3 2 4 8 10 (a+b+c==-7)
-25 -10 -7 -3 2 4 8 10 (a+b+c==-3)
-25 -10 -7 -3 2 4 8 10 (a+b+c==2)
-25 -10 -7 -3 2 4 8 10 (a+b+c==0)
The correctness of the algorithm can be seen as follows. Suppose we have a solution a + b + c = 0. Since the pointers only move in one direction, we can run the algorithm until the leftmost pointer points to a. Run the algorithm until either one of the remaining pointers points to b or c, whichever occurs first. Then the algorithm will run until the last pointer points to the remaining term, giving the affirmative solution.
Variants
Non-zero sum
Instead of looking for numbers whose sum is 0, it is possible to look for numbers whose sum is any constant C. The simplest way would be to modify the original algorithm to search the hash table for the integer $(C-(S[i]+S[j]))$.
Another method:
• Subtract C/3 from all elements of the input array.
• In the modified array, find 3 elements whose sum is 0.
For example, if A=[1,2,3,4] and if you are asked to find 3SUM for C=4, then subtract 4/3 from all the elements of A, and solve it in the usual 3sum way, i.e., $(a-C/3)+(b-C/3)+(c-C/3)=0$.
Three different arrays
Instead of searching for the 3 numbers in a single array, we can search for them in 3 different arrays. I.e., given three arrays X, Y and Z, find three numbers a∈X, b∈Y, c∈Z, such that $a+b+c=0$. Call the 1-array variant 3SUM×1 and the 3-array variant 3SUM×3.
Given a solver for 3SUM×1, the 3SUM×3 problem can be solved in the following way (assuming all elements are integers):
• For every element in X, Y and Z, set: $X[i]\gets X[i]*10+1$, $Y[i]\gets Y[i]*10+2$, $Z[i]\gets Z[i]*10-3$.
• Let S be a concatenation of the arrays X, Y and Z.
• Use the 3SUM×1 oracle to find three elements $a'\in S,\ b'\in S,\ c'\in S$ such that $a'+b'+c'=0$.
• Return $a\gets (a'-1)/10,\ b\gets (b'-2)/10,\ c\gets (c'+3)/10$.
By the way we transformed the arrays, it is guaranteed that a∈X, b∈Y, c∈Z.[9]
Convolution sum
Instead of looking for arbitrary elements of the array such that:
$S[k]=S[i]+S[j]$
the convolution 3sum problem (Conv3SUM) looks for elements in specific locations:[10]
$S[i+j]=S[i]+S[j]$
Reduction from Conv3SUM to 3SUM
Given a solver for 3SUM, the Conv3SUM problem can be solved in the following way.[10]
• Define a new array T, such that for every index i: $T[i]=2nS[i]+i$ (where n is the number of elements in the array, and the indices run from 0 to n-1).
• Solve 3SUM on the array T.
Correctness proof:
• If in the original array there is a triple with $S[i+j]=S[i]+S[j]$, then $T[i+j]=2nS[i+j]+i+j=(2nS[i]+i)+(2nS[j]+j)=T[i]+T[j]$, so this solution will be found by 3SUM on T.
• Conversely, if in the new array there is a triple with $T[k]=T[i]+T[j]$, then $2nS[k]+k=2n(S[i]+S[j])+(i+j)$. Because $i+j<2n$, necessarily $S[k]=S[i]+S[j]$ and $k=i+j$, so this is a valid solution for Conv3SUM on S.
Reduction from 3SUM to Conv3SUM
Given a solver for Conv3SUM, the 3SUM problem can be solved in the following way.[6][10]
The reduction uses a hash function. As a first approximation, assume that we have a linear hash function, i.e. a function h such that:
$h(x+y)=h(x)+h(y)$
Suppose that all elements are integers in the range: 0...N-1, and that the function h maps each element to an element in the smaller range of indices: 0...n-1. Create a new array T and send each element of S to its hash value in T, i.e., for every x in S($\forall x\in S$):
$T[h(x)]=x$
Initially, suppose that the mappings are unique (i.e. each cell in T accepts only a single element from S). Solve Conv3SUM on T. Now:
• If there is a solution for 3SUM: $z=x+y$, then: $T[h(z)]=T[h(x)]+T[h(y)]$ and $h(z)=h(x)+h(y)$, so this solution will be found by the Conv3SUM solver on T.
• Conversely, if a Conv3SUM is found on T, then obviously it corresponds to a 3SUM solution on S since T is just a permutation of S.
This idealized solution doesn't work, because any hash function might map several distinct elements of S to the same cell of T. The trick is to create an array $T^{*}$ by selecting a single random element from each cell of T, and run Conv3SUM on $T^{*}$. If a solution is found, then it is a correct solution for 3SUM on S. If no solution is found, then create a different random $T^{*}$ and try again. Suppose there are at most R elements in each cell of T. Then the probability of finding a solution (if a solution exists) is the probability that the random selection will select the correct element from each cell, which is $(1/R)^{3}$. By running Conv3SUM $R^{3}$ times, the solution will be found with a high probability.
Unfortunately, we do not have linear perfect hashing, so we have to use an almost linear hash function, i.e. a function h such that:
$h(x+y)=h(x)+h(y)$ or
$h(x+y)=h(x)+h(y)+1$
This requires to duplicate the elements of S when copying them into T, i.e., put every element $x\in S$ both in $T[h(x)]$ (as before) and in $T[h(x)]-1$. So each cell will have 2R elements, and we will have to run Conv3SUM $(2R)^{3}$ times.
3SUM-hardness
A problem is called 3SUM-hard if solving it in subquadratic time implies a subquadratic-time algorithm for 3SUM. The concept of 3SUM-hardness was introduced by Gajentaan & Overmars (1995). They proved that a large class of problems in computational geometry are 3SUM-hard, including the following ones. (The authors acknowledge that many of these problems are contributed by other researchers.)
• Given a set of lines in the plane, are there three that meet in a point?
• Given a set of non-intersecting axis-parallel line segments, is there a line that separates them into two non-empty subsets?
• Given a set of infinite strips in the plane, do they fully cover a given rectangle?
• Given a set of triangles in the plane, compute their measure.
• Given a set of triangles in the plane, does their union have a hole?
• A number of visibility and motion planning problems, e.g.,
• Given a set of horizontal triangles in space, can a particular triangle be seen from a particular point?
• Given a set of non-intersecting axis-parallel line segment obstacles in the plane, can a given rod be moved by translations and rotations between a start and finish positions without colliding with the obstacles?
By now there are a multitude of other problems that fall into this category. An example is the decision version of X + Y sorting: given sets of numbers X and Y of n elements each, are there n² distinct x + y for x ∈ X, y ∈ Y?[11]
See also
• Subset sum problem
Notes
1. Grønlund & Pettie 2014.
2. Freund 2017.
3. Gold & Sharir 2017.
4. Chan 2018.
5. Kane, Lovett & Moran 2018.
6. Kopelowitz, Tsvi; Pettie, Seth; Porat, Ely (2014). "3SUM Hardness in (Dynamic) Data Structures". arXiv:1407.6756 [cs.DS].
7. Cormen, Thomas H.; Leiserson, Charles E.; Rivest, Ronald L.; Stein, Clifford (2009) [1990]. Introduction to Algorithms (3rd ed.). MIT Press and McGraw-Hill. ISBN 0-262-03384-4. Ex. 30.1–7, p. 906.
8. Visibility Graphs and 3-Sum by Michael Hoffmann
9. For a reduction in the other direction, see Variants of the 3-sum problem.
10. Patrascu, M. (2010). Towards polynomial lower bounds for dynamic problems. Proceedings of the 42nd ACM symposium on Theory of computing - STOC '10. p. 603. doi:10.1145/1806689.1806772. ISBN 9781450300506.
11. Demaine, Erik; Erickson, Jeff; O'Rourke, Joseph (20 August 2006). "Problem 41: Sorting X + Y (Pairwise Sums)". The Open Problems Project. Retrieved 23 September 2014.
References
• Kane, Daniel M.; Lovett, Shachar; Moran, Shay (2018). "Near-optimal linear decision trees for k-SUM and related problems". Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing. pp. 554–563. arXiv:1705.01720. doi:10.1145/3188745.3188770. ISBN 9781450355599. S2CID 30368541.{{cite book}}: CS1 maint: date and year (link)
• Chan, Timothy M. (2018), "More Logarithmic-Factor Speedups for 3SUM, (Median,+)-Convolution, and Some Geometric 3SUM-Hard Problems", Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 881–897, doi:10.1137/1.9781611975031.57, ISBN 978-1-61197-503-1
• Grønlund, A.; Pettie, S. (2014). Threesomes, Degenerates, and Love Triangles. 2014 IEEE 55th Annual Symposium on Foundations of Computer Science. p. 621. arXiv:1404.0799. Bibcode:2014arXiv1404.0799G. doi:10.1109/FOCS.2014.72. ISBN 978-1-4799-6517-5.
• Freund, Ari (2017), "Improved Subquadratic 3SUM", Algorithmica, 44 (2): 440–458, doi:10.1007/s00453-015-0079-6, S2CID 253979651.
• Gold, Omer; Sharir, Micha (2017), "Improved Bounds for 3SUM, k-SUM, and Linear Degeneracy", In Proc. 25th Annual European Symposium on Algorithms (ESA), LIPIcs, 87: 42:1–42:13, doi:10.4230/LIPIcs.ESA.2017.42, S2CID 691387
• Baran, Ilya; Demaine, Erik D.; Pătraşcu, Mihai (2008), "Subquadratic algorithms for 3SUM", Algorithmica, 50 (4): 584–596, doi:10.1007/s00453-007-9036-3, S2CID 9855995.
• Demaine, Erik D.; Mitchell, Joseph S. B.; O'Rourke, Joseph (July 2005), "Problem 11: 3SUM Hard Problems", The Open Problems Project, archived from the original on 2012-12-15, retrieved 2008-09-02.
• Erickson, Jeff (1999), "Lower bounds for linear satisfiability problems", Chicago Journal of Theoretical Computer Science, MIT Press, 1999.
• Gajentaan, Anka; Overmars, Mark H. (1995), "On a class of O(n2) problems in computational geometry", Computational Geometry: Theory and Applications, 5 (3): 165–185, doi:10.1016/0925-7721(95)00022-2, hdl:1874/17058.
• King, James (2004), A survey of 3SUM-hard problems (PDF).
| Wikipedia |
4-manifold
In mathematics, a 4-manifold is a 4-dimensional topological manifold. A smooth 4-manifold is a 4-manifold with a smooth structure. In dimension four, in marked contrast with lower dimensions, topological and smooth manifolds are quite different. There exist some topological 4-manifolds which admit no smooth structure, and even if there exists a smooth structure, it need not be unique (i.e. there are smooth 4-manifolds which are homeomorphic but not diffeomorphic).
4-manifolds are important in physics because in General Relativity, spacetime is modeled as a pseudo-Riemannian 4-manifold.
Topological 4-manifolds
The homotopy type of a simply connected compact 4-manifold only depends on the intersection form on the middle dimensional homology. A famous theorem of Michael Freedman (1982) implies that the homeomorphism type of the manifold only depends on this intersection form, and on a $\mathbb {Z} /2\mathbb {Z} $ invariant called the Kirby–Siebenmann invariant, and moreover that every combination of unimodular form and Kirby–Siebenmann invariant can arise, except that if the form is even, then the Kirby–Siebenmann invariant must be the signature/8 (mod 2).
Examples:
• In the special case when the form is 0, this implies the 4-dimensional topological Poincaré conjecture.
• If the form is the E8 lattice, this gives a manifold called the E8 manifold, a manifold not homeomorphic to any simplicial complex.
• If the form is $\mathbb {Z} $, there are two manifolds depending on the Kirby–Siebenmann invariant: one is 2-dimensional complex projective space, and the other is a fake projective space, with the same homotopy type but not homeomorphic (and with no smooth structure).
• When the rank of the form is greater than about 28, the number of positive definite unimodular forms starts to increase extremely rapidly with the rank, so there are huge numbers of corresponding simply connected topological 4-manifolds (most of which seem to be of almost no interest).
Freedman's classification can be extended to some cases when the fundamental group is not too complicated; for example, when it is $\mathbb {Z} $, there is a classification similar to the one above using Hermitian forms over the group ring of $\mathbb {Z} $. If the fundamental group is too large (for example, a free group on 2 generators), then Freedman's techniques seem to fail and very little is known about such manifolds.
For any finitely presented group it is easy to construct a (smooth) compact 4-manifold with it as its fundamental group. As there is no algorithm to tell whether two finitely presented groups are isomorphic (even if one is known to be trivial) there is no algorithm to tell if two 4-manifolds have the same fundamental group. This is one reason why much of the work on 4-manifolds just considers the simply connected case: the general case of many problems is already known to be intractable.
Smooth 4-manifolds
For manifolds of dimension at most 6, any piecewise linear (PL) structure can be smoothed in an essentially unique way,[1] so in particular the theory of 4 dimensional PL manifolds is much the same as the theory of 4 dimensional smooth manifolds.
A major open problem in the theory of smooth 4-manifolds is to classify the simply connected compact ones. As the topological ones are known, this breaks up into two parts:
1. Which topological manifolds are smoothable?
2. Classify the different smooth structures on a smoothable manifold.
There is an almost complete answer to the first problem of which simply connected compact 4-manifolds have smooth structures. First, the Kirby–Siebenmann class must vanish.
• If the intersection form is definite Donaldson's theorem (Donaldson 1983) gives a complete answer: there is a smooth structure if and only if the form is diagonalizable.
• If the form is indefinite and odd there is a smooth structure.
• If the form is indefinite and even we may as well assume that it is of nonpositive signature by changing orientations if necessary, in which case it is isomorphic to a sum of m copies of II1,1 and 2n copies of E8(−1) for some m and n. If m ≥ 3n (so that the dimension is at least 11/8 times the |signature|) then there is a smooth structure, given by taking a connected sum of n K3 surfaces and m − 3n copies of S2×S2. If m ≤ 2n (so the dimension is at most 10/8 times the |signature|) then Furuta proved that no smooth structure exists (Furuta 2001). This leaves a small gap between 10/8 and 11/8 where the answer is mostly unknown. (The smallest case not covered above has n=2 and m=5, but this has also been ruled out, so the smallest lattice for which the answer is not currently known is the lattice II7,55 of rank 62 with n=3 and m=7. See [2] for recent (as of 2019) progress in this area.) The "11/8 conjecture" states that smooth structures do not exist if the dimension is less than 11/8 times the |signature|.
In contrast, very little is known about the second question of classifying the smooth structures on a smoothable 4-manifold; in fact, there is not a single smoothable 4-manifold where the answer is known. Donaldson showed that there are some simply connected compact 4-manifolds, such as Dolgachev surfaces, with a countably infinite number of different smooth structures. There are an uncountable number of different smooth structures on R4; see exotic R4. Fintushel and Stern showed how to use surgery to construct large numbers of different smooth structures (indexed by arbitrary integral polynomials) on many different manifolds, using Seiberg–Witten invariants to show that the smooth structures are different. Their results suggest that any classification of simply connected smooth 4-manifolds will be very complicated. There are currently no plausible conjectures about what this classification might look like. (Some early conjectures that all simply connected smooth 4-manifolds might be connected sums of algebraic surfaces, or symplectic manifolds, possibly with orientations reversed, have been disproved.)
Special phenomena in 4 dimensions
There are several fundamental theorems about manifolds that can be proved by low-dimensional methods in dimensions at most 3, and by completely different high-dimensional methods in dimension at least 5, but which are false in dimension 4. Here are some examples:
• In dimensions other than 4, the Kirby–Siebenmann invariant provides the obstruction to the existence of a PL structure; in other words a compact topological manifold has a PL structure if and only if its Kirby–Siebenmann invariant in H4(M,Z/2Z) vanishes. In dimension 3 and lower, every topological manifold admits an essentially unique PL structure. In dimension 4 there are many examples with vanishing Kirby–Siebenmann invariant but no PL structure.
• In any dimension other than 4, a compact topological manifold has only a finite number of essentially distinct PL or smooth structures. In dimension 4, compact manifolds can have a countably infinite number of non-diffeomorphic smooth structures.
• Four is the only dimension n for which Rn can have an exotic smooth structure. R4 has an uncountable number of exotic smooth structures; see exotic R4.
• The solution to the smooth Poincaré conjecture is known in all dimensions other than 4 (it is usually false in dimensions at least 7; see exotic sphere). The Poincaré conjecture for PL manifolds has been proved for all dimensions other than 4, but it is not known whether it is true in 4 dimensions (it is equivalent to the smooth Poincaré conjecture in 4 dimensions).
• The smooth h-cobordism theorem holds for cobordisms provided that neither the cobordism nor its boundary has dimension 4. It can fail if the boundary of the cobordism has dimension 4 (as shown by Donaldson).[3] If the cobordism has dimension 4, then it is unknown whether the h-cobordism theorem holds.
• A topological manifold of dimension not equal to 4 has a handlebody decomposition. Manifolds of dimension 4 have a handlebody decomposition if and only if they are smoothable.
• There are compact 4-dimensional topological manifolds that are not homeomorphic to any simplicial complex. In dimension at least 5 the existence of topological manifolds not homeomorphic to a simplicial complex was an open problem. Ciprian Manolescu showed that there are manifolds in each dimension greater than or equal to 5, that are not homeomorphic to a simplicial complex.[4]
Failure of the Whitney trick in dimension 4
According to Frank Quinn, "Two n-dimensional submanifolds of a manifold of dimension 2n will usually intersect themselves and each other in isolated points. The "Whitney trick" uses an isotopy across an embedded 2-disk to simplify these intersections. Roughly speaking this reduces the study of n-dimensional embeddings to embeddings of 2-disks. But this is not a reduction when the dimension is 4: the 2-disks themselves are middle-dimensional, so trying to embed them encounters exactly the same problems they are supposed to solve. This is the phenomenon that separates dimension 4 from others."[5]
See also
• Kirby calculus
• Algebraic surface
• 3-manifold
• 5-manifold
• Enriques–Kodaira classification
• Casson handle
• Akbulut cork
References
1. Milnor, John (2011), "Differential topology forty-six years later" (PDF), Notices of the American Mathematical Society, 58 (6): 804–809, MR 2839925.
2. Hopkins, Michael J.; Lin, Jianfeng; Shi, XiaoLin; Xu, Zhouli (2019), "Intersection Forms of Spin 4-Manifolds and the Pin(2)-Equivariant Mahowald Invariant", arXiv:1812.04052 [math.AT].
3. Donaldson, Simon K. (1987). "Irrationality and the h-cobordism conjecture". J. Differential Geom. 26 (1): 141–168. doi:10.4310/jdg/1214441179. MR 0892034.
4. Manolescu, Ciprian (2016). "Pin(2)-equivariant Seiberg–Witten Floer homology and the Triangulation Conjecture". J. Amer. Math. Soc. 29: 147–176. arXiv:1303.2354. doi:10.1090/jams829. S2CID 16403004.
5. Quinn, F. (1996). "Problems in low-dimensional topology". In Ranicki, A.; Yamasaki, M. (eds.). Surgery and Geometric Topology: Proceedings of a conference held at Josai University, Sakado, Sept. 1996 (PDF). pp. 97–104.
• Donaldson, Simon K. (1983), "An application of gauge theory to four-dimensional topology", Journal of Differential Geometry, 18 (2): 279–315, doi:10.4310/jdg/1214437665
• Donaldson, Simon K.; Kronheimer, Peter B. (1997), The Geometry of Four-Manifolds, Oxford Mathematical Monographs, Oxford: Clarendon Press, ISBN 0-19-850269-9
• Freed, Daniel S.; Uhlenbeck, Karen K. (1984), Instantons and four-manifolds, Mathematical Sciences Research Institute Publications, vol. 1, Springer-Verlag, New York, doi:10.1007/978-1-4684-0258-2, ISBN 0-387-96036-8, MR 0757358
• Freedman, Michael Hartley (1982), "The topology of four-dimensional manifolds", Journal of Differential Geometry, 17 (3): 357–453, doi:10.4310/jdg/1214437136, MR 0679066
• Freedman, Michael H.; Quinn, Frank (1990), Topology of 4-manifolds, Princeton, N.J.: Princeton University Press, ISBN 0-691-08577-3
• Furuta, Mikio (2001), "Monopole Equation and the 11/8-Conjecture", Mathematical Research Letters, 8: 279–291, doi:10.4310/mrl.2001.v8.n3.a5, MR 1839478
• Kirby, Robion C. (1989), The topology of 4-manifolds, Lecture Notes in Mathematics, vol. 1374, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0089031, ISBN 978-3-540-51148-9, MR 1001966
• Gompf, Robert E.; Stipsicz, András I. (1999), 4-Manifolds and Kirby Calculus, Grad. Studies in Math., vol. 20, American Mathematical Society, MR 1707327
• Kirby, R. C.; Taylor, L. R. (1998). "A survey of 4-manifolds through the eyes of surgery". arXiv:math.GT/9803101.
• Mandelbaum, R. (1980), "Four-dimensional topology: an introduction", Bull. Amer. Math. Soc., 2: 1–159, doi:10.1090/S0273-0979-1980-14687-X
• Matveev, S. V. (2001) [1994], "Four-dimensional manifolds", Encyclopedia of Mathematics, EMS Press
• Scorpan, A. (2005), The wild world of 4-manifolds, Providence, R.I.: American Mathematical Society, ISBN 0-8218-3749-4
External links
• Media related to 4-manifolds at Wikimedia Commons
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• Israel
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| Wikipedia |
4104
4104 (four thousand one hundred [and] four) is the natural number following 4103 and preceding 4105. It is the second positive integer which can be expressed as the sum of two positive cubes in two different ways. The first such number, 1729, is called the "Ramanujan–Hardy number".
← 4103 4104 4105 →
• List of numbers
• Integers
← 0 1k 2k 3k 4k 5k 6k 7k 8k 9k →
Cardinalfour thousand one hundred four
Ordinal4104th
(four thousand one hundred fourth)
Factorization23 × 33 × 19
Greek numeral,ΔΡΔ´
Roman numeralMVCIV, or IVCIV
Binary10000000010002
Ternary121220003
Senary310006
Octal100108
Duodecimal246012
Hexadecimal100816
4104 is the sum of 4096 + 8 (that is, 163 + 23), and also the sum of 3375 + 729 (that is, 153 + 93).
See also
• Taxicab number
• 1729
External links
• MathWorld: Hardy–Ramanujan Number
| Wikipedia |
Special right triangle
A special right triangle is a right triangle with some regular feature that makes calculations on the triangle easier, or for which simple formulas exist. For example, a right triangle may have angles that form simple relationships, such as 45°–45°–90°. This is called an "angle-based" right triangle. A "side-based" right triangle is one in which the lengths of the sides form ratios of whole numbers, such as 3 : 4 : 5, or of other special numbers such as the golden ratio. Knowing the relationships of the angles or ratios of sides of these special right triangles allows one to quickly calculate various lengths in geometric problems without resorting to more advanced methods.
Angle-based
Angle-based special right triangles are specified by the relationships of the angles of which the triangle is composed. The angles of these triangles are such that the larger (right) angle, which is 90 degrees or π/2 radians, is equal to the sum of the other two angles.
The side lengths are generally deduced from the basis of the unit circle or other geometric methods. This approach may be used to rapidly reproduce the values of trigonometric functions for the angles 30°, 45°, and 60°.
Special triangles are used to aid in calculating common trigonometric functions, as below:
degreesradiansgonsturnssincostancotan
0°00g0√0/2 = 0√4/2 = 10undefined
30°π/633+1/3g1/12√1/2 = 1/2√3/21/√3√3
45°π/450g1/8√2/2 = 1/√2√2/2 = 1/√211
60°π/366+2/3g1/6√3/2√1/2 = 1/2√31/√3
90°π/2100g1/4√4/2 = 1√0/2 = 0undefined0
45°–45°–90°
30°–60°–90°
The 45°–45°–90° triangle, the 30°–60°–90° triangle, and the equilateral/equiangular (60°–60°–60°) triangle are the three Möbius triangles in the plane, meaning that they tessellate the plane via reflections in their sides; see Triangle group.
45° - 45° - 90° triangle
In plane geometry, constructing the diagonal of a square results in a triangle whose three angles are in the ratio 1 : 1 : 2, adding up to 180° or π radians. Hence, the angles respectively measure 45° (π/4), 45° (π/4), and 90° (π/2). The sides in this triangle are in the ratio 1 : 1 : √2, which follows immediately from the Pythagorean theorem.
Of all right triangles, the 45° - 45° - 90° degree triangle has the smallest ratio of the hypotenuse to the sum of the legs, namely √2/2.[1]: p.282, p.358 and the greatest ratio of the altitude from the hypotenuse to the sum of the legs, namely √2/4.[1]: p.282
Triangles with these angles are the only possible right triangles that are also isosceles triangles in Euclidean geometry. However, in spherical geometry and hyperbolic geometry, there are infinitely many different shapes of right isosceles triangles.
30° - 60° - 90° triangle
This is a triangle whose three angles are in the ratio 1 : 2 : 3 and respectively measure 30° (π/6), 60° (π/3), and 90° (π/2). The sides are in the ratio 1 : √3 : 2.
The proof of this fact is clear using trigonometry. The geometric proof is:
Draw an equilateral triangle ABC with side length 2 and with point D as the midpoint of segment BC. Draw an altitude line from A to D. Then ABD is a 30°–60°–90° triangle with hypotenuse of length 2, and base BD of length 1.
The fact that the remaining leg AD has length √3 follows immediately from the Pythagorean theorem.
The 30°–60°–90° triangle is the only right triangle whose angles are in an arithmetic progression. The proof of this fact is simple and follows on from the fact that if α, α + δ, α + 2δ are the angles in the progression then the sum of the angles 3α + 3δ = 180°. After dividing by 3, the angle α + δ must be 60°. The right angle is 90°, leaving the remaining angle to be 30°.
Side-based
Right triangles whose sides are of integer lengths, with the sides collectively known as Pythagorean triples, possess angles that cannot all be rational numbers of degrees.[2] (This follows from Niven's theorem.) They are most useful in that they may be easily remembered and any multiple of the sides produces the same relationship. Using Euclid's formula for generating Pythagorean triples, the sides must be in the ratio
m2 − n2 : 2mn : m2 + n2
where m and n are any positive integers such that m > n.
Common Pythagorean triples
Main article: Pythagorean triple
There are several Pythagorean triples which are well-known, including those with sides in the ratios:
3:4:5
5:12:13
8:15:17
7:24:25
9:40:41
The 3 : 4 : 5 triangles are the only right triangles with edges in arithmetic progression. Triangles based on Pythagorean triples are Heronian, meaning they have integer area as well as integer sides.
The possible use of the 3 : 4 : 5 triangle in Ancient Egypt, with the supposed use of a knotted rope to lay out such a triangle, and the question whether Pythagoras' theorem was known at that time, have been much debated.[3] It was first conjectured by the historian Moritz Cantor in 1882.[3] It is known that right angles were laid out accurately in Ancient Egypt; that their surveyors did use ropes for measurement;[3] that Plutarch recorded in Isis and Osiris (around 100 AD) that the Egyptians admired the 3 : 4 : 5 triangle;[3] and that the Berlin Papyrus 6619 from the Middle Kingdom of Egypt (before 1700 BC) stated that "the area of a square of 100 is equal to that of two smaller squares. The side of one is ½ + ¼ the side of the other."[4] The historian of mathematics Roger L. Cooke observes that "It is hard to imagine anyone being interested in such conditions without knowing the Pythagorean theorem."[3] Against this, Cooke notes that no Egyptian text before 300 BC actually mentions the use of the theorem to find the length of a triangle's sides, and that there are simpler ways to construct a right angle. Cooke concludes that Cantor's conjecture remains uncertain: he guesses that the Ancient Egyptians probably did know the Pythagorean theorem, but that "there is no evidence that they used it to construct right angles".[3]
The following are all the Pythagorean triple ratios expressed in lowest form (beyond the five smallest ones in lowest form in the list above) with both non-hypotenuse sides less than 256:
11:60:61
12:35:37
13:84:85
15:112:113
16:63:65
17:144:145
19:180:181
20:21:29
20:99:101
21:220:221
24:143:145
28:45:53
28:195:197
32:255:257
33:56:65
36:77:85
39:80:89
44:117:125
48:55:73
51:140:149
52:165:173
57:176:185
60:91:109
60:221:229
65:72:97
84:187:205
85:132:157
88:105:137
95:168:193
96:247:265
104:153:185
105:208:233
115:252:277
119:120:169
120:209:241
133:156:205
140:171:221
160:231:281
161:240:289
204:253:325
207:224:305
Almost-isosceles Pythagorean triples
Isosceles right-angled triangles cannot have sides with integer values, because the ratio of the hypotenuse to either other side is √2 and √2 cannot be expressed as a ratio of two integers. However, infinitely many almost-isosceles right triangles do exist. These are right-angled triangles with integer sides for which the lengths of the non-hypotenuse edges differ by one.[5][6] Such almost-isosceles right-angled triangles can be obtained recursively,
a0 = 1, b0 = 2
an = 2bn−1 + an−1
bn = 2an + bn−1
an is length of hypotenuse, n = 1, 2, 3, .... Equivalently,
$({\tfrac {x-1}{2}})^{2}+({\tfrac {x+1}{2}})^{2}=y^{2}$
where {x, y} are solutions to the Pell equation x2 − 2y2 = −1, with the hypotenuse y being the odd terms of the Pell numbers 1, 2, 5, 12, 29, 70, 169, 408, 985, 2378... (sequence A000129 in the OEIS).. The smallest Pythagorean triples resulting are:[7]
3 :4: 5
20 :21: 29
119 :120: 169
696 :697: 985
4,059 :4,060: 5,741
23,660 :23,661: 33,461
137,903 :137,904: 195,025
803,760 :803,761: 1,136,689
4,684,659 : 4,684,660 : 6,625,109
Alternatively, the same triangles can be derived from the square triangular numbers.[8]
Arithmetic and geometric progressions
Main article: Kepler triangle
The Kepler triangle is a right triangle whose sides are in geometric progression. If the sides are formed from the geometric progression a, ar, ar2 then its common ratio r is given by r = √φ where φ is the golden ratio. Its sides are therefore in the ratio 1 : √φ : φ. Thus, the shape of the Kepler triangle is uniquely determined (up to a scale factor) by the requirement that its sides be in geometric progression.
The 3–4–5 triangle is the unique right triangle (up to scaling) whose sides are in arithmetic progression.[9]
Sides of regular polygons
Let
$a=2\sin {\frac {\pi }{10}}={\frac {-1+{\sqrt {5}}}{2}}={\frac {1}{\varphi }}\approx 0.618$
be the side length of a regular decagon inscribed in the unit circle, where $\varphi $ is the golden ratio. Let
$b=2\sin {\frac {\pi }{6}}=1$
be the side length of a regular hexagon in the unit circle, and let
$c=2\sin {\frac {\pi }{5}}={\sqrt {\frac {5-{\sqrt {5}}}{2}}}\approx 1.176$
be the side length of a regular pentagon in the unit circle. Then $a^{2}+b^{2}=c^{2}$, so these three lengths form the sides of a right triangle.[10] The same triangle forms half of a golden rectangle. It may also be found within a regular icosahedron of side length $c$: the shortest line segment from any vertex $V$ to the plane of its five neighbors has length $a$, and the endpoints of this line segment together with any of the neighbors of $V$ form the vertices of a right triangle with sides $a$, $b$, and $c$.[11]
See also
• Integer triangle
• Spiral of Theodorus
References
1. Posamentier, Alfred S., and Lehman, Ingmar. The Secrets of Triangles. Prometheus Books, 2012.
2. Weisstein, Eric W. "Rational Triangle". MathWorld.
3. Cooke, Roger L. (2011). The History of Mathematics: A Brief Course (2nd ed.). John Wiley & Sons. pp. 237–238. ISBN 978-1-118-03024-0.
4. Gillings, Richard J. (1982). Mathematics in the Time of the Pharaohs. Dover. p. 161.
5. Forget, T. W.; Larkin, T. A. (1968), "Pythagorean triads of the form x, x + 1, z described by recurrence sequences" (PDF), Fibonacci Quarterly, 6 (3): 94–104.
6. Chen, C. C.; Peng, T. A. (1995), "Almost-isosceles right-angled triangles" (PDF), The Australasian Journal of Combinatorics, 11: 263–267, MR 1327342.
7. (sequence A001652 in the OEIS)
8. Nyblom, M. A. (1998), "A note on the set of almost-isosceles right-angled triangles" (PDF), The Fibonacci Quarterly, 36 (4): 319–322, MR 1640364.
9. Beauregard, Raymond A.; Suryanarayan, E. R. (1997), "Arithmetic triangles", Mathematics Magazine, 70 (2): 105–115, doi:10.2307/2691431, JSTOR 2691431, MR 1448883.
10. Euclid's Elements, Book XIII, Proposition 10.
11. nLab: pentagon decagon hexagon identity.
External links
• 3 : 4 : 5 triangle
• 30–60–90 triangle
• 45–45–90 triangle – with interactive animations
| Wikipedia |
4D vector
In computer science, a 4D vector is a 4-component vector data type. Uses include homogeneous coordinates for 3-dimensional space in computer graphics, and red green blue alpha (RGBA) values for bitmap images with a color and alpha channel (as such they are widely used in computer graphics). They may also represent quaternions (useful for rotations) although the algebra they define is different.
Computer hardware support
Some microprocessors have hardware support for 4D vectors with instructions dealing with 4 lane single instruction, multiple data (SIMD) instructions, usually with a 128-bit data path and 32-bit floating point fields.[1]
Specific instructions (e.g., 4 element dot product) may facilitate the use of one 128-bit register to represent a 4D vector. For example, in chronological order: Hitachi SH4, PowerPC VMX128 extension,[2] and Intel x86 SSE4.[3]
Some 4-element vector engines (e.g., the PS2 vector units) went further with the ability to broadcast components as multiply sources, and cross product support.[4][5] Earlier generations of graphics processing unit (GPU) shader pipelines used very long instruction word (VLIW) instruction sets tailored for similar operations.
Software support
SIMD use for 4D vectors can be conveniently wrapped in a vector maths library (commonly implemented in C or C++)[6][7][8] commonly used in video game development, along with 4×4 matrix support. These are distinct from more general linear algebra libraries in other domains focussing on matrices of arbitrary size. Such libraries sometimes support 3D vectors padded to 4D or loading 3D data into 4D registers, with arithmetic mapped efficiently to SIMD operations by per platform intrinsic function implementations. There is choice between AOS and SOA approaches given the availability of 4 element registers, versus SIMD instructions that are usually tailored toward homogenous data.
Shading languages for graphics processing unit (GPU) programming usually have a 4D datatypes (along with 2D, 3D) with x-y-z-w accessors including permutes or swizzle access, e.g., allowing easy swapping of RGBA or ARGB formats, accessing two 2D vectors packed into one 4D vector, etc.[9] Modern GPUs have since moved to scalar single instruction, multiple threads (SIMT) pipelines (for more efficiency in general-purpose computing on graphics processing units (GPGPU)) but still support this programming model.[10]
See also
• Euclidean space
• Four-dimensional space
• Quaternion
• Dimension
• RGBA color space
• Tesseract
• 4×4 matrix
References
1. "intel SSE intrinsics".
2. "Putting It All Together: Anatomy of the XBox 360 Game Console (see VMX128 dot product)" (PDF).
3. "intel SSE4 dot product".
4. "VU0 user manual" (PDF).
5. "feasibility study on using the playstation 2 for scientific computing" (PDF).
6. "sce vectormath". GitHub. 19 April 2022.
7. "GLM (vector maths library)".
8. "Microsoft DirectX Maths".
9. "GLSL data types & swizzling".
10. "AMD graphics core next".
| Wikipedia |
Order-8 hexagonal tiling
In geometry, the order-8 hexagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {6,8}.
Order-8 hexagonal tiling
Poincaré disk model of the hyperbolic plane
TypeHyperbolic regular tiling
Vertex configuration68
Schläfli symbol{6,8}
Wythoff symbol8 | 6 2
Coxeter diagram
Symmetry group[8,6], (*862)
DualOrder-6 octagonal tiling
PropertiesVertex-transitive, edge-transitive, face-transitive
Uniform constructions
There are four uniform constructions of this tiling, three of them as constructed by mirror removal from the [8,6] kaleidoscope. Removing the mirror between the order 2 and 6 points, [6,8,1+], gives [(6,6,4)], (*664). Removing the mirror between the order 8 and 6 points, [6,1+,8], gives (*4232). Removing two mirrors as [6,8*], leaves remaining mirrors (*33333333).
Four uniform constructions of 6.6.6.6.6.6.6.6
Uniform
Coloring
Symmetry [6,8]
(*862)
[6,8,1+] = [(6,6,4)]
(*664)
=
[6,1+,8]
(*4232)
=
[6,8*]
(*33333333)
Symbol {6,8} {6,8}1⁄2 r(8,6,8) {6,8}1⁄8
Coxeter
diagram
= =
Symmetry
This tiling represents a hyperbolic kaleidoscope of 4 mirrors meeting as edges of a square, with eight squares around every vertex. This symmetry by orbifold notation is called (*444444) with 6 order-4 mirror intersections. In Coxeter notation can be represented as [8,6*], removing two of three mirrors (passing through the square center) in the [8,6] symmetry.
Related polyhedra and tiling
Uniform octagonal/hexagonal tilings
Symmetry: [8,6], (*862)
{8,6} t{8,6}
r{8,6} 2t{8,6}=t{6,8} 2r{8,6}={6,8} rr{8,6} tr{8,6}
Uniform duals
V86 V6.16.16 V(6.8)2 V8.12.12 V68 V4.6.4.8 V4.12.16
Alternations
[1+,8,6]
(*466)
[8+,6]
(8*3)
[8,1+,6]
(*4232)
[8,6+]
(6*4)
[8,6,1+]
(*883)
[(8,6,2+)]
(2*43)
[8,6]+
(862)
h{8,6} s{8,6} hr{8,6} s{6,8} h{6,8} hrr{8,6} sr{8,6}
Alternation duals
V(4.6)6 V3.3.8.3.8.3 V(3.4.4.4)2 V3.4.3.4.3.6 V(3.8)8 V3.45 V3.3.6.3.8
See also
Wikimedia Commons has media related to Order-8 hexagonal tiling.
• Uniform tilings in hyperbolic plane
• List of regular polytopes
References
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 19, The Hyperbolic Archimedean Tessellations)
• "Chapter 10: Regular honeycombs in hyperbolic space". The Beauty of Geometry: Twelve Essays. Dover Publications. 1999. ISBN 0-486-40919-8. LCCN 99035678.
External links
• Weisstein, Eric W. "Hyperbolic tiling". MathWorld.
• Weisstein, Eric W. "Poincaré hyperbolic disk". MathWorld.
• Hyperbolic and Spherical Tiling Gallery
• KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
• Hyperbolic Planar Tessellations, Don Hatch
Tessellation
Periodic
• Pythagorean
• Rhombille
• Schwarz triangle
• Rectangle
• Domino
• Uniform tiling and honeycomb
• Coloring
• Convex
• Kisrhombille
• Wallpaper group
• Wythoff
Aperiodic
• Ammann–Beenker
• Aperiodic set of prototiles
• List
• Einstein problem
• Socolar–Taylor
• Gilbert
• Penrose
• Pentagonal
• Pinwheel
• Quaquaversal
• Rep-tile and Self-tiling
• Sphinx
• Socolar
• Truchet
Other
• Anisohedral and Isohedral
• Architectonic and catoptric
• Circle Limit III
• Computer graphics
• Honeycomb
• Isotoxal
• List
• Packing
• Problems
• Domino
• Wang
• Heesch's
• Squaring
• Dividing a square into similar rectangles
• Prototile
• Conway criterion
• Girih
• Regular Division of the Plane
• Regular grid
• Substitution
• Voronoi
• Voderberg
By vertex type
Spherical
• 2n
• 33.n
• V33.n
• 42.n
• V42.n
Regular
• 2∞
• 36
• 44
• 63
Semi-
regular
• 32.4.3.4
• V32.4.3.4
• 33.42
• 33.∞
• 34.6
• V34.6
• 3.4.6.4
• (3.6)2
• 3.122
• 42.∞
• 4.6.12
• 4.82
Hyper-
bolic
• 32.4.3.5
• 32.4.3.6
• 32.4.3.7
• 32.4.3.8
• 32.4.3.∞
• 32.5.3.5
• 32.5.3.6
• 32.6.3.6
• 32.6.3.8
• 32.7.3.7
• 32.8.3.8
• 33.4.3.4
• 32.∞.3.∞
• 34.7
• 34.8
• 34.∞
• 35.4
• 37
• 38
• 3∞
• (3.4)3
• (3.4)4
• 3.4.62.4
• 3.4.7.4
• 3.4.8.4
• 3.4.∞.4
• 3.6.4.6
• (3.7)2
• (3.8)2
• 3.142
• 3.162
• (3.∞)2
• 3.∞2
• 42.5.4
• 42.6.4
• 42.7.4
• 42.8.4
• 42.∞.4
• 45
• 46
• 47
• 48
• 4∞
• (4.5)2
• (4.6)2
• 4.6.12
• 4.6.14
• V4.6.14
• 4.6.16
• V4.6.16
• 4.6.∞
• (4.7)2
• (4.8)2
• 4.8.10
• V4.8.10
• 4.8.12
• 4.8.14
• 4.8.16
• 4.8.∞
• 4.102
• 4.10.12
• 4.122
• 4.12.16
• 4.142
• 4.162
• 4.∞2
• (4.∞)2
• 54
• 55
• 56
• 5∞
• 5.4.6.4
• (5.6)2
• 5.82
• 5.102
• 5.122
• (5.∞)2
• 64
• 65
• 66
• 68
• 6.4.8.4
• (6.8)2
• 6.82
• 6.102
• 6.122
• 6.162
• 73
• 74
• 77
• 7.62
• 7.82
• 7.142
• 83
• 84
• 86
• 88
• 8.62
• 8.122
• 8.162
• ∞3
• ∞4
• ∞5
• ∞∞
• ∞.62
• ∞.82
| Wikipedia |
Four fours
Four fours is a mathematical puzzle, the goal of which is to find the simplest mathematical expression for every whole number from 0 to some maximum, using only common mathematical symbols and the digit four. No other digit is allowed. Most versions of the puzzle require that each expression have exactly four fours, but some variations require that each expression have some minimum number of fours. The puzzle requires skill and mathematical reasoning.
The first printed occurrence of the specific problem of four fours is in Knowledge: An Illustrated Magazine of Science in 1881.[1] A similar problem involving arranging four identical digits to equal a certain amount was given in Thomas Dilworth's popular 1734 textbook The Schoolmaster's Assistant, Being a Compendium of Arithmetic Both Practical and Theoretical.[2]
W. W. Rouse Ball described it in the 6th edition (1914) of his Mathematical Recreations and Essays. In this book it is described as a "traditional recreation".[3]
Rules
There are many variations of four fours; their primary difference is which mathematical symbols are allowed. Essentially all variations at least allow addition ("+"), subtraction ("−"), multiplication ("×"), division ("÷"), and parentheses, as well as concatenation (e.g., "44" is allowed). Most also allow the factorial ("!"), exponentiation (e.g. "444"), the decimal point (".") and the square root ("√") operation. Other operations allowed by some variations include the reciprocal function ("1/x"), subfactorial ("!" before the number: !4 equals 9), overline (an infinitely repeated digit), an arbitrary root, the square function ("sqr"), the cube function ("cube"), the cube root, the gamma function (Γ(), where Γ(x) = (x − 1)!), and percent ("%"). Thus
$4\%=0.04$
$sqr(4)=16$
$cube(4)=64$
${\sqrt {4}}=2$
$4!=24$
$\Gamma (4)=6$
$!4=9$
$4'={\frac {1}{4}}=0.25$
$.4=0.4={\frac {4}{10}}={\frac {2}{5}}$
$4.4=4{\frac {2}{5}}$
$.{\overline {4}}=.4444...={\frac {4}{9}}$
etc.
A common use of the overline in this problem is for this value:
$.{\overline {4}}=.4444...={\frac {4}{9}}$
Typically the successor function is not allowed since any integer above 4 is trivially reachable with it. Similarly, "log" operators are usually not allowed as they allow a general method to produce any non-negative integer. This works by noticing three things:
1) It is possible to take square roots repeatedly without using any additional 4s
2) A square root can also be written as the exponent (^(1/2))
3) Exponents have logarithms as their inverse.
$\underbrace {\sqrt {\sqrt {\cdots {\sqrt {4}}}}} _{n}=4^{(1/2)^{n}}$
Writing repeated square root in this form we can isolate n, which is the number of square roots:
$4^{(1/2)^{n}}$
We can isolate both exponents by using the base 4 logarithm:
$\log _{4}4^{(1/2)^{n}}$
This logarithm can be thought of as the answer to the question: "4 to what power gives me 4 to the half power to the n power?"
$4^{x}=4^{(1/2)^{n}}$
so we are now left with:
$(1/2)^{n}$
and now we can take a logarithm to isolate the exponent, n:
$n=\log _{(1/2)}(1/2)^{n}$
so, putting it all together:
$n=\log _{(1/2)}\log _{4}4^{(1/2)^{n}}$
Now, we can rewrite the base (1/2) with only 4s and the exponent (1/2) back to a square root:
$n=\log _{{\sqrt {4}}/4}\log _{4}\underbrace {\sqrt {\sqrt {\cdots {\sqrt {4}}}}} _{n}$
We have used four fours and now the number of square roots we add equals whatever non-negative integer we wanted.
Paul Bourke credits Ben Rudiak-Gould with a different description of how four fours can be solved using natural logarithms (ln(n)) to represent any positive integer n as:
$n=-{\sqrt {4}}{\frac {\ln \left[\left(\ln \underbrace {\sqrt {\sqrt {\cdots {\sqrt {4}}}}} _{n}\right)/\ln 4\right]}{\ln {4}}}$
Additional variants (usually no longer called "four fours") replace the set of digits ("4, 4, 4, 4") with some other set of digits, say of the birthyear of someone. For example, a variant using "1975" would require each expression to use one 1, one 9, one 7, and one 5.
Solutions
Here is a set of four fours solutions for the numbers 0 through 32, using typical rules. Some alternate solutions are listed here, although there are actually many more correct solutions. The entries in blue are those that use four integers 4 (rather than four digits 4) and the basic arithmetic operations. Numbers without blue entries have no solution under these constraints. Additionally, solutions that repeat operators are marked in italics.
0 = 4 ÷ 4 × 4 − 4 = 44 − 44
1 = 4 ÷ 4 + 4 − 4 = 44 ÷ 44
2 = 4 −(4 + 4)÷ 4 = (44 + 4)÷ 4!
3 = (4 × 4 − 4)÷ 4 = (4 + 4 + 4)÷ 4
4 = 4 + 4 ×(4 − 4) = −44 + 4!+ 4!
5 = (4 × 4 + 4)÷ 4 = (44 − 4!)÷ 4
6 = (4 + 4)÷ 4 + 4 = 4.4 + 4 ×.4
7 = 4 + 4 − 4 ÷ 4 = 44 ÷ 4 − 4
8 = 4 ÷ 4 × 4 + 4 = 4.4 −.4 + 4
9 = 4 ÷ 4 + 4 + 4 = 44 ÷ 4 −√4
10 = (4 + 4 + 4)−√4 = (44 − 4)÷ 4
11 = (4!×√4 − 4)÷ 4 = √4 ×(4!−√4)÷ 4
12 = 4 ×(4 − 4 ÷ 4) = (44 + 4)÷ 4
13 = (4!×√4 + 4)÷ 4 = (4 −.4)÷.4 + 4
14 = 4 × 4 − 4 ÷√4 = 4 ×(√4 +√4)−√4
15 = 4 × 4 − 4 ÷ 4 = 44 ÷ 4 + 4
16 = 4 × 4 + 4 − 4 = (44 − 4)×.4
17 = 4 × 4 + 4 ÷ 4 = (44 + 4!)÷ 4
18 = 4 × 4 + 4 −√4 = (44 ÷√4) − 4
19 = 4!−(4 + 4 ÷ 4) = (4 + 4 −.4)÷.4
20 = 4 ×(4 ÷ 4 + 4) = (44 − 4)÷√4
21 = 4!− 4 + 4 ÷ 4 = (44 −√4)÷√4
22 = 4!÷ 4 + 4 × 4 = 44 ÷(4 −√4)
23 = 4!+ 4 ÷ 4 −√4 = (44 +√4)÷√4
24 = 4 × 4 + 4 + 4 = (44 + 4)÷√4
25 = 4!− 4 ÷ 4 +√4 = (4 + 4 +√4)÷.4
26 = 4!+√4 + 4 - 4
27 = 4!+√4 +(4 ÷ 4)
28 = (4 + 4)× 4 − 4 = 4!+ 4 + 4 - 4
29 = 4!+ 4 +(4 ÷ 4)
30 = 4!+ 4 + 4 -√4
31 = 4!+(4!+ 4)÷ 4
32 = 4 × 4 + 4 × 4
Note that numbers with values less than one are not usually written with a leading zero. For example, "0.4" is usually written as ".4". This is because "0" is a digit, and in this puzzle only the digit "4" can be used.
There are also many other ways to find the answer for all of these. A given number will generally have a few possible solutions; any solution that meets the rules is acceptable. Some variations prefer the "fewest" number of operations, or prefer some operations to others. Others simply prefer "interesting" solutions, i.e., a surprising way to reach the goal.
Certain numbers, such as 113, 157, and 347, are particularly difficult to solve under typical rules. For 113, Wheeler suggests $\Gamma (\Gamma (4))-{\frac {4!+4}{4}}$.[4] A non-standard solution is $4(4!+4+4')$, where 4' is the multiplicative inverse of 4. (i.e. ${\frac {1}{4}}$) Another possible solution is ${\frac {((4!)!_{14})!_{127}}{(4!)!_{14}}}$, where $!_{14}$ and $!_{127}$ represent the 14th and 127th multifactorials respectively, and should technically be denoted with that many exclamation marks to adhere to the rules of the problem. Note that the number 113/16 can be written by three 4’s, but this does not help for 113 unless the square function (i.e. sq(4) = 16) is allowed.
The use of percent ("%") admits solutions for a much greater proportion of numbers; for example, 113 = (√4 + (√4 + 4!)%) ÷ (√4)%.
Algorithmics of the problem
This problem and its generalizations (like the five fives and the six sixes problem, both shown below) may be solved by a simple algorithm. The basic ingredients are hash tables that map rationals to strings. In these tables, the keys are the numbers being represented by some admissible combination of operators and the chosen digit d, e.g. four, and the values are strings that contain the actual formula. There is one table for each number n of occurrences of d. For example, when d=4, the hash table for two occurrences of d would contain the key-value pair 8 and 4+4, and the one for three occurrences, the key-value pair 2 and (4+4)/4 (strings shown in bold).
The task is then reduced to recursively computing these hash tables for increasing n, starting from n=1 and continuing up to e.g. n=4. The tables for n=1 and n=2 are special, because they contain primitive entries that are not the combination of other, smaller formulas, and hence they must be initialized properly, like so (for n=1)
T[4] := "4";
T[4/10] := ".4";
T[4/9] := ".4...";
and
T[44] := "44";.
(for n=2). Now there are two ways in which new entries may arise, either as a combination of existing ones through a binary operator, or by applying the factorial or square root operators (which does not use additional instances of d). The first case is treated by iterating over all pairs of subexpressions that use a total of n instances of d. For example, when n=4, we would check pairs (a,b) with a containing one instance of d and b three, and with a containing two instances of d and b two as well. We would then enter a+b, a-b, b-a, a*b, a/b, b/a) into the hash table, including parenthesis, for n=4. Here the sets A and B that contain a and b are calculated recursively, with n=1 and n=2 being the base case. Memoization is used to ensure that every hash table is only computed once.
The second case (factorials and roots) is treated with the help of an auxiliary function, which is invoked every time a value v is recorded. This function computes nested factorials and roots of v up to some maximum depth, restricted to rationals.
The last phase of the algorithm consists in iterating over the keys of the table for the desired value of n and extracting and sorting those keys that are integers. This algorithm was used to calculate the five fives and six sixes examples shown below. The more compact formula (in the sense of number of characters in the corresponding value) was chosen every time a key occurred more than once.
Excerpt from the solution to the five fives problem
139 = (((5+(5/5))!/5)-5)
140 = (.5*(5+(5*55)))
141 = ((5)!+((5+(5+.5))/.5))
142 = ((5)!+((55/.5)/5))
143 = ((((5+(5/5)))!-5)/5)
144 = ((((55/5)-5))!/5)
145 = ((5*(5+(5*5)))-5)
146 = ((5)!+((5/5)+(5*5)))
147 = ((5)!+((.5*55)-.5))
148 = ((5)!+(.5+(.5*55)))
149 = (5+(((5+(5/5)))!+5))
Excerpt from the solution to the six sixes problem
In the table below, the notation .6... represents the value 6/9 or 2/3 (recurring decimal 6).
241 = ((.6+((6+6)*(6+6)))/.6)
242 = ((6*(6+(6*6)))-(6/.6))
243 = (6+((6*(.6*66))-.6))
244 = (.6...*(6+(6*(66-6))))
245 = ((((6)!+((6)!+66))/6)-6)
246 = (66+(6*((6*6)-6)))
247 = (66+((6+((6)!/.6...))/6))
248 = (6*(6+(6*(6-(.6.../6)))))
249 = (.6+(6*(6+((6*6)-.6))))
250 = (((6*(6*6))-66)/.6)
251 = ((6*(6+(6*6)))-(6/6))
See also
• Krypto (game)
References
1. Pat Ballew, Before there were Four-Fours, there were four threes, and several others, Pat'sBlog, 30 December 2018.
2. Bellos, Alex (2016). Can You Solve My Problems?: A casebook of ingenious, perplexing and totally satisfying puzzles. Faber & Faber. p. 104. ISBN 978-1615193882. ...It contains the following puzzle. 'Says Jack to his brother Harry, "I can place four threes in such manner that they shall make just 34; can you do so too?"'
3. Ball, Walter William Rouse (1914). Mathematical Recreations and Essays, page 14 (6th ed.).
4. "The Definitive Four Fours Answer Key (by David A. Wheeler)". Dwheeler.com.
External links
• Bourke, Paul. "Four Fours Problem".
• Carver, Ruth. "Four Fours Puzzle". at MathForum.org
• "4444 (Four Fours)". Archived from the original on 2011-08-02. Retrieved 2010-06-04. Eyegate Gallery.
• Four fours
• four4s on GitHub
• "Online Implementation of the Four Fours Game".
| Wikipedia |
5-Con triangles
In geometry, two triangles are said to be 5-Con or almost congruent if they are not congruent triangles but they are similar triangles and share two side lengths (of non-corresponding sides). The 5-Con triangles are important examples for understanding the solution of triangles. Indeed, knowing three angles and two sides (but not their sequence) is not enough to determine a triangle up to congruence. A triangle is said to be 5-Con capable if there is another triangle which is almost congruent to it.
The 5-Con triangles have been discussed by Pawley:,[1] and later by Jones and Peterson.[2] They are briefly mentioned by Martin Gardner in his book Mathematical Circus. Another reference is the following exercise[3]
Explain how two triangles can have five parts (sides, angles) of one triangle congruent to five parts of the other triangle, but not be congruent triangles.
A similar exercise dates back to 1955,[4] and there an earlier reference is mentioned. It is however not possible to date the first occurrence of such standard exercises about triangles.
Examples
There are infinitely many pairs of 5-Con triangles, even up to scaling.
• The smallest 5-Con triangles with integer sides have side lengths (8; 12; 18) and (12; 18; 27). This is an example with obtuse triangles.
• An example of acute 5-Con triangles is (1000; 1100; 1210) and (1100; 1210; 1331).
• The 5-Con right triangles are exactly those obtained from scaling the pair $(1;m;m^{2})$ and $(m;m^{2};m^{3})$ with $m={\sqrt {\frac {1+{\sqrt {5}}}{2}}}={\sqrt {\phi }}$ where φ is the golden ratio. Consequently, these are Kepler triangles and there can be no right 5-Con triangles with integer sides.
• There are no 5-Con triangles that are equilateral or isosceles because that would require m = 1 and the 5-Con triangles would be congruent.
• There are no integer 5-Con triangles that are Heronian because the sides of integer 5-Con triangles are in a geometric progression.[5]
Results
1. Consider 5-Con triangles with side lengths $(a;b;c)$ and $(ma;mb;mc)$ where $m$ is the scaling factor, which we may suppose to be greater than $1$. We may also suppose $a\leq b\leq c$. Then we must have $b=ma$ and $c=mb$. The two triples of side lengths are then of the form:
$a(1;m;m^{2})\qquad \mathrm {and} \qquad a(m;m^{2};m^{3}).$
Conversely, for any $a>0$ and $1<m<{\frac {1+{\sqrt {5}}}{2}}$, such triples are the side lengths for 5-Con triangles. (Supposing without loss of generality that $a=1$, the greatest number in the first triple is $m^{2}$ and we only need to ensure $m^{2}<1+m$; the second triple is obtained from the first by scaling with $m$. So we have two triangles: They are clearly similar and exactly two of the three side lengths coincide.) Some references work with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle m^{-1}<1} instead, which leads to the inequalities ${\frac {{\sqrt {5}}-1}{2}}<m^{-1}<1$.
2. Any 5-Con capable triangle has different side lengths and the middle one is the geometric mean of the other two. The ratio between the largest and the middle side length is then equal to that between the middle and the smallest side length. We can use both this ratio and its inverse for scaling and obtaining an almost congruent triangle.
3. To study the possible shapes of 5-Con triangles, we may restrict to studying the triangles with side lengths
$(1;m;m^{2})\qquad \mathrm {where} \qquad 1<m<{\frac {1+{\sqrt {5}}}{2}}.$
The greatest angle is a strictly increasing continuous function of $m$ and varies from 60° to 180° (the limit cases are excluded). The right triangle corresponds to the value $m={\sqrt {\frac {1+{\sqrt {5}}}{2}}}$. For convenience, scale the triangle to obtain $(m^{-2};m^{-1};1)$, so that the largest side is fixed: The opposite vertex then moves along a curve as $m$ is varied, as shown in the figure.
4. Having two 5-Con triangles with integral sides amounts (in the above notation) to taking any rational number $1<m<{\frac {1+{\sqrt {5}}}{2}}$ and then choosing $a>0$ in such a way that $am^{3}$is an integer. The four involved integral side lengths $(a;am;am^{2};am^{3})$ do not share any common factor (the 4-tuple is then called primitive) if and only if they are of the form $(x^{3};x^{2}y;xy^{2};y^{3})$where $x,y$ are coprime positive integers.
Further remarks
Defining almost congruent triangles gives a binary relation on the set of triangles. This relation is clearly not reflexive, but it is symmetric. It is not transitive: As a counterexample, consider the three triangles with side lengths (8;12;18), (12;18;27), and (18;27;40.5).
There are infinite sequences of triangles such that any two subsequent terms are 5-Con triangles. It is easy to construct such a sequence from any 5-Con capable triangle: To get an ascending (respectively, descending) sequence, keep the two greatest (respectively, smallest) side lengths and simply choose a third greater (respectively, smaller) side length to obtain a similar triangle. One may easily arrange the triangles in the sequence in a neat way, for example in a spiral.[1]
One generalization is considering 7-Con quadrilaterals, i.e. non-congruent (and not necessarily similar) quadrilaterals where four angles and three sides coincide or, more generally, (2n-1)-Con n-gons.[1]
References
1. Pawley, Richard G. (1967). "5-Con triangles". The Mathematics Teacher. National Council of Teachers of Mathematics. 60 (5, May 1967): 438–443. doi:10.5951/MT.60.5.0438. JSTOR 27957592.
2. Jones, Robert T.; Peterson, Bruce B. (1974). "Almost Congruent Triangles". Mathematics Magazine. Mathematical Association of America. 47 (4, Sep. 1974): 180–189. doi:10.1080/0025570X.1974.11976393. JSTOR 2689207.
3. School Mathematics Study Group. (1960). Mathematics for high school--Geometry. Student's text. Geometry. Vol. 2. New Haven: Yale University Press. p. 382.
4. Thebault, Victor; Pinzka, C. F. (1955). "E1162". The American Mathematical Monthly. Mathematical Association of America. 62 (10): 729–730. doi:10.1080/00029890.1955.11988730. JSTOR 2307084.
5. Buchholz, R. H.; MacDougall, J. A. (1999). "Heron Quadrilaterals with sides in Arithmetic or Geometric progression". Bulletin of the Australian Mathematical Society. 59 (2): 263–269. doi:10.1017/s0004972700032883.
| Wikipedia |
Uniform 5-polytope
In geometry, a uniform 5-polytope is a five-dimensional uniform polytope. By definition, a uniform 5-polytope is vertex-transitive and constructed from uniform 4-polytope facets.
Graphs of regular and uniform 5-polytopes.
5-simplex
Rectified 5-simplex
Truncated 5-simplex
Cantellated 5-simplex
Runcinated 5-simplex
Stericated 5-simplex
5-orthoplex
Truncated 5-orthoplex
Rectified 5-orthoplex
Cantellated 5-orthoplex
Runcinated 5-orthoplex
Cantellated 5-cube
Runcinated 5-cube
Stericated 5-cube
5-cube
Truncated 5-cube
Rectified 5-cube
5-demicube
Truncated 5-demicube
Cantellated 5-demicube
Runcinated 5-demicube
The complete set of convex uniform 5-polytopes has not been determined, but many can be made as Wythoff constructions from a small set of symmetry groups. These construction operations are represented by the permutations of rings of the Coxeter diagrams.
History of discovery
• Regular polytopes: (convex faces)
• 1852: Ludwig Schläfli proved in his manuscript Theorie der vielfachen Kontinuität that there are exactly 3 regular polytopes in 5 or more dimensions.
• Convex semiregular polytopes: (Various definitions before Coxeter's uniform category)
• 1900: Thorold Gosset enumerated the list of nonprismatic semiregular convex polytopes with regular facets (convex regular 4-polytopes) in his publication On the Regular and Semi-Regular Figures in Space of n Dimensions.[1]
• Convex uniform polytopes:
• 1940-1988: The search was expanded systematically by H.S.M. Coxeter in his publication Regular and Semi-Regular Polytopes I, II, and III.
• 1966: Norman W. Johnson completed his Ph.D. Dissertation under Coxeter, The Theory of Uniform Polytopes and Honeycombs, University of Toronto
• Non-convex uniform polytopes:
• 1966: Johnson describes two non-convex uniform antiprisms in 5-space in his dissertation.[2]
• 2000-2023: Jonathan Bowers and other researchers search for other non-convex uniform 5-polytopes,[3] with a current count of 1297 known uniform 5-polytopes outside infinite families (convex and non-convex), excluding the prisms of the uniform 4-polytopes. The list is not proven complete.[4][5]
Regular 5-polytopes
Main article: List of regular polytopes § Five Dimensions
Regular 5-polytopes can be represented by the Schläfli symbol {p,q,r,s}, with s {p,q,r} 4-polytope facets around each face. There are exactly three such regular polytopes, all convex:
• {3,3,3,3} - 5-simplex
• {4,3,3,3} - 5-cube
• {3,3,3,4} - 5-orthoplex
There are no nonconvex regular polytopes in 5 dimensions or above.
Convex uniform 5-polytopes
Unsolved problem in mathematics:
What is the complete set of convex uniform 5-polytopes?[6]
(more unsolved problems in mathematics)
There are 104 known convex uniform 5-polytopes, plus a number of infinite families of duoprism prisms, and polygon-polyhedron duoprisms. All except the grand antiprism prism are based on Wythoff constructions, reflection symmetry generated with Coxeter groups.
Symmetry of uniform 5-polytopes in four dimensions
The 5-simplex is the regular form in the A5 family. The 5-cube and 5-orthoplex are the regular forms in the B5 family. The bifurcating graph of the D5 family contains the 5-orthoplex, as well as a 5-demicube which is an alternated 5-cube.
Each reflective uniform 5-polytope can be constructed in one or more reflective point group in 5 dimensions by a Wythoff construction, represented by rings around permutations of nodes in a Coxeter diagram. Mirror hyperplanes can be grouped, as seen by colored nodes, separated by even-branches. Symmetry groups of the form [a,b,b,a], have an extended symmetry, [[a,b,b,a]], like [3,3,3,3], doubling the symmetry order. Uniform polytopes in these group with symmetric rings contain this extended symmetry.
If all mirrors of a given color are unringed (inactive) in a given uniform polytope, it will have a lower symmetry construction by removing all of the inactive mirrors. If all the nodes of a given color are ringed (active), an alternation operation can generate a new 5-polytope with chiral symmetry, shown as "empty" circled nodes", but the geometry is not generally adjustable to create uniform solutions.
Fundamental families[7]
Group
symbol
OrderCoxeter
graph
Bracket
notation
Commutator
subgroup
Coxeter
number
(h)
Reflections
m=5/2 h[8]
A5 720[3,3,3,3][3,3,3,3]+615
D5 1920[3,3,31,1][3,3,31,1]+820
B5 3840[4,3,3,3]105 20
Uniform prisms
There are 5 finite categorical uniform prismatic families of polytopes based on the nonprismatic uniform 4-polytopes. There is one infinite family of 5-polytopes based on prisms of the uniform duoprisms {p}×{q}×{ }.
Coxeter
group
Order Coxeter
diagram
Coxeter
notation
Commutator
subgroup
Reflections
A4A1 120[3,3,3,2] = [3,3,3]×[ ][3,3,3]+10 1
D4A1 384[31,1,1,2] = [31,1,1]×[ ][31,1,1]+12 1
B4A1 768[4,3,3,2] = [4,3,3]×[ ]4 12 1
F4A1 2304[3,4,3,2] = [3,4,3]×[ ][3+,4,3+]12 12 1
H4A1 28800[5,3,3,2] = [3,4,3]×[ ][5,3,3]+60 1
Duoprismatic prisms (use 2p and 2q for evens)
I2(p)I2(q)A1 8pq[p,2,q,2] = [p]×[q]×[ ][p+,2,q+]p q 1
I2(2p)I2(q)A1 16pq[2p,2,q,2] = [2p]×[q]×[ ]p p q 1
I2(2p)I2(2q)A1 32pq[2p,2,2q,2] = [2p]×[2q]×[ ]p p q q 1
Uniform duoprisms
There are 3 categorical uniform duoprismatic families of polytopes based on Cartesian products of the uniform polyhedra and regular polygons: {q,r}×{p}.
Coxeter
group
Order Coxeter
diagram
Coxeter
notation
Commutator
subgroup
Reflections
Prismatic groups (use 2p for even)
A3I2(p) 48p[3,3,2,p] = [3,3]×[p][(3,3)+,2,p+]6 p
A3I2(2p) 96p[3,3,2,2p] = [3,3]×[2p]6 p p
B3I2(p) 96p[4,3,2,p] = [4,3]×[p]3 6p
B3I2(2p) 192p[4,3,2,2p] = [4,3]×[2p]3 6 p p
H3I2(p) 240p[5,3,2,p] = [5,3]×[p][(5,3)+,2,p+]15 p
H3I2(2p) 480p[5,3,2,2p] = [5,3]×[2p]15 p p
Enumerating the convex uniform 5-polytopes
• Simplex family: A5 [34]
• 19 uniform 5-polytopes
• Hypercube/Orthoplex family: B5 [4,33]
• 31 uniform 5-polytopes
• Demihypercube D5/E5 family: [32,1,1]
• 23 uniform 5-polytopes (8 unique)
• Polychoral prisms:
• 56 uniform 5-polytope (45 unique) constructions based on prismatic families: [3,3,3]×[ ], [4,3,3]×[ ], [5,3,3]×[ ], [31,1,1]×[ ].
• One non-Wythoffian - The grand antiprism prism is the only known non-Wythoffian convex uniform 5-polytope, constructed from two grand antiprisms connected by polyhedral prisms.
That brings the tally to: 19+31+8+45+1=104
In addition there are:
• Infinitely many uniform 5-polytope constructions based on duoprism prismatic families: [p]×[q]×[ ].
• Infinitely many uniform 5-polytope constructions based on duoprismatic families: [3,3]×[p], [4,3]×[p], [5,3]×[p].
The A5 family
Further information: A5 polytope
There are 19 forms based on all permutations of the Coxeter diagrams with one or more rings. (16+4-1 cases)
They are named by Norman Johnson from the Wythoff construction operations upon regular 5-simplex (hexateron).
The A5 family has symmetry of order 720 (6 factorial). 7 of the 19 figures, with symmetrically ringed Coxeter diagrams have doubled symmetry, order 1440.
The coordinates of uniform 5-polytopes with 5-simplex symmetry can be generated as permutations of simple integers in 6-space, all in hyperplanes with normal vector (1,1,1,1,1,1).
# Base point Johnson naming system
Bowers name and (acronym)
Coxeter diagram
k-face element counts Vertex
figure
Facet counts by location: [3,3,3,3]
4 3 2 1 0
[3,3,3]
(6)
[3,3,2]
(15)
[3,2,3]
(20)
[2,3,3]
(15)
[3,3,3]
(6)
Alt
1 (0,0,0,0,0,1) or (0,1,1,1,1,1) 5-simplex
hexateron (hix)
6 15 20 15 6
{3,3,3}
{3,3,3}
- - - -
2 (0,0,0,0,1,1) or (0,0,1,1,1,1) Rectified 5-simplex
rectified hexateron (rix)
12 45 80 60 15
t{3,3}×{ }
r{3,3,3}
- - -
{3,3,3}
3 (0,0,0,0,1,2) or (0,1,2,2,2,2) Truncated 5-simplex
truncated hexateron (tix)
12 45 80 75 30
Tetrah.pyr
t{3,3,3}
- - -
{3,3,3}
4 (0,0,0,1,1,2) or (0,1,1,2,2,2) Cantellated 5-simplex
small rhombated hexateron (sarx)
27 135 290 240 60
prism-wedge
rr{3,3,3}
- -
{ }×{3,3}
r{3,3,3}
5 (0,0,0,1,2,2) or (0,0,1,2,2,2) Bitruncated 5-simplex
bitruncated hexateron (bittix)
12 60 140 150 60
2t{3,3,3}
- - -
t{3,3,3}
6 (0,0,0,1,2,3) or (0,1,2,3,3,3) Cantitruncated 5-simplex
great rhombated hexateron (garx)
27 135 290 300 120
tr{3,3,3}
- -
{ }×{3,3}
t{3,3,3}
7 (0,0,1,1,1,2) or (0,1,1,1,2,2) Runcinated 5-simplex
small prismated hexateron (spix)
47 255 420 270 60
t0,3{3,3,3}
-
{3}×{3}
{ }×r{3,3}
r{3,3,3}
8 (0,0,1,1,2,3) or (0,1,2,2,3,3) Runcitruncated 5-simplex
prismatotruncated hexateron (pattix)
47 315 720 630 180
t0,1,3{3,3,3}
-
{6}×{3}
{ }×r{3,3}
rr{3,3,3}
9 (0,0,1,2,2,3) or (0,1,1,2,3,3) Runcicantellated 5-simplex
prismatorhombated hexateron (pirx)
47 255 570 540 180
t0,1,3{3,3,3}
-
{3}×{3}
{ }×t{3,3}
2t{3,3,3}
10 (0,0,1,2,3,4) or (0,1,2,3,4,4) Runcicantitruncated 5-simplex
great prismated hexateron (gippix)
47 315 810 900 360
Irr.5-cell
t0,1,2,3{3,3,3}
-
{3}×{6}
{ }×t{3,3}
tr{3,3,3}
11 (0,1,1,1,2,3) or (0,1,2,2,2,3) Steritruncated 5-simplex
celliprismated hexateron (cappix)
62 330 570 420 120
t{3,3,3}
{ }×t{3,3}
{3}×{6}
{ }×{3,3}
t0,3{3,3,3}
12 (0,1,1,2,3,4) or (0,1,2,3,3,4) Stericantitruncated 5-simplex
celligreatorhombated hexateron (cograx)
62 480 1140 1080 360
tr{3,3,3}
{ }×tr{3,3}
{3}×{6}
{ }×rr{3,3}
t0,1,3{3,3,3}
13 (0,0,0,1,1,1) Birectified 5-simplex
dodecateron (dot)
12 60 120 90 20
{3}×{3}
r{3,3,3}
- - -
r{3,3,3}
14 (0,0,1,1,2,2) Bicantellated 5-simplex
small birhombated dodecateron (sibrid)
32 180 420 360 90
rr{3,3,3}
-
{3}×{3}
-
rr{3,3,3}
15 (0,0,1,2,3,3) Bicantitruncated 5-simplex
great birhombated dodecateron (gibrid)
32 180 420 450 180
tr{3,3,3}
-
{3}×{3}
-
tr{3,3,3}
16 (0,1,1,1,1,2) Stericated 5-simplex
small cellated dodecateron (scad)
62 180 210 120 30
Irr.16-cell
{3,3,3}
{ }×{3,3}
{3}×{3}
{ }×{3,3}
{3,3,3}
17 (0,1,1,2,2,3) Stericantellated 5-simplex
small cellirhombated dodecateron (card)
62 420 900 720 180
rr{3,3,3}
{ }×rr{3,3}
{3}×{3}
{ }×rr{3,3}
rr{3,3,3}
18 (0,1,2,2,3,4) Steriruncitruncated 5-simplex
celliprismatotruncated dodecateron (captid)
62 450 1110 1080 360
t0,1,3{3,3,3}
{ }×t{3,3}
{6}×{6}
{ }×t{3,3}
t0,1,3{3,3,3}
19 (0,1,2,3,4,5) Omnitruncated 5-simplex
great cellated dodecateron (gocad)
62 540 1560 1800 720
Irr. {3,3,3}
t0,1,2,3{3,3,3}
{ }×tr{3,3}
{6}×{6}
{ }×tr{3,3}
t0,1,2,3{3,3,3}
Nonuniform Omnisnub 5-simplex
snub dodecateron (snod)
snub hexateron (snix)
422 2340 4080 2520 360 ht0,1,2,3{3,3,3} ht0,1,2,3{3,3,2} ht0,1,2,3{3,2,3} ht0,1,2,3{3,3,2} ht0,1,2,3{3,3,3} (360)
Irr. {3,3,3}
The B5 family
Further information: B5 polytope
The B5 family has symmetry of order 3840 (5!×25).
This family has 25−1=31 Wythoffian uniform polytopes generated by marking one or more nodes of the Coxeter diagram. Also added are 8 uniform polytopes generated as alternations with half the symmetry, which form a complete duplicate of the D5 family as ... = ..... (There are more alternations that are not listed because they produce only repetitions, as ... = .... and ... = .... These would give a complete duplication of the uniform 5-polytopes numbered 20 through 34 with symmetry broken in half.)
For simplicity it is divided into two subgroups, each with 12 forms, and 7 "middle" forms which equally belong in both.
The 5-cube family of 5-polytopes are given by the convex hulls of the base points listed in the following table, with all permutations of coordinates and sign taken. Each base point generates a distinct uniform 5-polytope. All coordinates correspond with uniform 5-polytopes of edge length 2.
# Base point Name
Coxeter diagram
Element counts Vertex
figure
Facet counts by location: [4,3,3,3]
43210
[4,3,3]
(10)
[4,3,2]
(40)
[4,2,3]
(80)
[2,3,3]
(80)
[3,3,3]
(32)
Alt
20 (0,0,0,0,1)√25-orthoplex
triacontaditeron (tac)
3280804010
{3,3,4}
----
{3,3,3}
21 (0,0,0,1,1)√2Rectified 5-orthoplex
rectified triacontaditeron (rat)
4224040024040
{ }×{3,4}
{3,3,4}
---
r{3,3,3}
22 (0,0,0,1,2)√2Truncated 5-orthoplex
truncated triacontaditeron (tot)
4224040028080
(Octah.pyr)
{3,3,4}
---
t{3,3,3}
23 (0,0,1,1,1)√2Birectified 5-cube
penteractitriacontaditeron (nit)
(Birectified 5-orthoplex)
4228064048080
{4}×{3}
r{3,3,4}
---
r{3,3,3}
24 (0,0,1,1,2)√2Cantellated 5-orthoplex
small rhombated triacontaditeron (sart)
8264015201200240
Prism-wedge
r{3,3,4}
{ }×{3,4}
--
rr{3,3,3}
25 (0,0,1,2,2)√2Bitruncated 5-orthoplex
bitruncated triacontaditeron (bittit)
42280720720240
t{3,3,4}
---
2t{3,3,3}
26 (0,0,1,2,3)√2Cantitruncated 5-orthoplex
great rhombated triacontaditeron (gart)
8264015201440480
t{3,3,4}
{ }×{3,4}
--
t0,1,3{3,3,3}
27 (0,1,1,1,1)√2Rectified 5-cube
rectified penteract (rin)
4220040032080
{3,3}×{ }
r{4,3,3}
---
{3,3,3}
28 (0,1,1,1,2)√2Runcinated 5-orthoplex
small prismated triacontaditeron (spat)
162120021601440320
r{4,3,3}
{ }×r{3,4}
{3}×{4}
t0,3{3,3,3}
29 (0,1,1,2,2)√2Bicantellated 5-cube
small birhombated penteractitriacontaditeron (sibrant)
(Bicantellated 5-orthoplex)
12284021601920480
rr{3,3,4}
-
{4}×{3}
-
rr{3,3,3}
30 (0,1,1,2,3)√2Runcitruncated 5-orthoplex
prismatotruncated triacontaditeron (pattit)
162144036803360960
rr{3,3,4}
{ }×r{3,4}
{6}×{4}
-
t0,1,3{3,3,3}
31 (0,1,2,2,2)√2Bitruncated 5-cube
bitruncated penteract (bittin)
42280720800320
2t{4,3,3}
---
t{3,3,3}
32 (0,1,2,2,3)√2Runcicantellated 5-orthoplex
prismatorhombated triacontaditeron (pirt)
162120029602880960
2t{4,3,3}
{ }×t{3,4}
{3}×{4}
-
t0,1,3{3,3,3}
33 (0,1,2,3,3)√2Bicantitruncated 5-cube
great birhombated triacontaditeron (gibrant)
(Bicantitruncated 5-orthoplex)
12284021602400960
tr{3,3,4}
-
{4}×{3}
-
rr{3,3,3}
34 (0,1,2,3,4)√2Runcicantitruncated 5-orthoplex
great prismated triacontaditeron (gippit)
1621440416048001920
tr{3,3,4}
{ }×t{3,4}
{6}×{4}
-
t0,1,2,3{3,3,3}
35 (1,1,1,1,1)5-cube
penteract (pent)
1040808032
{3,3,3}
{4,3,3}
----
36 (1,1,1,1,1)
+ (0,0,0,0,1)√2
Stericated 5-cube
small cellated penteractitriacontaditeron (scant)
(Stericated 5-orthoplex)
2428001040640160
Tetr.antiprm
{4,3,3}
{4,3}×{ }
{4}×{3}
{ }×{3,3}
{3,3,3}
37 (1,1,1,1,1)
+ (0,0,0,1,1)√2
Runcinated 5-cube
small prismated penteract (span)
202124021601440320
t0,3{4,3,3}
-
{4}×{3}
{ }×r{3,3}
r{3,3,3}
38 (1,1,1,1,1)
+ (0,0,0,1,2)√2
Steritruncated 5-orthoplex
celliprismated triacontaditeron (cappin)
242152028802240640
t0,3{4,3,3}
{4,3}×{ }
{6}×{4}
{ }×t{3,3}
t{3,3,3}
39 (1,1,1,1,1)
+ (0,0,1,1,1)√2
Cantellated 5-cube
small rhombated penteract (sirn)
12268015201280320
Prism-wedge
rr{4,3,3}
--
{ }×{3,3}
r{3,3,3}
40 (1,1,1,1,1)
+ (0,0,1,1,2)√2
Stericantellated 5-cube
cellirhombated penteractitriacontaditeron (carnit)
(Stericantellated 5-orthoplex)
242208047203840960
rr{4,3,3}
rr{4,3}×{ }
{4}×{3}
{ }×rr{3,3}
rr{3,3,3}
41 (1,1,1,1,1)
+ (0,0,1,2,2)√2
Runcicantellated 5-cube
prismatorhombated penteract (prin)
202124029602880960
t0,2,3{4,3,3}
-
{4}×{3}
{ }×t{3,3}
2t{3,3,3}
42 (1,1,1,1,1)
+ (0,0,1,2,3)√2
Stericantitruncated 5-orthoplex
celligreatorhombated triacontaditeron (cogart)
2422320592057601920
t0,2,3{4,3,3}
rr{4,3}×{ }
{6}×{4}
{ }×tr{3,3}
tr{3,3,3}
43 (1,1,1,1,1)
+ (0,1,1,1,1)√2
Truncated 5-cube
truncated penteract (tan)
42200400400160
Tetrah.pyr
t{4,3,3}
---
{3,3,3}
44 (1,1,1,1,1)
+ (0,1,1,1,2)√2
Steritruncated 5-cube
celliprismated triacontaditeron (capt)
242160029602240640
t{4,3,3}
t{4,3}×{ }
{8}×{3}
{ }×{3,3}
t0,3{3,3,3}
45 (1,1,1,1,1)
+ (0,1,1,2,2)√2
Runcitruncated 5-cube
prismatotruncated penteract (pattin)
202156037603360960
t0,1,3{4,3,3}
-
{8}×{3}
{ }×r{3,3}
rr{3,3,3}
46 (1,1,1,1,1)
+ (0,1,1,2,3)√2
Steriruncitruncated 5-cube
celliprismatotruncated penteractitriacontaditeron (captint)
(Steriruncitruncated 5-orthoplex)
2422160576057601920
t0,1,3{4,3,3}
t{4,3}×{ }
{8}×{6}
{ }×t{3,3}
t0,1,3{3,3,3}
47 (1,1,1,1,1)
+ (0,1,2,2,2)√2
Cantitruncated 5-cube
great rhombated penteract (girn)
12268015201600640
tr{4,3,3}
--
{ }×{3,3}
t{3,3,3}
48 (1,1,1,1,1)
+ (0,1,2,2,3)√2
Stericantitruncated 5-cube
celligreatorhombated penteract (cogrin)
2422400600057601920
tr{4,3,3}
tr{4,3}×{ }
{8}×{3}
{ }×rr{3,3}
t0,1,3{3,3,3}
49 (1,1,1,1,1)
+ (0,1,2,3,3)√2
Runcicantitruncated 5-cube
great prismated penteract (gippin)
2021560424048001920
t0,1,2,3{4,3,3}
-
{8}×{3}
{ }×t{3,3}
tr{3,3,3}
50 (1,1,1,1,1)
+ (0,1,2,3,4)√2
Omnitruncated 5-cube
great cellated penteractitriacontaditeron (gacnet)
(omnitruncated 5-orthoplex)
2422640816096003840
Irr. {3,3,3}
tr{4,3}×{ }
tr{4,3}×{ }
{8}×{6}
{ }×tr{3,3}
t0,1,2,3{3,3,3}
51 5-demicube
hemipenteract (hin)
=
26 120 160 80 16
r{3,3,3}
h{4,3,3}
- - - - (16)
{3,3,3}
52 Cantic 5-cube
Truncated hemipenteract (thin)
=
42 280 640 560 160
h2{4,3,3}
- - - (16)
r{3,3,3}
(16)
t{3,3,3}
53 Runcic 5-cube
Small rhombated hemipenteract (sirhin)
=
42 360 880 720 160
h3{4,3,3}
- - - (16)
r{3,3,3}
(16)
rr{3,3,3}
54 Steric 5-cube
Small prismated hemipenteract (siphin)
=
82 480 720 400 80
h{4,3,3}
h{4,3}×{}
- - (16)
{3,3,3}
(16)
t0,3{3,3,3}
55 Runcicantic 5-cube
Great rhombated hemipenteract (girhin)
=
42 360 1040 1200 480
h2,3{4,3,3}
- - - (16)
2t{3,3,3}
(16)
tr{3,3,3}
56 Stericantic 5-cube
Prismatotruncated hemipenteract (pithin)
=
82 720 1840 1680 480
h2{4,3,3}
h2{4,3}×{}
- - (16)
rr{3,3,3}
(16)
t0,1,3{3,3,3}
57 Steriruncic 5-cube
Prismatorhombated hemipenteract (pirhin)
=
82 560 1280 1120 320
h3{4,3,3}
h{4,3}×{}
- - (16)
t{3,3,3}
(16)
t0,1,3{3,3,3}
58 Steriruncicantic 5-cube
Great prismated hemipenteract (giphin)
=
82 720 2080 2400 960
h2,3{4,3,3}
h2{4,3}×{}
- - (16)
tr{3,3,3}
(16)
t0,1,2,3{3,3,3}
Nonuniform Alternated runcicantitruncated 5-orthoplex
Snub prismatotriacontaditeron (snippit)
Snub hemipenteract (snahin)
=
1122 6240 10880 6720 960
sr{3,3,4}
sr{2,3,4} sr{3,2,4} - ht0,1,2,3{3,3,3} (960)
Irr. {3,3,3}
Nonuniform Edge-snub 5-orthoplex
Pyritosnub penteract (pysnan)
1202 7920 15360 10560 1920 sr3{3,3,4} sr3{2,3,4} sr3{3,2,4}
s{3,3}×{ }
ht0,1,2,3{3,3,3} (960)
Irr. {3,3}×{ }
Nonuniform Snub 5-cube
Snub penteract (snan)
2162 12240 21600 13440 960 ht0,1,2,3{3,3,4} ht0,1,2,3{2,3,4} ht0,1,2,3{3,2,4} ht0,1,2,3{3,3,2} ht0,1,2,3{3,3,3} (1920)
Irr. {3,3,3}
The D5 family
Further information: D5 polytope
The D5 family has symmetry of order 1920 (5! x 24).
This family has 23 Wythoffian uniform polytopes, from 3×8-1 permutations of the D5 Coxeter diagram with one or more rings. 15 (2×8-1) are repeated from the B5 family and 8 are unique to this family, though even those 8 duplicate the alternations from the B5 family.
In the 15 repeats, both of the nodes terminating the length-1 branches are ringed, so the two kinds of element are identical and the symmetry doubles: the relations are ... = .... and ... = ..., creating a complete duplication of the uniform 5-polytopes 20 through 34 above. The 8 new forms have one such node ringed and one not, with the relation ... = ... duplicating uniform 5-polytopes 51 through 58 above.
# Coxeter diagram
Schläfli symbol symbols
Johnson and Bowers names
Element counts Vertex
figure
Facets by location: [31,2,1]
4 3 2 1 0
[3,3,3]
(16)
[31,1,1]
(10)
[3,3]×[ ]
(40)
[ ]×[3]×[ ]
(80)
[3,3,3]
(16)
Alt
[51] =
h{4,3,3,3}, 5-demicube
Hemipenteract (hin)
26 120 160 80 16
r{3,3,3}
{3,3,3}
h{4,3,3}
- - -
[52] =
h2{4,3,3,3}, cantic 5-cube
Truncated hemipenteract (thin)
42 280 640 560 160
t{3,3,3}
h2{4,3,3}
- -
r{3,3,3}
[53] =
h3{4,3,3,3}, runcic 5-cube
Small rhombated hemipenteract (sirhin)
42 360 880 720 160
rr{3,3,3}
h3{4,3,3}
- -
r{3,3,3}
[54] =
h4{4,3,3,3}, steric 5-cube
Small prismated hemipenteract (siphin)
82 480 720 400 80
t0,3{3,3,3}
h{4,3,3}
h{4,3}×{}
-
{3,3,3}
[55] =
h2,3{4,3,3,3}, runcicantic 5-cube
Great rhombated hemipenteract (girhin)
42 360 1040 1200 480
2t{3,3,3}
h2,3{4,3,3}
- -
tr{3,3,3}
[56] =
h2,4{4,3,3,3}, stericantic 5-cube
Prismatotruncated hemipenteract (pithin)
82 720 1840 1680 480
t0,1,3{3,3,3}
h2{4,3,3}
h2{4,3}×{}
-
rr{3,3,3}
[57] =
h3,4{4,3,3,3}, steriruncic 5-cube
Prismatorhombated hemipenteract (pirhin)
82 560 1280 1120 320
t0,1,3{3,3,3}
h3{4,3,3}
h{4,3}×{}
-
t{3,3,3}
[58] =
h2,3,4{4,3,3,3}, steriruncicantic 5-cube
Great prismated hemipenteract (giphin)
82 720 2080 2400 960
t0,1,2,3{3,3,3}
h2,3{4,3,3}
h2{4,3}×{}
-
tr{3,3,3}
Nonuniform =
ht0,1,2,3{3,3,3,4}, alternated runcicantitruncated 5-orthoplex
Snub hemipenteract (snahin)
1122 6240 10880 6720 960 ht0,1,2,3{3,3,3}
sr{3,3,4}
sr{2,3,4} sr{3,2,4} ht0,1,2,3{3,3,3} (960)
Irr. {3,3,3}
Uniform prismatic forms
There are 5 finite categorical uniform prismatic families of polytopes based on the nonprismatic uniform 4-polytopes. For simplicity, most alternations are not shown.
A4 × A1
This prismatic family has 9 forms:
The A1 x A4 family has symmetry of order 240 (2*5!).
# Coxeter diagram
and Schläfli
symbols
Name
Element counts
FacetsCellsFacesEdgesVertices
59 = {3,3,3}×{ }
5-cell prism (penp)
720302510
60 = r{3,3,3}×{ }
Rectified 5-cell prism (rappip)
1250907020
61 = t{3,3,3}×{ }
Truncated 5-cell prism (tippip)
125010010040
62 = rr{3,3,3}×{ }
Cantellated 5-cell prism (srippip)
2212025021060
63 = t0,3{3,3,3}×{ }
Runcinated 5-cell prism (spiddip)
3213020014040
64 = 2t{3,3,3}×{ }
Bitruncated 5-cell prism (decap)
126014015060
65 = tr{3,3,3}×{ }
Cantitruncated 5-cell prism (grippip)
22120280300120
66 = t0,1,3{3,3,3}×{ }
Runcitruncated 5-cell prism (prippip)
32180390360120
67 = t0,1,2,3{3,3,3}×{ }
Omnitruncated 5-cell prism (gippiddip)
32210540600240
B4 × A1
This prismatic family has 16 forms. (Three are shared with [3,4,3]×[ ] family)
The A1×B4 family has symmetry of order 768 (254!).
The last three snubs can be realised with equal-length edges, but turn out nonuniform anyway because some of their 4-faces are not uniform 4-polytopes.
# Coxeter diagram
and Schläfli
symbols
Name
Element counts
FacetsCellsFacesEdgesVertices
[16] = {4,3,3}×{ }
Tesseractic prism (pent)
(Same as 5-cube)
1040808032
68 = r{4,3,3}×{ }
Rectified tesseractic prism (rittip)
2613627222464
69 = t{4,3,3}×{ }
Truncated tesseractic prism (tattip)
26136304320128
70 = rr{4,3,3}×{ }
Cantellated tesseractic prism (srittip)
58360784672192
71 = t0,3{4,3,3}×{ }
Runcinated tesseractic prism (sidpithip)
82368608448128
72 = 2t{4,3,3}×{ }
Bitruncated tesseractic prism (tahp)
26168432480192
73 = tr{4,3,3}×{ }
Cantitruncated tesseractic prism (grittip)
58360880960384
74 = t0,1,3{4,3,3}×{ }
Runcitruncated tesseractic prism (prohp)
8252812161152384
75 = t0,1,2,3{4,3,3}×{ }
Omnitruncated tesseractic prism (gidpithip)
8262416961920768
76 = {3,3,4}×{ }
16-cell prism (hexip)
1864885616
77 = r{3,3,4}×{ }
Rectified 16-cell prism (icope)
(Same as 24-cell prism)
2614428821648
78 = t{3,3,4}×{ }
Truncated 16-cell prism (thexip)
2614431228896
79 = rr{3,3,4}×{ }
Cantellated 16-cell prism (ricope)
(Same as rectified 24-cell prism)
50336768672192
80 = tr{3,3,4}×{ }
Cantitruncated 16-cell prism (ticope)
(Same as truncated 24-cell prism)
50336864960384
81 = t0,1,3{3,3,4}×{ }
Runcitruncated 16-cell prism (prittip)
8252812161152384
82 = sr{3,3,4}×{ }
snub 24-cell prism (sadip)
1467681392960192
Nonuniform
rectified tesseractic alterprism (rita)
5028846428864
Nonuniform
truncated 16-cell alterprism (thexa)
2616838433696
Nonuniform
bitruncated tesseractic alterprism (taha)
50288624576192
F4 × A1
This prismatic family has 10 forms.
The A1 x F4 family has symmetry of order 2304 (2*1152). Three polytopes 85, 86 and 89 (green background) have double symmetry [[3,4,3],2], order 4608. The last one, snub 24-cell prism, (blue background) has [3+,4,3,2] symmetry, order 1152.
# Coxeter diagram
and Schläfli
symbols
Name
Element counts
FacetsCellsFacesEdgesVertices
[77] = {3,4,3}×{ }
24-cell prism (icope)
2614428821648
[79] = r{3,4,3}×{ }
rectified 24-cell prism (ricope)
50336768672192
[80] = t{3,4,3}×{ }
truncated 24-cell prism (ticope)
50336864960384
83 = rr{3,4,3}×{ }
cantellated 24-cell prism (sricope)
146100823042016576
84 = t0,3{3,4,3}×{ }
runcinated 24-cell prism (spiccup)
242115219201296288
85 = 2t{3,4,3}×{ }
bitruncated 24-cell prism (contip)
5043212481440576
86 = tr{3,4,3}×{ }
cantitruncated 24-cell prism (gricope)
1461008259228801152
87 = t0,1,3{3,4,3}×{ }
runcitruncated 24-cell prism (pricope)
2421584364834561152
88 = t0,1,2,3{3,4,3}×{ }
omnitruncated 24-cell prism (gippiccup)
2421872508857602304
[82] = s{3,4,3}×{ }
snub 24-cell prism (sadip)
1467681392960192
H4 × A1
This prismatic family has 15 forms:
The A1 x H4 family has symmetry of order 28800 (2*14400).
# Coxeter diagram
and Schläfli
symbols
Name
Element counts
FacetsCellsFacesEdgesVertices
89 = {5,3,3}×{ }
120-cell prism (hipe)
122960264030001200
90 = r{5,3,3}×{ }
Rectified 120-cell prism (rahipe)
7224560984084002400
91 = t{5,3,3}×{ }
Truncated 120-cell prism (thipe)
722456011040120004800
92 = rr{5,3,3}×{ }
Cantellated 120-cell prism (srahip)
19221296029040252007200
93 = t0,3{5,3,3}×{ }
Runcinated 120-cell prism (sidpixhip)
26421272022080168004800
94 = 2t{5,3,3}×{ }
Bitruncated 120-cell prism (xhip)
722576015840180007200
95 = tr{5,3,3}×{ }
Cantitruncated 120-cell prism (grahip)
192212960326403600014400
96 = t0,1,3{5,3,3}×{ }
Runcitruncated 120-cell prism (prixip)
264218720448804320014400
97 = t0,1,2,3{5,3,3}×{ }
Omnitruncated 120-cell prism (gidpixhip)
264222320628807200028800
98 = {3,3,5}×{ }
600-cell prism (exip)
602240031201560240
99 = r{3,3,5}×{ }
Rectified 600-cell prism (roxip)
72250401080079201440
100 = t{3,3,5}×{ }
Truncated 600-cell prism (texip)
722504011520100802880
101 = rr{3,3,5}×{ }
Cantellated 600-cell prism (srixip)
14421152028080252007200
102 = tr{3,3,5}×{ }
Cantitruncated 600-cell prism (grixip)
144211520316803600014400
103 = t0,1,3{3,3,5}×{ }
Runcitruncated 600-cell prism (prahip)
264218720448804320014400
Duoprism prisms
Uniform duoprism prisms, {p}×{q}×{ }, form an infinite class for all integers p,q>2. {4}×{4}×{ } makes a lower symmetry form of the 5-cube.
The extended f-vector of {p}×{q}×{ } is computed as (p,p,1)*(q,q,1)*(2,1) = (2pq,5pq,4pq+2p+2q,3pq+3p+3q,p+q+2,1).
Coxeter diagram Names Element counts
4-faces Cells Faces Edges Vertices
{p}×{q}×{ }[9]p+q+23pq+3p+3q4pq+2p+2q5pq2pq
{p}2×{ }2(p+1)3p(p+1)4p(p+1)5p22p2
{3}2×{ }836484518
{4}2×{ } = 5-cube1040808032
Grand antiprism prism
The grand antiprism prism is the only known convex non-Wythoffian uniform 5-polytope. It has 200 vertices, 1100 edges, 1940 faces (40 pentagons, 500 squares, 1400 triangles), 1360 cells (600 tetrahedra, 40 pentagonal antiprisms, 700 triangular prisms, 20 pentagonal prisms), and 322 hypercells (2 grand antiprisms , 20 pentagonal antiprism prisms , and 300 tetrahedral prisms ).
# Name Element counts
FacetsCellsFacesEdgesVertices
104grand antiprism prism (gappip)[10]322136019401100200
Notes on the Wythoff construction for the uniform 5-polytopes
Construction of the reflective 5-dimensional uniform polytopes are done through a Wythoff construction process, and represented through a Coxeter diagram, where each node represents a mirror. Nodes are ringed to imply which mirrors are active. The full set of uniform polytopes generated are based on the unique permutations of ringed nodes. Uniform 5-polytopes are named in relation to the regular polytopes in each family. Some families have two regular constructors and thus may have two ways of naming them.
Here are the primary operators available for constructing and naming the uniform 5-polytopes.
The last operation, the snub, and more generally the alternation, are the operation that can create nonreflective forms. These are drawn with "hollow rings" at the nodes.
The prismatic forms and bifurcating graphs can use the same truncation indexing notation, but require an explicit numbering system on the nodes for clarity.
Operation Extended
Schläfli symbol
Coxeter diagram Description
Parent t0{p,q,r,s} {p,q,r,s} Any regular 5-polytope
Rectified t1{p,q,r,s}r{p,q,r,s} The edges are fully truncated into single points. The 5-polytope now has the combined faces of the parent and dual.
Birectified t2{p,q,r,s}2r{p,q,r,s} Birectification reduces faces to points, cells to their duals.
Trirectified t3{p,q,r,s}3r{p,q,r,s} Trirectification reduces cells to points. (Dual rectification)
Quadrirectified t4{p,q,r,s}4r{p,q,r,s} Quadrirectification reduces 4-faces to points. (Dual)
Truncated t0,1{p,q,r,s}t{p,q,r,s} Each original vertex is cut off, with a new face filling the gap. Truncation has a degree of freedom, which has one solution that creates a uniform truncated 5-polytope. The 5-polytope has its original faces doubled in sides, and contains the faces of the dual.
Cantellated t0,2{p,q,r,s}rr{p,q,r,s} In addition to vertex truncation, each original edge is beveled with new rectangular faces appearing in their place.
Runcinated t0,3{p,q,r,s} Runcination reduces cells and creates new cells at the vertices and edges.
Stericated t0,4{p,q,r,s}2r2r{p,q,r,s} Sterication reduces facets and creates new facets (hypercells) at the vertices and edges in the gaps. (Same as expansion operation for 5-polytopes.)
Omnitruncated t0,1,2,3,4{p,q,r,s} All four operators, truncation, cantellation, runcination, and sterication are applied.
Half h{2p,3,q,r} Alternation, same as
Cantic h2{2p,3,q,r} Same as
Runcic h3{2p,3,q,r} Same as
Runcicantic h2,3{2p,3,q,r} Same as
Steric h4{2p,3,q,r} Same as
Steriruncic h3,4{2p,3,q,r} Same as
Stericantic h2,4{2p,3,q,r} Same as
Steriruncicantic h2,3,4{2p,3,q,r} Same as
Snub s{p,2q,r,s} Alternated truncation
Snub rectified sr{p,q,2r,s} Alternated truncated rectification
ht0,1,2,3{p,q,r,s} Alternated runcicantitruncation
Full snub ht0,1,2,3,4{p,q,r,s} Alternated omnitruncation
Regular and uniform honeycombs
There are five fundamental affine Coxeter groups, and 13 prismatic groups that generate regular and uniform tessellations in Euclidean 4-space.[11][12]
Fundamental groups
# Coxeter group Coxeter diagram Forms
1${\tilde {A}}_{4}$[3[5]][(3,3,3,3,3)]7
2${\tilde {C}}_{4}$[4,3,3,4]19
3${\tilde {B}}_{4}$[4,3,31,1][4,3,3,4,1+] = 23 (8 new)
4${\tilde {D}}_{4}$[31,1,1,1][1+,4,3,3,4,1+] = 9 (0 new)
5${\tilde {F}}_{4}$[3,4,3,3]31 (21 new)
There are three regular honeycombs of Euclidean 4-space:
• tesseractic honeycomb, with symbols {4,3,3,4}, = . There are 19 uniform honeycombs in this family.
• 24-cell honeycomb, with symbols {3,4,3,3}, . There are 31 reflective uniform honeycombs in this family, and one alternated form.
• Truncated 24-cell honeycomb with symbols t{3,4,3,3},
• Snub 24-cell honeycomb, with symbols s{3,4,3,3}, and constructed by four snub 24-cell, one 16-cell, and five 5-cells at each vertex.
• 16-cell honeycomb, with symbols {3,3,4,3},
Other families that generate uniform honeycombs:
• There are 23 uniquely ringed forms, 8 new ones in the 16-cell honeycomb family. With symbols h{4,32,4} it is geometrically identical to the 16-cell honeycomb, =
• There are 7 uniquely ringed forms from the ${\tilde {A}}_{4}$, family, all new, including:
• 4-simplex honeycomb
• Truncated 4-simplex honeycomb
• Omnitruncated 4-simplex honeycomb
• There are 9 uniquely ringed forms in the ${\tilde {D}}_{4}$: [31,1,1,1] family, two new ones, including the quarter tesseractic honeycomb, = , and the bitruncated tesseractic honeycomb, = .
Non-Wythoffian uniform tessellations in 4-space also exist by elongation (inserting layers), and gyration (rotating layers) from these reflective forms.
Prismatic groups
# Coxeter group Coxeter diagram
1${\tilde {C}}_{3}$×${\tilde {I}}_{1}$[4,3,4,2,∞]
2${\tilde {B}}_{3}$×${\tilde {I}}_{1}$[4,31,1,2,∞]
3${\tilde {A}}_{3}$×${\tilde {I}}_{1}$[3[4],2,∞]
4${\tilde {C}}_{2}$×${\tilde {I}}_{1}$x${\tilde {I}}_{1}$[4,4,2,∞,2,∞]
5${\tilde {H}}_{2}$×${\tilde {I}}_{1}$x${\tilde {I}}_{1}$[6,3,2,∞,2,∞]
6${\tilde {A}}_{2}$×${\tilde {I}}_{1}$x${\tilde {I}}_{1}$[3[3],2,∞,2,∞]
7${\tilde {I}}_{1}$×${\tilde {I}}_{1}$x${\tilde {I}}_{1}$x${\tilde {I}}_{1}$[∞,2,∞,2,∞,2,∞]
8${\tilde {A}}_{2}$x${\tilde {A}}_{2}$[3[3],2,3[3]]
9${\tilde {A}}_{2}$×${\tilde {B}}_{2}$[3[3],2,4,4]
10${\tilde {A}}_{2}$×${\tilde {G}}_{2}$[3[3],2,6,3]
11${\tilde {B}}_{2}$×${\tilde {B}}_{2}$[4,4,2,4,4]
12${\tilde {B}}_{2}$×${\tilde {G}}_{2}$[4,4,2,6,3]
13${\tilde {G}}_{2}$×${\tilde {G}}_{2}$[6,3,2,6,3]
Regular and uniform hyperbolic honeycombs
Hyperbolic compact groups
There are 5 compact hyperbolic Coxeter groups of rank 5, each generating uniform honeycombs in hyperbolic 4-space as permutations of rings of the Coxeter diagrams.
${\widehat {AF}}_{4}$ = [(3,3,3,3,4)]:
${\bar {DH}}_{4}$ = [5,3,31,1]:
${\bar {H}}_{4}$ = [3,3,3,5]:
${\bar {BH}}_{4}$ = [4,3,3,5]:
${\bar {K}}_{4}$ = [5,3,3,5]:
There are 5 regular compact convex hyperbolic honeycombs in H4 space:[13]
Compact regular convex hyperbolic honeycombs
Honeycomb name Schläfli
Symbol
{p,q,r,s}
Coxeter diagram Facet
type
{p,q,r}
Cell
type
{p,q}
Face
type
{p}
Face
figure
{s}
Edge
figure
{r,s}
Vertex
figure
{q,r,s}
Dual
Order-5 5-cell (pente){3,3,3,5}{3,3,3}{3,3}{3}{5}{3,5}{3,3,5}{5,3,3,3}
Order-3 120-cell (hitte){5,3,3,3}{5,3,3}{5,3}{5}{3}{3,3}{3,3,3}{3,3,3,5}
Order-5 tesseractic (pitest){4,3,3,5}{4,3,3}{4,3}{4}{5}{3,5}{3,3,5}{5,3,3,4}
Order-4 120-cell (shitte){5,3,3,4}{5,3,3}{5,3}{5}{4}{3,4}{3,3,4}{4,3,3,5}
Order-5 120-cell (phitte){5,3,3,5}{5,3,3}{5,3}{5}{5}{3,5}{3,3,5}Self-dual
There are also 4 regular compact hyperbolic star-honeycombs in H4 space:
Compact regular hyperbolic star-honeycombs
Honeycomb name Schläfli
Symbol
{p,q,r,s}
Coxeter diagram Facet
type
{p,q,r}
Cell
type
{p,q}
Face
type
{p}
Face
figure
{s}
Edge
figure
{r,s}
Vertex
figure
{q,r,s}
Dual
Order-3 small stellated 120-cell{5/2,5,3,3}{5/2,5,3}{5/2,5}{5}{5}{3,3}{5,3,3}{3,3,5,5/2}
Order-5/2 600-cell{3,3,5,5/2}{3,3,5}{3,3}{3}{5/2}{5,5/2}{3,5,5/2}{5/2,5,3,3}
Order-5 icosahedral 120-cell{3,5,5/2,5}{3,5,5/2}{3,5}{3}{5}{5/2,5}{5,5/2,5}{5,5/2,5,3}
Order-3 great 120-cell{5,5/2,5,3}{5,5/2,5}{5,5/2}{5}{3}{5,3}{5/2,5,3}{3,5,5/2,5}
Hyperbolic paracompact groups
There are 9 paracompact hyperbolic Coxeter groups of rank 5, each generating uniform honeycombs in 4-space as permutations of rings of the Coxeter diagrams. Paracompact groups generate honeycombs with infinite facets or vertex figures.
${\bar {P}}_{4}$ = [3,3[4]]:
${\bar {BP}}_{4}$ = [4,3[4]]:
${\bar {FR}}_{4}$ = [(3,3,4,3,4)]:
${\bar {DP}}_{4}$ = [3[3]×[]]:
${\bar {N}}_{4}$ = [4,/3\,3,4]:
${\bar {O}}_{4}$ = [3,4,31,1]:
${\bar {S}}_{4}$ = [4,32,1]:
${\bar {M}}_{4}$ = [4,31,1,1]:
${\bar {R}}_{4}$ = [3,4,3,4]:
Notes
1. T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
2. Multidimensional Glossary, George Olshevsky
3. Bowers, Jonathan (2000). "Uniform Polychora" (PDF). In Reza Sarhagi (ed.). Bridges 2000. Bridges Conference. pp. 239–246.
4. Uniform Polytera, Jonathan Bowers
5. Uniform polytope
6. ACW (May 24, 2012), "Convex uniform 5-polytopes", Open Problem Garden, archived from the original on October 5, 2016, retrieved 2016-10-04
7. Regular and semi-regular polytopes III, p.315 Three finite groups of 5-dimensions
8. Coxeter, Regular polytopes, §12.6 The number of reflections, equation 12.61
9. "N,k-dippip".
10. "Gappip".
11. Regular polytopes, p.297. Table IV, Fundamental regions for irreducible groups generated by reflections.
12. Regular and Semiregular polytopes, II, pp.298-302 Four-dimensional honeycombs
13. Coxeter, The Beauty of Geometry: Twelve Essays, Chapter 10: Regular honeycombs in hyperbolic space, Summary tables IV p213
References
• T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900 (3 regular and one semiregular 4-polytope)
• A. Boole Stott: Geometrical deduction of semiregular from regular polytopes and space fillings, Verhandelingen of the Koninklijke academy van Wetenschappen width unit Amsterdam, Eerste Sectie 11,1, Amsterdam, 1910
• H.S.M. Coxeter:
• H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973 (p. 297 Fundamental regions for irreducible groups generated by reflections, Spherical and Euclidean)
• H.S.M. Coxeter, The Beauty of Geometry: Twelve Essays (Chapter 10: Regular honeycombs in hyperbolic space, Summary tables IV p213)
• Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6
• (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
• (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591] (p. 287 5D Euclidean groups, p. 298 Four-dimensionsal honeycombs)
• (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
• N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
• James E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge studies in advanced mathematics, 29 (1990) (Page 141, 6.9 List of hyperbolic Coxeter groups, figure 2)
External links
• Klitzing, Richard. "5D uniform polytopes (polytera)". – includes nonconvex forms as well as the duplicate constructions from the B5 and D5 families
Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron
Uniform polychoron Pentachoron 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell
Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221
Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321
Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421
Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds
Fundamental convex regular and uniform honeycombs in dimensions 2–9
Space Family ${\tilde {A}}_{n-1}$ ${\tilde {C}}_{n-1}$ ${\tilde {B}}_{n-1}$ ${\tilde {D}}_{n-1}$ ${\tilde {G}}_{2}$ / ${\tilde {F}}_{4}$ / ${\tilde {E}}_{n-1}$
E2 Uniform tiling {3[3]} δ3 hδ3 qδ3 Hexagonal
E3 Uniform convex honeycomb {3[4]} δ4 hδ4 qδ4
E4 Uniform 4-honeycomb {3[5]} δ5 hδ5 qδ5 24-cell honeycomb
E5 Uniform 5-honeycomb {3[6]} δ6 hδ6 qδ6
E6 Uniform 6-honeycomb {3[7]} δ7 hδ7 qδ7 222
E7 Uniform 7-honeycomb {3[8]} δ8 hδ8 qδ8 133 • 331
E8 Uniform 8-honeycomb {3[9]} δ9 hδ9 qδ9 152 • 251 • 521
E9 Uniform 9-honeycomb {3[10]} δ10 hδ10 qδ10
E10 Uniform 10-honeycomb {3[11]} δ11 hδ11 qδ11
En-1 Uniform (n-1)-honeycomb {3[n]} δn hδn qδn 1k2 • 2k1 • k21
| Wikipedia |
5-orthoplex
In five-dimensional geometry, a 5-orthoplex, or 5-cross polytope, is a five-dimensional polytope with 10 vertices, 40 edges, 80 triangle faces, 80 tetrahedron cells, 32 5-cell 4-faces.
Regular 5-orthoplex
(pentacross)
Orthogonal projection
inside Petrie polygon
TypeRegular 5-polytope
Familyorthoplex
Schläfli symbol{3,3,3,4}
{3,3,31,1}
Coxeter-Dynkin diagrams
4-faces32 {33}
Cells80 {3,3}
Faces80 {3}
Edges40
Vertices10
Vertex figure
16-cell
Petrie polygondecagon
Coxeter groupsBC5, [3,3,3,4]
D5, [32,1,1]
Dual5-cube
Propertiesconvex, Hanner polytope
It has two constructed forms, the first being regular with Schläfli symbol {33,4}, and the second with alternately labeled (checkerboarded) facets, with Schläfli symbol {3,3,31,1} or Coxeter symbol 211.
It is a part of an infinite family of polytopes, called cross-polytopes or orthoplexes. The dual polytope is the 5-hypercube or 5-cube.
Alternate names
• pentacross, derived from combining the family name cross polytope with pente for five (dimensions) in Greek.
• Triacontaditeron (or triacontakaiditeron) - as a 32-facetted 5-polytope (polyteron).
As a configuration
This configuration matrix represents the 5-orthoplex. The rows and columns correspond to vertices, edges, faces, cells and 4-faces. The diagonal numbers say how many of each element occur in the whole 5-orthoplex. The nondiagonal numbers say how many of the column's element occur in or at the row's element.[1][2]
${\begin{bmatrix}{\begin{matrix}10&8&24&32&16\\2&40&6&12&8\\3&3&80&4&4\\4&6&4&80&2\\5&10&10&5&32\end{matrix}}\end{bmatrix}}$
Cartesian coordinates
Cartesian coordinates for the vertices of a 5-orthoplex, centered at the origin are
(±1,0,0,0,0), (0,±1,0,0,0), (0,0,±1,0,0), (0,0,0,±1,0), (0,0,0,0,±1)
Construction
There are three Coxeter groups associated with the 5-orthoplex, one regular, dual of the penteract with the C5 or [4,3,3,3] Coxeter group, and a lower symmetry with two copies of 5-cell facets, alternating, with the D5 or [32,1,1] Coxeter group, and the final one as a dual 5-orthotope, called a 5-fusil which can have a variety of subsymmetries.
Name Coxeter diagram Schläfli symbol Symmetry Order Vertex figure(s)
regular 5-orthoplex {3,3,3,4} [3,3,3,4]3840
Quasiregular 5-orthoplex {3,3,31,1} [3,3,31,1]1920
5-fusil
{3,3,3,4}[4,3,3,3]3840
{3,3,4}+{}[4,3,3,2]768
{3,4}+{4}[4,3,2,4]384
{3,4}+2{}[4,3,2,2]192
2{4}+{}[4,2,4,2]128
{4}+3{}[4,2,2,2]64
5{} [2,2,2,2]32
Other images
orthographic projections
Coxeter plane B5 B4 / D5 B3 / D4 / A2
Graph
Dihedral symmetry [10] [8] [6]
Coxeter plane B2 A3
Graph
Dihedral symmetry [4] [4]
The perspective projection (3D to 2D) of a stereographic projection (4D to 3D) of the Schlegel diagram (5D to 4D) of the 5-orthoplex. 10 sets of 4 edges form 10 circles in the 4D Schlegel diagram: two of these circles are straight lines in the stereographic projection because they contain the center of projection.
Related polytopes and honeycombs
2k1 figures in n dimensions
Space Finite Euclidean Hyperbolic
n 3 4 5 6 7 8 9 10
Coxeter
group
E3=A2A1 E4=A4 E5=D5 E6 E7 E8 E9 = ${\tilde {E}}_{8}$ = E8+ E10 = ${\bar {T}}_{8}$ = E8++
Coxeter
diagram
Symmetry [3−1,2,1] [30,2,1] [[31,2,1]] [32,2,1] [33,2,1] [34,2,1] [35,2,1] [36,2,1]
Order 12 120 384 51,840 2,903,040 696,729,600 ∞
Graph - -
Name 2−1,1 201 211 221 231 241 251 261
This polytope is one of 31 uniform 5-polytopes generated from the B5 Coxeter plane, including the regular 5-cube and 5-orthoplex.
B5 polytopes
β5
t1β5
t2γ5
t1γ5
γ5
t0,1β5
t0,2β5
t1,2β5
t0,3β5
t1,3γ5
t1,2γ5
t0,4γ5
t0,3γ5
t0,2γ5
t0,1γ5
t0,1,2β5
t0,1,3β5
t0,2,3β5
t1,2,3γ5
t0,1,4β5
t0,2,4γ5
t0,2,3γ5
t0,1,4γ5
t0,1,3γ5
t0,1,2γ5
t0,1,2,3β5
t0,1,2,4β5
t0,1,3,4γ5
t0,1,2,4γ5
t0,1,2,3γ5
t0,1,2,3,4γ5
References
1. Coxeter, Regular Polytopes, sec 1.8 Configurations
2. Coxeter, Complex Regular Polytopes, p.117
• H.S.M. Coxeter:
• H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
• Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6
• (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
• (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
• (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
• Norman Johnson Uniform Polytopes, Manuscript (1991)
• N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. (1966)
• Klitzing, Richard. "5D uniform polytopes (polytera) x3o3o3o4o - tac".
External links
• Olshevsky, George. "Cross polytope". Glossary for Hyperspace. Archived from the original on 4 February 2007.
• Polytopes of Various Dimensions
• Multi-dimensional Glossary
Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron
Uniform polychoron Pentachoron 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell
Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221
Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321
Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421
Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds
| Wikipedia |
5-cubic honeycomb
In geometry, the 5-cubic honeycomb or penteractic honeycomb is the only regular space-filling tessellation (or honeycomb) in Euclidean 5-space. Four 5-cubes meet at each cubic cell, and it is more explicitly called an order-4 penteractic honeycomb.
5-cubic honeycomb
(no image)
TypeRegular 5-space honeycomb
Uniform 5-honeycomb
FamilyHypercube honeycomb
Schläfli symbol{4,33,4}
t0,5{4,33,4}
{4,3,3,31,1}
{4,3,4}×{∞}
{4,3,4}×{4,4}
{4,3,4}×{∞}(2)
{4,4}(2)×{∞}
{∞}(5)
Coxeter-Dynkin diagrams
5-face type{4,33} (5-cube)
4-face type{4,3,3} (tesseract)
Cell type{4,3} (cube)
Face type{4} (square)
Face figure{4,3} (octahedron)
Edge figure8 {4,3,3} (16-cell)
Vertex figure32 {4,33} (5-orthoplex)
Coxeter group${\tilde {C}}_{5}$
[4,33,4]
Dualself-dual
Propertiesvertex-transitive, edge-transitive, face-transitive, cell-transitive
It is analogous to the square tiling of the plane and to the cubic honeycomb of 3-space, and the tesseractic honeycomb of 4-space.
Constructions
There are many different Wythoff constructions of this honeycomb. The most symmetric form is regular, with Schläfli symbol {4,33,4}. Another form has two alternating 5-cube facets (like a checkerboard) with Schläfli symbol {4,3,3,31,1}. The lowest symmetry Wythoff construction has 32 types of facets around each vertex and a prismatic product Schläfli symbol {∞}(5).
Related polytopes and honeycombs
The [4,33,4], , Coxeter group generates 63 permutations of uniform tessellations, 35 with unique symmetry and 34 with unique geometry. The expanded 5-cubic honeycomb is geometrically identical to the 5-cubic honeycomb.
The 5-cubic honeycomb can be alternated into the 5-demicubic honeycomb, replacing the 5-cubes with 5-demicubes, and the alternated gaps are filled by 5-orthoplex facets.
It is also related to the regular 6-cube which exists in 6-space with 3 5-cubes on each cell. This could be considered as a tessellation on the 5-sphere, an order-3 penteractic honeycomb, {4,34}.
Tritruncated 5-cubic honeycomb
A tritruncated 5-cubic honeycomb, , contains all bitruncated 5-orthoplex facets and is the Voronoi tessellation of the D5* lattice. Facets can be identically colored from a doubled ${\tilde {C}}_{5}$×2, [[4,33,4]] symmetry, alternately colored from ${\tilde {C}}_{5}$, [4,33,4] symmetry, three colors from ${\tilde {B}}_{5}$, [4,3,3,31,1] symmetry, and 4 colors from ${\tilde {D}}_{5}$, [31,1,3,31,1] symmetry.
See also
• List of regular polytopes
Regular and uniform honeycombs in 5-space:
• 5-demicubic honeycomb
• 5-simplex honeycomb
• Truncated 5-simplex honeycomb
• Omnitruncated 5-simplex honeycomb
References
• Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8 p. 296, Table II: Regular honeycombs
• Kaleidoscopes: Selected Writings of H. S. M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6
• (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
Fundamental convex regular and uniform honeycombs in dimensions 2–9
Space Family ${\tilde {A}}_{n-1}$ ${\tilde {C}}_{n-1}$ ${\tilde {B}}_{n-1}$ ${\tilde {D}}_{n-1}$ ${\tilde {G}}_{2}$ / ${\tilde {F}}_{4}$ / ${\tilde {E}}_{n-1}$
E2 Uniform tiling {3[3]} δ3 hδ3 qδ3 Hexagonal
E3 Uniform convex honeycomb {3[4]} δ4 hδ4 qδ4
E4 Uniform 4-honeycomb {3[5]} δ5 hδ5 qδ5 24-cell honeycomb
E5 Uniform 5-honeycomb {3[6]} δ6 hδ6 qδ6
E6 Uniform 6-honeycomb {3[7]} δ7 hδ7 qδ7 222
E7 Uniform 7-honeycomb {3[8]} δ8 hδ8 qδ8 133 • 331
E8 Uniform 8-honeycomb {3[9]} δ9 hδ9 qδ9 152 • 251 • 521
E9 Uniform 9-honeycomb {3[10]} δ10 hδ10 qδ10
E10 Uniform 10-honeycomb {3[11]} δ11 hδ11 qδ11
En-1 Uniform (n-1)-honeycomb {3[n]} δn hδn qδn 1k2 • 2k1 • k21
| Wikipedia |
5-manifold
In mathematics, a 5-manifold is a 5-dimensional topological manifold, possibly with a piecewise linear or smooth structure.
Non-simply connected 5-manifolds are impossible to classify, as this is harder than solving the word problem for groups.[1] Simply connected compact 5-manifolds were first classified by Stephen Smale[2] and then in full generality by Dennis Barden,[3] while another proof was later given by Aleksey V. Zhubr.[4] This turns out to be easier than the 3- or 4-dimensional case: the 3-dimensional case is the Thurston geometrisation conjecture, and the 4-dimensional case was solved by Michael Freedman (1982) in the topological case,[5] but is a very hard unsolved problem in the smooth case.
In dimension 5, the smooth classification of simply connected manifolds is governed by classical algebraic topology. Namely, two simply connected, smooth 5-manifolds are diffeomorphic if and only if there exists an isomorphism of their second homology groups with integer coefficients, preserving the linking form and the second Stiefel–Whitney class. Moreover, any such isomorphism in second homology is induced by some diffeomorphism. It is undecidable if a given 5-manifold is homeomorphic to $S^{5}$, the 5-sphere.[1]
Examples
Here are some examples of smooth, closed, simply connected 5-manifolds:
• $S^{5}$, the 5-sphere.
• $S^{2}\times S^{3}$, the product of a 2-sphere with a 3-sphere.
• $S^{2}{\widetilde {\times }}S^{3}$, the total space of the non-trivial $S^{3}$-bundle over $S^{2}$.
• $\operatorname {SU} (3)/\operatorname {SO} (3)$, the homogeneous space obtained as the quotient of the special unitary group SU(3) by the rotation subgroup SO(3).
References
1. Stillwell, John (1993), Classical Topology and Combinatorial Group Theory, Graduate Texts in Mathematics, vol. 72, Springer, p. 247, ISBN 9780387979700.
2. Smale, Stephen (1962). "On the structure of 5-manifolds". Annals of Mathematics. 2. 75: 38–46. doi:10.2307/1970417. MR 0141133.
3. Barden, Dennis (1965). "Simply Connected Five-Manifolds". Annals of Mathematics. 2nd Ser. 82 (3): 365–385. doi:10.2307/1970702. JSTOR 1970702. MR 0184241.
4. Zhubr, Aleksey Viktorovich (2004). "On a paper of Barden". Journal of Mathematical Sciences. 119 (1): 35–44. doi:10.1023/B:JOTH.0000008739.46142.89. MR 1846073.
5. Freedman, Michael Hartley (1982). "The topology of four-dimensional manifolds". Journal of Differential Geometry. 17 (3): 357–453. ISSN 0022-040X. MR 0679066.
External links
• "5-manifolds: 1-connected". Manifold Atlas Project.
• "Linking form". Manifold Atlas Project.
| Wikipedia |
5-orthoplex honeycomb
In the geometry of hyperbolic 5-space, the 5-orthoplex honeycomb is one of five paracompact regular space-filling tessellations (or honeycombs). It is called paracompact because it has infinite vertex figures, with all vertices as ideal points at infinity. With Schläfli symbol {3,3,3,4,3}, it has three 5-orthoplexes around each cell. It is dual to the 24-cell honeycomb honeycomb.
5-orthoplex honeycomb
(No image)
TypeHyperbolic regular honeycomb
Schläfli symbol{3,3,3,4,3}
Coxeter diagram
=
5-faces {3,3,3,4}
4-faces {3,3,3}
Cells {3,3}
Faces {3}
Cell figure {3}
Face figure {4,3}
Edge figure {3,4,3}
Vertex figure {3,3,4,3}
Dual24-cell honeycomb honeycomb
Coxeter groupU5, [3,3,3,4,3]
PropertiesRegular
Related honeycombs
It is related to the regular Euclidean 4-space 16-cell honeycomb, {3,3,4,3}, with 16-cell (4-orthoplex) facets, and the regular 4-polytope 24-cell, {3,4,3} with octahedral (3-orthoplex) cell, and cube {4,3}, with (2-orthoplex) square faces.
See also
• List of regular polytopes
References
• Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. ISBN 0-486-61480-8. (Tables I and II: Regular polytopes and honeycombs, pp. 294–296)
• Coxeter, The Beauty of Geometry: Twelve Essays, Dover Publications, 1999 ISBN 0-486-40919-8 (Chapter 10: Regular honeycombs in hyperbolic space, Summary tables II,III,IV,V, p212-213)
| Wikipedia |
Regular number
Regular numbers are numbers that evenly divide powers of 60 (or, equivalently, powers of 30). Equivalently, they are the numbers whose only prime divisors are 2, 3, and 5. As an example, 602 = 3600 = 48 × 75, so as divisors of a power of 60 both 48 and 75 are regular.
Not to be confused with regular prime.
These numbers arise in several areas of mathematics and its applications, and have different names coming from their different areas of study.
• In number theory, these numbers are called 5-smooth, because they can be characterized as having only 2, 3, or 5 as their prime factors. This is a specific case of the more general k-smooth numbers, the numbers that have no prime factor greater than k.
• In the study of Babylonian mathematics, the divisors of powers of 60 are called regular numbers or regular sexagesimal numbers, and are of great importance in this area because of the sexagesimal (base 60) number system that the Babylonians used for writing their numbers, and that was central to Babylonian mathematics.
• In music theory, regular numbers occur in the ratios of tones in five-limit just intonation. In connection with music theory and related theories of architecture, these numbers have been called the harmonic whole numbers.
• In computer science, regular numbers are often called Hamming numbers, after Richard Hamming, who proposed the problem of finding computer algorithms for generating these numbers in ascending order. This problem has been used as a test case for functional programming.
Number theory
Formally, a regular number is an integer of the form $2^{i}\cdot 3^{j}\cdot 5^{k}$, for nonnegative integers $i$, $j$, and $k$. Such a number is a divisor of $60^{\max(\lceil i\,/2\rceil ,j,k)}$. The regular numbers are also called 5-smooth, indicating that their greatest prime factor is at most 5.[2] More generally, a k-smooth number is a number whose greatest prime factor is at most k.[3]
The first few regular numbers are[2]
1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 25, 27, 30, 32, 36, 40, 45, 48, 50, 54, 60, ... (sequence A051037 in the OEIS)
Several other sequences at the On-Line Encyclopedia of Integer Sequences have definitions involving 5-smooth numbers.[4]
Although the regular numbers appear dense within the range from 1 to 60, they are quite sparse among the larger integers. A regular number $n=2^{i}\cdot 3^{j}\cdot 5^{k}$ is less than or equal to some threshold $N$ if and only if the point $(i,j,k)$ belongs to the tetrahedron bounded by the coordinate planes and the plane
$i\ln 2+j\ln 3+k\ln 5\leq \ln N,$
as can be seen by taking logarithms of both sides of the inequality $2^{i}\cdot 3^{j}\cdot 5^{k}\leq N$. Therefore, the number of regular numbers that are at most $N$ can be estimated as the volume of this tetrahedron, which is
${\frac {\log _{2}N\,\log _{3}N\,\log _{5}N}{6}}.$
Even more precisely, using big O notation, the number of regular numbers up to $N$ is
${\frac {\left(\ln(N{\sqrt {30}})\right)^{3}}{6\ln 2\ln 3\ln 5}}+O(\log N),$
and it has been conjectured that the error term of this approximation is actually $O(\log \log N)$.[2] A similar formula for the number of 3-smooth numbers up to $N$ is given by Srinivasa Ramanujan in his first letter to G. H. Hardy.[5]
Babylonian mathematics
In the Babylonian sexagesimal notation, the reciprocal of a regular number has a finite representation. If $n$ divides $60^{k}$, then the sexagesimal representation of $1/n$ is just that for $60^{k}/n$, shifted by some number of places. This allows for easy division by these numbers: to divide by $n$, multiply by $1/n$, then shift.[6]
For instance, consider division by the regular number 54 = 2133. 54 is a divisor of 603, and 603/54 = 4000, so dividing by 54 in sexagesimal can be accomplished by multiplying by 4000 and shifting three places. In sexagesimal 4000 = 1×3600 + 6×60 + 40×1, or (as listed by Joyce) 1:6:40. Thus, 1/54, in sexagesimal, is 1/60 + 6/602 + 40/603, also denoted 1:6:40 as Babylonian notational conventions did not specify the power of the starting digit. Conversely 1/4000 = 54/603, so division by 1:6:40 = 4000 can be accomplished by instead multiplying by 54 and shifting three sexagesimal places.
The Babylonians used tables of reciprocals of regular numbers, some of which still survive.[7] These tables existed relatively unchanged throughout Babylonian times.[6] One tablet from Seleucid times, by someone named Inaqibıt-Anu, contains the reciprocals of 136 of the 231 six-place regular numbers whose first place is 1 or 2, listed in order. It also includes reciprocals of some numbers of more than six places, such as 323 (2 1 4 8 3 0 7 in sexagesimal), whose reciprocal has 17 sexagesimal digits. Noting the difficulty of both calculating these numbers and sorting them, Donald Knuth in 1972 hailed Inaqibıt-Anu as "the first man in history to solve a computational problem that takes longer than one second of time on a modern electronic computer!" (Two tables are also known giving approximations of reciprocals of non-regular numbers, one of which gives reciprocals for all the numbers from 56 to 80.)[8][9]
Although the primary reason for preferring regular numbers to other numbers involves the finiteness of their reciprocals, some Babylonian calculations other than reciprocals also involved regular numbers. For instance, tables of regular squares have been found[6] and the broken tablet Plimpton 322 has been interpreted by Neugebauer as listing Pythagorean triples $(p^{2}-q^{2},\,2pq,\,p^{2}+q^{2})$ generated by $p$ and $q$ both regular and less than 60.[10] Fowler and Robson discuss the calculation of square roots, such as how the Babylonians found an approximation to the square root of 2, perhaps using regular number approximations of fractions such as 17/12.[9]
Music theory
In music theory, the just intonation of the diatonic scale involves regular numbers: the pitches in a single octave of this scale have frequencies proportional to the numbers in the sequence 24, 27, 30, 32, 36, 40, 45, 48 of nearly consecutive regular numbers.[11] Thus, for an instrument with this tuning, all pitches are regular-number harmonics of a single fundamental frequency. This scale is called a 5-limit tuning, meaning that the interval between any two pitches can be described as a product 2i3j5k of powers of the prime numbers up to 5, or equivalently as a ratio of regular numbers.[12]
5-limit musical scales other than the familiar diatonic scale of Western music have also been used, both in traditional musics of other cultures and in modern experimental music: Honingh & Bod (2005) list 31 different 5-limit scales, drawn from a larger database of musical scales. Each of these 31 scales shares with diatonic just intonation the property that all intervals are ratios of regular numbers.[12] Euler's tonnetz provides a convenient graphical representation of the pitches in any 5-limit tuning, by factoring out the octave relationships (powers of two) so that the remaining values form a planar grid.[12] Some music theorists have stated more generally that regular numbers are fundamental to tonal music itself, and that pitch ratios based on primes larger than 5 cannot be consonant.[13] However the equal temperament of modern pianos is not a 5-limit tuning,[14] and some modern composers have experimented with tunings based on primes larger than five.[15]
In connection with the application of regular numbers to music theory, it is of interest to find pairs of regular numbers that differ by one. There are exactly ten such pairs $(x,x+1)$ and each such pair defines a superparticular ratio ${\tfrac {x+1}{x}}$ that is meaningful as a musical interval. These intervals are 2/1 (the octave), 3/2 (the perfect fifth), 4/3 (the perfect fourth), 5/4 (the just major third), 6/5 (the just minor third), 9/8 (the just major tone), 10/9 (the just minor tone), 16/15 (the just diatonic semitone), 25/24 (the just chromatic semitone), and 81/80 (the syntonic comma).[16]
In the Renaissance theory of universal harmony, musical ratios were used in other applications, including the architecture of buildings. In connection with the analysis of these shared musical and architectural ratios, for instance in the architecture of Palladio, the regular numbers have also been called the harmonic whole numbers.[17]
Algorithms
Algorithms for calculating the regular numbers in ascending order were popularized by Edsger Dijkstra. Dijkstra (1976, 1981) attributes to Hamming the problem of building the infinite ascending sequence of all 5-smooth numbers; this problem is now known as Hamming's problem, and the numbers so generated are also called the Hamming numbers. Dijkstra's ideas to compute these numbers are the following:
• The sequence of Hamming numbers begins with the number 1.
• The remaining values in the sequence are of the form $2h$, $3h$, and $5h$, where $h$ is any Hamming number.
• Therefore, the sequence $H$ may be generated by outputting the value 1, and then merging the sequences $2H$, $3H$, and $5H$.
This algorithm is often used to demonstrate the power of a lazy functional programming language, because (implicitly) concurrent efficient implementations, using a constant number of arithmetic operations per generated value, are easily constructed as described above. Similarly efficient strict functional or imperative sequential implementations are also possible whereas explicitly concurrent generative solutions might be non-trivial.[18]
In the Python programming language, lazy functional code for generating regular numbers is used as one of the built-in tests for correctness of the language's implementation.[19]
A related problem, discussed by Knuth (1972), is to list all $k$-digit sexagesimal numbers in ascending order (see #Babylonian mathematics above). In algorithmic terms, this is equivalent to generating (in order) the subsequence of the infinite sequence of regular numbers, ranging from $60^{k}$ to $60^{k+1}$.[8] See Gingerich (1965) for an early description of computer code that generates these numbers out of order and then sorts them;[20] Knuth describes an ad hoc algorithm, which he attributes to Bruins (1970), for generating the six-digit numbers more quickly but that does not generalize in a straightforward way to larger values of $k$.[8] Eppstein (2007) describes an algorithm for computing tables of this type in linear time for arbitrary values of $k$.[21]
Other applications
Heninger, Rains & Sloane (2006) show that, when $n$ is a regular number and is divisible by 8, the generating function of an $n$-dimensional extremal even unimodular lattice is an $n$th power of a polynomial.[22]
As with other classes of smooth numbers, regular numbers are important as problem sizes in computer programs for performing the fast Fourier transform, a technique for analyzing the dominant frequencies of signals in time-varying data. For instance, the method of Temperton (1992) requires that the transform length be a regular number.[23]
Book VIII of Plato's Republic involves an allegory of marriage centered on the highly regular number 604 = 12,960,000 and its divisors (see Plato's number). Later scholars have invoked both Babylonian mathematics and music theory in an attempt to explain this passage.[24]
Certain species of bamboo release large numbers of seeds in synchrony (a process called masting) at intervals that have been estimated as regular numbers of years, with different intervals for different species, including examples with intervals of 10, 15, 16, 30, 32, 48, 60, and 120 years.[25] It has been hypothesized that the biological mechanism for timing and synchronizing this process lends itself to smooth numbers, and in particular in this case to 5-smooth numbers. Although the estimated masting intervals for some other species of bamboo are not regular numbers of years, this may be explainable as measurement error.[25]
Notes
1. Inspired by similar diagrams by Erkki Kurenniemi in "Chords, scales, and divisor lattices".
2. Sloane "A051037".
3. Pomerance (1995).
4. OEIS search for sequences involving 5-smoothness.
5. Berndt & Rankin (1995).
6. Aaboe (1965).
7. Sachs (1947).
8. Knuth (1972).
9. Fowler & Robson (1998).
10. See Conway & Guy (1996) for a popular treatment of this interpretation. Plimpton 322 has other interpretations, for which see its article, but all involve regular numbers.
11. Clarke (1877).
12. Honingh & Bod (2005).
13. Asmussen (2001), for instance, states that "within any piece of tonal music" all intervals must be ratios of regular numbers, echoing similar statements by much earlier writers such as Habens (1889). In the modern music theory literature this assertion is often attributed to Longuet-Higgins (1962), who used a graphical arrangement closely related to the tonnetz to organize 5-limit pitches.
14. Kopiez (2003).
15. Wolf (2003).
16. Halsey & Hewitt (1972) note that this follows from Størmer's theorem (Størmer 1897), and provide a proof for this case; see also Silver (1971).
17. Howard & Longair (1982).
18. See, e.g., Hemmendinger (1988) or Yuen (1992).
19. Function m235 in test_generators.py.
20. Gingerich (1965).
21. Eppstein (2007).
22. Heninger, Rains & Sloane (2006).
23. Temperton (1992).
24. Barton (1908); McClain (1974).
25. Veller, Nowak & Davis (2015).
References
• Aaboe, Asger (1965), "Some Seleucid mathematical tables (extended reciprocals and squares of regular numbers)", Journal of Cuneiform Studies, The American Schools of Oriental Research, 19 (3): 79–86, doi:10.2307/1359089, JSTOR 1359089, MR 0191779, S2CID 164195082.
• Asmussen, Robert (2001), Periodicity of sinusoidal frequencies as a basis for the analysis of Baroque and Classical harmony: a computer based study (PDF), Ph.D. thesis, University of Leeds.
• Barton, George A. (1908), "On the Babylonian origin of Plato's nuptial number", Journal of the American Oriental Society, American Oriental Society, 29: 210–219, doi:10.2307/592627, JSTOR 592627.
• Berndt, Bruce C.; Rankin, Robert Alexander, eds. (1995), Ramanujan: letters and commentary, History of mathematics, vol. 9, American Mathematical Society, p. 23, Bibcode:1995rlc..book.....B, ISBN 978-0-8218-0470-4.
• Bruins, E. M. (1970), "La construction de la grande table le valeurs réciproques AO 6456", in Finet, André (ed.), Actes de la XVIIe Rencontre Assyriologique Internationale, Comité belge de recherches en Mésopotamie, pp. 99–115.
• Clarke, A. R. (January 1877), "Just intonation", Nature, 15 (377): 253, Bibcode:1877Natur..15..253C, doi:10.1038/015253b0.
• Conway, John H.; Guy, Richard K. (1996), The Book of Numbers, Copernicus, pp. 172–176, ISBN 0-387-97993-X.
• Dijkstra, Edsger W. (1976), "17. An exercise attributed to R. W. Hamming", A Discipline of Programming, Prentice-Hall, pp. 129–134, ISBN 978-0132158718
• Dijkstra, Edsger W. (1981), Hamming's exercise in SASL (PDF), Report EWD792. Originally a privately circulated handwritten note.
• Eppstein, David (2007), The range-restricted Hamming problem.
• Fowler, David; Robson, Eleanor (1998), "Square Root Approximations in Old Babylonian Mathematics: YBC 7289 in Context" (PDF), Historia Mathematica, 25 (4): 366–378, doi:10.1006/hmat.1998.2209, page 375.
• Gingerich, Owen (1965), "Eleven-digit regular sexagesimals and their reciprocals", Transactions of the American Philosophical Society, American Philosophical Society, 55 (8): 3–38, doi:10.2307/1006080, JSTOR 1006080.
• Habens, Rev. W. J. (1889), "On the musical scale", Proceedings of the Musical Association, Royal Musical Association, 16: 16th Session, p. 1, JSTOR 765355.
• Halsey, G. D.; Hewitt, Edwin (1972), "More on the superparticular ratios in music", American Mathematical Monthly, Mathematical Association of America, 79 (10): 1096–1100, doi:10.2307/2317424, JSTOR 2317424, MR 0313189.
• Hemmendinger, David (1988), "The "Hamming problem" in Prolog", ACM SIGPLAN Notices, 23 (4): 81–86, doi:10.1145/44326.44335, S2CID 28906392.
• Heninger, Nadia; Rains, E. M.; Sloane, N. J. A. (2006), "On the integrality of nth roots of generating functions", Journal of Combinatorial Theory, Series A, 113 (8): 1732–1745, arXiv:math.NT/0509316, doi:10.1016/j.jcta.2006.03.018, MR 2269551, S2CID 15913795}.
• Honingh, Aline; Bod, Rens (2005), "Convexity and the well-formedness of musical objects", Journal of New Music Research, 34 (3): 293–303, doi:10.1080/09298210500280612, S2CID 16321292.
• Howard, Deborah; Longair, Malcolm (May 1982), "Harmonic proportion and Palladio's Quattro Libri", Journal of the Society of Architectural Historians, 41 (2): 116–143, doi:10.2307/989675, JSTOR 989675
• Knuth, D. E. (1972), "Ancient Babylonian algorithms" (PDF), Communications of the ACM, 15 (7): 671–677, doi:10.1145/361454.361514, S2CID 7829945. A correction appears in CACM 19(2), 1976, stating that the tablet does not contain all 231 of the numbers of interest. The article (corrected) with a brief addendum appears in Selected Papers on Computer Science, CSLI Lecture Notes 59, Cambridge Univ. Press, 1996, pp. 185–203, but without the Appendix that was included in the original paper.
• Kopiez, Reinhard (2003), "Intonation of harmonic intervals: adaptability of expert musicians to equal temperament and just intonation", Music Perception, 20 (4): 383–410, doi:10.1525/mp.2003.20.4.383
• Longuet-Higgins, H. C. (1962), "Letter to a musical friend", Music Review (August): 244–248.
• McClain, Ernest G. (1974), "Musical "Marriages" in Plato's "Republic"", Journal of Music Theory, Duke University Press, 18 (2): 242–272, doi:10.2307/843638, JSTOR 843638.
• Pomerance, Carl (1995), "The role of smooth numbers in number-theoretic algorithms", Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), Basel: Birkhäuser, pp. 411–422, MR 1403941.
• Sachs, A. J. (1947), "Babylonian mathematical texts. I. Reciprocals of regular sexagesimal numbers", Journal of Cuneiform Studies, The American Schools of Oriental Research, 1 (3): 219–240, doi:10.2307/1359434, JSTOR 1359434, MR 0022180, S2CID 163783242.
• Silver, A. L. Leigh (1971), "Musimatics or the nun's fiddle", American Mathematical Monthly, Mathematical Association of America, 78 (4): 351–357, doi:10.2307/2316896, JSTOR 2316896.
• Sloane, N. J. A. (ed.), "Sequence A051037 (5-smooth numbers)", The On-Line Encyclopedia of Integer Sequences, OEIS Foundation
• Størmer, Carl (1897), "Quelques théorèmes sur l'équation de Pell x2 − Dy2 = ±1 et leurs applications", Skrifter Videnskabs-selskabet (Christiania), Mat.-Naturv. Kl., I (2).
• Temperton, Clive (1992), "A generalized prime factor FFT algorithm for any N = 2p3q5r", SIAM Journal on Scientific and Statistical Computing, 13 (3): 676–686, doi:10.1137/0913039, S2CID 14764894.
• Veller, Carl; Nowak, Martin A.; Davis, Charles C. (May 2015), "Extended flowering intervals of bamboos evolved by discrete multiplication", Ecology Letters, 18 (7): 653–659, doi:10.1111/ele.12442, PMID 25963600
• Wolf, Daniel James (March 2003), "Alternative tunings, alternative tonalities", Contemporary Music Review, 22 (1–2): 3–14, doi:10.1080/0749446032000134715, S2CID 191457676
• Yuen, C. K. (1992), "Hamming numbers, lazy evaluation, and eager disposal", ACM SIGPLAN Notices, 27 (8): 71–75, doi:10.1145/142137.142151, S2CID 18283005.
External links
• Table of reciprocals of regular numbers up to 3600 from the web site of Professor David E. Joyce, Clark University.
• RosettaCode Generation of Hamming_numbers in ~ 50 programming languages
Divisibility-based sets of integers
Overview
• Integer factorization
• Divisor
• Unitary divisor
• Divisor function
• Prime factor
• Fundamental theorem of arithmetic
Factorization forms
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With many divisors
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Classes of natural numbers
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2-dimensional
centered
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| Wikipedia |
Cinquefoil knot
In knot theory, the cinquefoil knot, also known as Solomon's seal knot or the pentafoil knot, is one of two knots with crossing number five, the other being the three-twist knot. It is listed as the 51 knot in the Alexander-Briggs notation, and can also be described as the (5,2)-torus knot. The cinquefoil is the closed version of the double overhand knot.
Cinquefoil
Common nameDouble overhand knot
Arf invariant1
Braid length5
Braid no.2
Bridge no.2
Crosscap no.1
Crossing no.5
Genus2
Hyperbolic volume0
Stick no.8
Unknotting no.2
Conway notation[5]
A–B notation51
Dowker notation6, 8, 10, 2, 4
Last /Next41 / 52
Other
alternating, torus, fibered, prime, reversible
Properties
The cinquefoil is a prime knot. Its writhe is 5, and it is invertible but not amphichiral.[1] Its Alexander polynomial is
$\Delta (t)=t^{2}-t+1-t^{-1}+t^{-2}$,
its Conway polynomial is
$\nabla (z)=z^{4}+3z^{2}+1$,
and its Jones polynomial is
$V(q)=q^{-2}+q^{-4}-q^{-5}+q^{-6}-q^{-7}.$
These are the same as the Alexander, Conway, and Jones polynomials of the knot 10132. However, the Kauffman polynomial can be used to distinguish between these two knots.
History
The name “cinquefoil” comes from the five-petaled flowers of plants in the genus Potentilla.
See also
• Pentagram
• Trefoil knot
• 7₁ knot
• Skein relation
References
1. Weisstein, Eric W. "Solomon's Seal Knot". MathWorld.
Further reading
• A Pentafoil Knot at the Wayback Machine (archived June 4, 2004)
Knot theory (knots and links)
Hyperbolic
• Figure-eight (41)
• Three-twist (52)
• Stevedore (61)
• 62
• 63
• Endless (74)
• Carrick mat (818)
• Perko pair (10161)
• (−2,3,7) pretzel (12n242)
• Whitehead (52
1
)
• Borromean rings (63
2
)
• L10a140
• Conway knot (11n34)
Satellite
• Composite knots
• Granny
• Square
• Knot sum
Torus
• Unknot (01)
• Trefoil (31)
• Cinquefoil (51)
• Septafoil (71)
• Unlink (02
1
)
• Hopf (22
1
)
• Solomon's (42
1
)
Invariants
• Alternating
• Arf invariant
• Bridge no.
• 2-bridge
• Brunnian
• Chirality
• Invertible
• Crosscap no.
• Crossing no.
• Finite type invariant
• Hyperbolic volume
• Khovanov homology
• Genus
• Knot group
• Link group
• Linking no.
• Polynomial
• Alexander
• Bracket
• HOMFLY
• Jones
• Kauffman
• Pretzel
• Prime
• list
• Stick no.
• Tricolorability
• Unknotting no. and problem
Notation
and operations
• Alexander–Briggs notation
• Conway notation
• Dowker–Thistlethwaite notation
• Flype
• Mutation
• Reidemeister move
• Skein relation
• Tabulation
Other
• Alexander's theorem
• Berge
• Braid theory
• Conway sphere
• Complement
• Double torus
• Fibered
• Knot
• List of knots and links
• Ribbon
• Slice
• Sum
• Tait conjectures
• Twist
• Wild
• Writhe
• Surgery theory
• Category
• Commons
| Wikipedia |
Five lemma
In mathematics, especially homological algebra and other applications of abelian category theory, the five lemma is an important and widely used lemma about commutative diagrams. The five lemma is not only valid for abelian categories but also works in the category of groups, for example.
The five lemma can be thought of as a combination of two other theorems, the four lemmas, which are dual to each other.
Statements
Consider the following commutative diagram in any abelian category (such as the category of abelian groups or the category of vector spaces over a given field) or in the category of groups.
The five lemma states that, if the rows are exact, m and p are isomorphisms, l is an epimorphism, and q is a monomorphism, then n is also an isomorphism.
The two four-lemmas state:
1. If the rows in the commutative diagram
are exact and m and p are epimorphisms and q is a monomorphism, then n is an epimorphism.
2. If the rows in the commutative diagram
are exact and m and p are monomorphisms and l is an epimorphism, then n is a monomorphism.
Proof
The method of proof we shall use is commonly referred to as diagram chasing.[1] We shall prove the five lemma by individually proving each of the two four lemmas.
To perform diagram chasing, we assume that we are in a category of modules over some ring, so that we may speak of elements of the objects in the diagram and think of the morphisms of the diagram as functions (in fact, homomorphisms) acting on those elements. Then a morphism is a monomorphism if and only if it is injective, and it is an epimorphism if and only if it is surjective. Similarly, to deal with exactness, we can think of kernels and images in a function-theoretic sense. The proof will still apply to any (small) abelian category because of Mitchell's embedding theorem, which states that any small abelian category can be represented as a category of modules over some ring. For the category of groups, just turn all additive notation below into multiplicative notation, and note that commutativity of abelian group is never used.
So, to prove (1), assume that m and p are surjective and q is injective.
• Let c′ be an element of C′.
• Since p is surjective, there exists an element d in D with p(d) = t(c′).
• By commutativity of the diagram, u(p(d)) = q(j(d)).
• Since im t = ker u by exactness, 0 = u(t(c′)) = u(p(d)) = q(j(d)).
• Since q is injective, j(d) = 0, so d is in ker j = im h.
• Therefore, there exists c in C with h(c) = d.
• Then t(n(c)) = p(h(c)) = t(c′). Since t is a homomorphism, it follows that t(c′ − n(c)) = 0.
• By exactness, c′ − n(c) is in the image of s, so there exists b′ in B′ with s(b′) = c′ − n(c).
• Since m is surjective, we can find b in B such that b′ = m(b).
• By commutativity, n(g(b)) = s(m(b)) = c′ − n(c).
• Since n is a homomorphism, n(g(b) + c) = n(g(b)) + n(c) = c′ − n(c) + n(c) = c′.
• Therefore, n is surjective.
Then, to prove (2), assume that m and p are injective and l is surjective.
• Let c in C be such that n(c) = 0.
• t(n(c)) is then 0.
• By commutativity, p(h(c)) = 0.
• Since p is injective, h(c) = 0.
• By exactness, there is an element b of B such that g(b) = c.
• By commutativity, s(m(b)) = n(g(b)) = n(c) = 0.
• By exactness, there is then an element a′ of A′ such that r(a′) = m(b).
• Since l is surjective, there is a in A such that l(a) = a′.
• By commutativity, m(f(a)) = r(l(a)) = m(b).
• Since m is injective, f(a) = b.
• So c = g(f(a)).
• Since the composition of g and f is trivial, c = 0.
• Therefore, n is injective.
Combining the two four lemmas now proves the entire five lemma.
Applications
The five lemma is often applied to long exact sequences: when computing homology or cohomology of a given object, one typically employs a simpler subobject whose homology/cohomology is known, and arrives at a long exact sequence which involves the unknown homology groups of the original object. This alone is often not sufficient to determine the unknown homology groups, but if one can compare the original object and sub object to well-understood ones via morphisms, then a morphism between the respective long exact sequences is induced, and the five lemma can then be used to determine the unknown homology groups.
See also
• Short five lemma, a special case of the five lemma for short exact sequences
• Snake lemma, another lemma proved by diagram chasing
• Nine lemma
Notes
1. Massey (1991). A basic course in algebraic topology. p. 184.
References
• Scott, W.R. (1987) [1964]. Group Theory. Dover. ISBN 978-0-486-65377-8.
• Massey, William S. (1991), A basic course in algebraic topology, Graduate texts in mathematics, vol. 127 (3rd ed.), Springer, ISBN 978-0-387-97430-9
| Wikipedia |
Pentagon
In geometry, a pentagon (from the Greek πέντε pente meaning five and γωνία gonia meaning angle[1]) is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°.
Pentagon
A cyclic pentagon
Edges and vertices5
A pentagon may be simple or self-intersecting. A self-intersecting regular pentagon (or star pentagon) is called a pentagram.
Regular pentagons
Regular pentagon
A regular pentagon
TypeRegular polygon
Edges and vertices5
Schläfli symbol{5}
Coxeter–Dynkin diagrams
Symmetry groupDihedral (D5), order 2×5
Internal angle (degrees)108°
PropertiesConvex, cyclic, equilateral, isogonal, isotoxal
Dual polygonSelf
A regular pentagon has Schläfli symbol {5} and interior angles of 108°.
A regular pentagon has five lines of reflectional symmetry, and rotational symmetry of order 5 (through 72°, 144°, 216° and 288°). The diagonals of a convex regular pentagon are in the golden ratio to its sides. Given its side length $t,$ its height $H$ (distance from one side to the opposite vertex), width $W$ (distance between two farthest separated points, which equals the diagonal length $D$) and circumradius $R$ are given by:
${\begin{aligned}H&={\frac {\sqrt {5+2{\sqrt {5}}}}{2}}~t\approx 1.539~t,\\W=D&={\frac {1+{\sqrt {5}}}{2}}~t\approx 1.618~t,\\W&={\sqrt {2-{\frac {2}{\sqrt {5}}}}}\cdot H\approx 1.051~H,\\R&={\sqrt {\frac {5+{\sqrt {5}}}{10}}}t\approx 0.8507~t,\\D&=R\ {\sqrt {\frac {5+{\sqrt {5}}}{2}}}=2R\cos 18^{\circ }=2R\cos {\frac {\pi }{10}}\approx 1.902~R.\end{aligned}}$
The area of a convex regular pentagon with side length $t$ is given by
${\begin{aligned}A&={\frac {t^{2}{\sqrt {25+10{\sqrt {5}}}}}{4}}={\frac {5t^{2}\tan 54^{\circ }}{4}}\\&={\frac {{\sqrt {5(5+2{\sqrt {5}})}}\;t^{2}}{4}}\approx 1.720~t^{2}.\end{aligned}}$
If the circumradius $R$ of a regular pentagon is given, its edge length $t$ is found by the expression
$t=R\ {\sqrt {\frac {5-{\sqrt {5}}}{2}}}=2R\sin 36^{\circ }=2R\sin {\frac {\pi }{5}}\approx 1.176~R,$
and its area is
$A={\frac {5R^{2}}{4}}{\sqrt {\frac {5+{\sqrt {5}}}{2}}};$
since the area of the circumscribed circle is $\pi R^{2},$ the regular pentagon fills approximately 0.7568 of its circumscribed circle.
Derivation of the area formula
The area of any regular polygon is:
$A={\frac {1}{2}}Pr$
where P is the perimeter of the polygon, and r is the inradius (equivalently the apothem). Substituting the regular pentagon's values for P and r gives the formula
$A={\frac {1}{2}}\cdot 5t\cdot {\frac {t\tan {\mathord {\left({\frac {3\pi }{10}}\right)}}}{2}}={\frac {5t^{2}\tan {\mathord {\left({\frac {3\pi }{10}}\right)}}}{4}}$
with side length t.
Inradius
Similar to every regular convex polygon, the regular convex pentagon has an inscribed circle. The apothem, which is the radius r of the inscribed circle, of a regular pentagon is related to the side length t by
$r={\frac {t}{2\tan {\mathord {\left({\frac {\pi }{5}}\right)}}}}={\frac {t}{2{\sqrt {5-{\sqrt {20}}}}}}\approx 0.6882\cdot t.$
Chords from the circumscribed circle to the vertices
Like every regular convex polygon, the regular convex pentagon has a circumscribed circle. For a regular pentagon with successive vertices A, B, C, D, E, if P is any point on the circumcircle between points B and C, then PA + PD = PB + PC + PE.
Point in plane
For an arbitrary point in the plane of a regular pentagon with circumradius $R$, whose distances to the centroid of the regular pentagon and its five vertices are $L$ and $d_{i}$ respectively, we have [2]
${\begin{aligned}\textstyle \sum _{i=1}^{5}d_{i}^{2}&=5\left(R^{2}+L^{2}\right),\\\textstyle \sum _{i=1}^{5}d_{i}^{4}&=5\left(\left(R^{2}+L^{2}\right)^{2}+2R^{2}L^{2}\right),\\\textstyle \sum _{i=1}^{5}d_{i}^{6}&=5\left(\left(R^{2}+L^{2}\right)^{3}+6R^{2}L^{2}\left(R^{2}+L^{2}\right)\right),\\\textstyle \sum _{i=1}^{5}d_{i}^{8}&=5\left(\left(R^{2}+L^{2}\right)^{4}+12R^{2}L^{2}\left(R^{2}+L^{2}\right)^{2}+6R^{4}L^{4}\right).\end{aligned}}$
If $d_{i}$ are the distances from the vertices of a regular pentagon to any point on its circumcircle, then [2]
$3\left(\textstyle \sum _{i=1}^{5}d_{i}^{2}\right)^{2}=10\textstyle \sum _{i=1}^{5}d_{i}^{4}.$
Geometrical constructions
The regular pentagon is constructible with compass and straightedge, as 5 is a Fermat prime. A variety of methods are known for constructing a regular pentagon. Some are discussed below.
Richmond's method
One method to construct a regular pentagon in a given circle is described by Richmond[3] and further discussed in Cromwell's Polyhedra.[4]
The top panel shows the construction used in Richmond's method to create the side of the inscribed pentagon. The circle defining the pentagon has unit radius. Its center is located at point C and a midpoint M is marked halfway along its radius. This point is joined to the periphery vertically above the center at point D. Angle CMD is bisected, and the bisector intersects the vertical axis at point Q. A horizontal line through Q intersects the circle at point P, and chord PD is the required side of the inscribed pentagon.
To determine the length of this side, the two right triangles DCM and QCM are depicted below the circle. Using Pythagoras' theorem and two sides, the hypotenuse of the larger triangle is found as $\scriptstyle {\sqrt {5}}/2$. Side h of the smaller triangle then is found using the half-angle formula:
$\tan(\phi /2)={\frac {1-\cos(\phi )}{\sin(\phi )}}\ ,$
where cosine and sine of ϕ are known from the larger triangle. The result is:
$h={\frac {{\sqrt {5}}-1}{4}}\ .$
If DP is truly the side of a regular pentagon, $m\angle \mathrm {CDP} =54^{\circ }$, so DP = 2 cos(54°), QD = DP cos(54°) = 2cos2(54°), and CQ = 1 − 2cos2(54°), which equals −cos(108°) by the cosine double angle formula. This is the cosine of 72°, which equals $\left({\sqrt {5}}-1\right)/4$ as desired.
Carlyle circles
Main article: Carlyle circle
The Carlyle circle was invented as a geometric method to find the roots of a quadratic equation.[5] This methodology leads to a procedure for constructing a regular pentagon. The steps are as follows:[6]
1. Draw a circle in which to inscribe the pentagon and mark the center point O.
2. Draw a horizontal line through the center of the circle. Mark the left intersection with the circle as point B.
3. Construct a vertical line through the center. Mark one intersection with the circle as point A.
4. Construct the point M as the midpoint of O and B.
5. Draw a circle centered at M through the point A. Mark its intersection with the horizontal line (inside the original circle) as the point W and its intersection outside the circle as the point V.
6. Draw a circle of radius OA and center W. It intersects the original circle at two of the vertices of the pentagon.
7. Draw a circle of radius OA and center V. It intersects the original circle at two of the vertices of the pentagon.
8. The fifth vertex is the rightmost intersection of the horizontal line with the original circle.
Steps 6–8 are equivalent to the following version, shown in the animation:
6a. Construct point F as the midpoint of O and W.
7a. Construct a vertical line through F. It intersects the original circle at two of the vertices of the pentagon. The third vertex is the rightmost intersection of the horizontal line with the original circle.
8a. Construct the other two vertices using the compass and the length of the vertex found in step 7a.
Euclid's method
A regular pentagon is constructible using a compass and straightedge, either by inscribing one in a given circle or constructing one on a given edge. This process was described by Euclid in his Elements circa 300 BC.[7][8]
Physical construction methods
• A regular pentagon may be created from just a strip of paper by tying an overhand knot into the strip and carefully flattening the knot by pulling the ends of the paper strip. Folding one of the ends back over the pentagon will reveal a pentagram when backlit.
• Construct a regular hexagon on stiff paper or card. Crease along the three diameters between opposite vertices. Cut from one vertex to the center to make an equilateral triangular flap. Fix this flap underneath its neighbor to make a pentagonal pyramid. The base of the pyramid is a regular pentagon.
Symmetry
The regular pentagon has Dih5 symmetry, order 10. Since 5 is a prime number there is one subgroup with dihedral symmetry: Dih1, and 2 cyclic group symmetries: Z5, and Z1.
These 4 symmetries can be seen in 4 distinct symmetries on the pentagon. John Conway labels these by a letter and group order.[9] Full symmetry of the regular form is r10 and no symmetry is labeled a1. The dihedral symmetries are divided depending on whether they pass through vertices (d for diagonal) or edges (p for perpendiculars), and i when reflection lines path through both edges and vertices. Cyclic symmetries in the middle column are labeled as g for their central gyration orders.
Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the g5 subgroup has no degrees of freedom but can be seen as directed edges.
Regular pentagram
Main article: Pentagram
A pentagram or pentangle is a regular star pentagon. Its Schläfli symbol is {5/2}. Its sides form the diagonals of a regular convex pentagon – in this arrangement the sides of the two pentagons are in the golden ratio.
Equilateral pentagons
Main article: Equilateral pentagon
An equilateral pentagon is a polygon with five sides of equal length. However, its five internal angles can take a range of sets of values, thus permitting it to form a family of pentagons. In contrast, the regular pentagon is unique up to similarity, because it is equilateral and it is equiangular (its five angles are equal).
Cyclic pentagons
A cyclic pentagon is one for which a circle called the circumcircle goes through all five vertices. The regular pentagon is an example of a cyclic pentagon. The area of a cyclic pentagon, whether regular or not, can be expressed as one fourth the square root of one of the roots of a septic equation whose coefficients are functions of the sides of the pentagon.[10][11][12]
There exist cyclic pentagons with rational sides and rational area; these are called Robbins pentagons. It has been proven that the diagonals of a Robbins pentagon must be either all rational or all irrational, and it is conjectured that all the diagonals must be rational.[13]
General convex pentagons
For all convex pentagons, the sum of the squares of the diagonals is less than 3 times the sum of the squares of the sides.[14]: p.75, #1854
Pentagons in tiling
Main article: Pentagon tiling
A regular pentagon cannot appear in any tiling of regular polygons. First, to prove a pentagon cannot form a regular tiling (one in which all faces are congruent, thus requiring that all the polygons be pentagons), observe that 360° / 108° = 31⁄3 (where 108° Is the interior angle), which is not a whole number; hence there exists no integer number of pentagons sharing a single vertex and leaving no gaps between them. More difficult is proving a pentagon cannot be in any edge-to-edge tiling made by regular polygons:
The maximum known packing density of a regular pentagon is $(5-{\sqrt {5}})/3\approx 0.921$, achieved by the double lattice packing shown. In a preprint released in 2016, Thomas Hales and Wöden Kusner announced a proof that this double lattice packing of the regular pentagon (known as the "pentagonal ice-ray" Chinese lattice design, dating from around 1900) has the optimal density among all packings of regular pentagons in the plane.[15] As of 2022, their proof had not yet been published in a peer-reviewed journal.
There are no combinations of regular polygons with 4 or more meeting at a vertex that contain a pentagon. For combinations with 3, if 3 polygons meet at a vertex and one has an odd number of sides, the other 2 must be congruent. The reason for this is that the polygons that touch the edges of the pentagon must alternate around the pentagon, which is impossible because of the pentagon's odd number of sides. For the pentagon, this results in a polygon whose angles are all (360 − 108) / 2 = 126°. To find the number of sides this polygon has, the result is 360 / (180 − 126) = 62⁄3, which is not a whole number. Therefore, a pentagon cannot appear in any tiling made by regular polygons.
There are 15 classes of pentagons that can monohedrally tile the plane. None of the pentagons have any symmetry in general, although some have special cases with mirror symmetry.
15 monohedral pentagonal tiles
12345
678910
1112131415
Pentagons in polyhedra
Ih Th Td O I D5d
Dodecahedron Pyritohedron Tetartoid Pentagonal icositetrahedron Pentagonal hexecontahedron Truncated trapezohedron
Pentagons in nature
Plants
• Pentagonal cross-section of okra.
• Morning glories, like many other flowers, have a pentagonal shape.
• Perigone tube of Rafflesia flower.
• The gynoecium of an apple contains five carpels, arranged in a five-pointed star
• Starfruit is another fruit with fivefold symmetry.
Animals
• A sea star. Many echinoderms have fivefold radial symmetry.
• Another example of echinoderm, a sea urchin endoskeleton.
• An illustration of brittle stars, also echinoderms with a pentagonal shape.
Minerals
• A Ho-Mg-Zn icosahedral quasicrystal formed as a pentagonal dodecahedron. The faces are true regular pentagons.
• A pyritohedral crystal of pyrite. A pyritohedron has 12 identical pentagonal faces that are not constrained to be regular.
Other examples
• The Pentagon, headquarters of the United States Department of Defense.
• Home plate of a baseball field
See also
• Associahedron; A pentagon is an order-4 associahedron
• Dodecahedron, a polyhedron whose regular form is composed of 12 pentagonal faces
• Golden ratio
• List of geometric shapes
• Pentagonal numbers
• Pentagram
• Pentagram map
• Pentastar, the Chrysler logo
• Pythagoras' theorem#Similar figures on the three sides
• Trigonometric constants for a pentagon
In-line notes and references
1. "pentagon, adj. and n." OED Online. Oxford University Press, June 2014. Web. 17 August 2014.
2. Meskhishvili, Mamuka (2020). "Cyclic Averages of Regular Polygons and Platonic Solids". Communications in Mathematics and Applications. 11: 335–355.
3. Richmond, Herbert W. (1893). "A Construction for a Regular Polygon of Seventeen Sides". The Quarterly Journal of Pure and Applied Mathematics. 26: 206–207.
4. Peter R. Cromwell (22 July 1999). Polyhedra. p. 63. ISBN 0-521-66405-5.
5. Eric W. Weisstein (2003). CRC concise encyclopedia of mathematics (2nd ed.). CRC Press. p. 329. ISBN 1-58488-347-2.
6. DeTemple, Duane W. (Feb 1991). "Carlyle circles and Lemoine simplicity of polygon constructions" (PDF). The American Mathematical Monthly. 98 (2): 97–108. doi:10.2307/2323939. JSTOR 2323939. Archived from the original (PDF) on 2015-12-21.
7. George Edward Martin (1998). Geometric constructions. Springer. p. 6. ISBN 0-387-98276-0.
8. Fitzpatrick, Richard (2008). Euklid's Elements of Geometry, Book 4, Proposition 11 (PDF). Translated by Richard Fitzpatrick. p. 119. ISBN 978-0-615-17984-1.
9. John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, (2008) The Symmetries of Things, ISBN 978-1-56881-220-5 (Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275-278)
10. Weisstein, Eric W. "Cyclic Pentagon." From MathWorld--A Wolfram Web Resource.
11. Robbins, D. P. (1994). "Areas of Polygons Inscribed in a Circle". Discrete and Computational Geometry. 12 (2): 223–236. doi:10.1007/bf02574377.
12. Robbins, D. P. (1995). "Areas of Polygons Inscribed in a Circle". The American Mathematical Monthly. 102 (6): 523–530. doi:10.2307/2974766. JSTOR 2974766.
• Buchholz, Ralph H.; MacDougall, James A. (2008), "Cyclic polygons with rational sides and area", Journal of Number Theory, 128 (1): 17–48, doi:10.1016/j.jnt.2007.05.005, MR 2382768.
13. Inequalities proposed in “Crux Mathematicorum”, .
14. Hales, Thomas; Kusner, Wöden (September 2016), Packings of regular pentagons in the plane, arXiv:1602.07220
External links
Look up pentagon in Wiktionary, the free dictionary.
Wikimedia Commons has media related to Pentagons.
• Weisstein, Eric W. "Pentagon". MathWorld.
• Animated demonstration constructing an inscribed pentagon with compass and straightedge.
• How to construct a regular pentagon with only a compass and straightedge.
• How to fold a regular pentagon using only a strip of paper
• Definition and properties of the pentagon, with interactive animation
• Renaissance artists' approximate constructions of regular pentagons
• Pentagon. How to calculate various dimensions of regular pentagons.
Polygons (List)
Triangles
• Acute
• Equilateral
• Ideal
• Isosceles
• Kepler
• Obtuse
• Right
Quadrilaterals
• Antiparallelogram
• Bicentric
• Crossed
• Cyclic
• Equidiagonal
• Ex-tangential
• Harmonic
• Isosceles trapezoid
• Kite
• Orthodiagonal
• Parallelogram
• Rectangle
• Right kite
• Right trapezoid
• Rhombus
• Square
• Tangential
• Tangential trapezoid
• Trapezoid
By number
of sides
1–10 sides
• Monogon (1)
• Digon (2)
• Triangle (3)
• Quadrilateral (4)
• Pentagon (5)
• Hexagon (6)
• Heptagon (7)
• Octagon (8)
• Nonagon (Enneagon, 9)
• Decagon (10)
11–20 sides
• Hendecagon (11)
• Dodecagon (12)
• Tridecagon (13)
• Tetradecagon (14)
• Pentadecagon (15)
• Hexadecagon (16)
• Heptadecagon (17)
• Octadecagon (18)
• Icosagon (20)
>20 sides
• Icositrigon (23)
• Icositetragon (24)
• Triacontagon (30)
• 257-gon
• Chiliagon (1000)
• Myriagon (10,000)
• 65537-gon
• Megagon (1,000,000)
• Apeirogon (∞)
Star polygons
• Pentagram
• Hexagram
• Heptagram
• Octagram
• Enneagram
• Decagram
• Hendecagram
• Dodecagram
Classes
• Concave
• Convex
• Cyclic
• Equiangular
• Equilateral
• Infinite skew
• Isogonal
• Isotoxal
• Magic
• Pseudotriangle
• Rectilinear
• Regular
• Reinhardt
• Simple
• Skew
• Star-shaped
• Tangential
• Weakly simple
Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron
Uniform polychoron Pentachoron 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell
Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221
Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321
Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421
Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds
| Wikipedia |
Duoprism
In geometry of 4 dimensions or higher, a double prism[1] or duoprism is a polytope resulting from the Cartesian product of two polytopes, each of two dimensions or higher. The Cartesian product of an n-polytope and an m-polytope is an (n+m)-polytope, where n and m are dimensions of 2 (polygon) or higher.
Set of uniform p-q duoprisms
TypePrismatic uniform 4-polytopes
Schläfli symbol{p}×{q}
Coxeter-Dynkin diagram
Cellsp q-gonal prisms,
q p-gonal prisms
Facespq squares,
p q-gons,
q p-gons
Edges2pq
Verticespq
Vertex figure
disphenoid
Symmetry[p,2,q], order 4pq
Dualp-q duopyramid
Propertiesconvex, vertex-uniform
Set of uniform p-p duoprisms
TypePrismatic uniform 4-polytope
Schläfli symbol{p}×{p}
Coxeter-Dynkin diagram
Cells2p p-gonal prisms
Facesp2 squares,
2p p-gons
Edges2p2
Verticesp2
Symmetry[p,2,p] = [2p,2+,2p], order 8p2
Dualp-p duopyramid
Propertiesconvex, vertex-uniform, Facet-transitive
The lowest-dimensional duoprisms exist in 4-dimensional space as 4-polytopes being the Cartesian product of two polygons in 2-dimensional Euclidean space. More precisely, it is the set of points:
$P_{1}\times P_{2}=\{(x,y,z,w)|(x,y)\in P_{1},(z,w)\in P_{2}\}$
where P1 and P2 are the sets of the points contained in the respective polygons. Such a duoprism is convex if both bases are convex, and is bounded by prismatic cells.
Nomenclature
Four-dimensional duoprisms are considered to be prismatic 4-polytopes. A duoprism constructed from two regular polygons of the same edge length is a uniform duoprism.
A duoprism made of n-polygons and m-polygons is named by prefixing 'duoprism' with the names of the base polygons, for example: a triangular-pentagonal duoprism is the Cartesian product of a triangle and a pentagon.
An alternative, more concise way of specifying a particular duoprism is by prefixing with numbers denoting the base polygons, for example: 3,5-duoprism for the triangular-pentagonal duoprism.
Other alternative names:
• q-gonal-p-gonal prism
• q-gonal-p-gonal double prism
• q-gonal-p-gonal hyperprism
The term duoprism is coined by George Olshevsky, shortened from double prism. John Horton Conway proposed a similar name proprism for product prism, a Cartesian product of two or more polytopes of dimension at least two. The duoprisms are proprisms formed from exactly two polytopes.
Example 16-16 duoprism
Schlegel diagram
Projection from the center of one 16-gonal prism, and all but one of the opposite 16-gonal prisms are shown.
net
The two sets of 16-gonal prisms are shown. The top and bottom faces of the vertical cylinder are connected when folded together in 4D.
Geometry of 4-dimensional duoprisms
A 4-dimensional uniform duoprism is created by the product of a regular n-sided polygon and a regular m-sided polygon with the same edge length. It is bounded by n m-gonal prisms and m n-gonal prisms. For example, the Cartesian product of a triangle and a hexagon is a duoprism bounded by 6 triangular prisms and 3 hexagonal prisms.
• When m and n are identical, the resulting duoprism is bounded by 2n identical n-gonal prisms. For example, the Cartesian product of two triangles is a duoprism bounded by 6 triangular prisms.
• When m and n are identically 4, the resulting duoprism is bounded by 8 square prisms (cubes), and is identical to the tesseract.
The m-gonal prisms are attached to each other via their m-gonal faces, and form a closed loop. Similarly, the n-gonal prisms are attached to each other via their n-gonal faces, and form a second loop perpendicular to the first. These two loops are attached to each other via their square faces, and are mutually perpendicular.
As m and n approach infinity, the corresponding duoprisms approach the duocylinder. As such, duoprisms are useful as non-quadric approximations of the duocylinder.
Nets
3-3
3-4
4-4
3-5
4-5
5-5
3-6
4-6
5-6
6-6
3-7
4-7
5-7
6-7
7-7
3-8
4-8
5-8
6-8
7-8
8-8
3-9
4-9
5-9
6-9
7-9
8-9
9-9
3-10
4-10
5-10
6-10
7-10
8-10
9-10
10-10
Perspective projections
A cell-centered perspective projection makes a duoprism look like a torus, with two sets of orthogonal cells, p-gonal and q-gonal prisms.
Schlegel diagrams
6-prism 6-6 duoprism
A hexagonal prism, projected into the plane by perspective, centered on a hexagonal face, looks like a double hexagon connected by (distorted) squares. Similarly a 6-6 duoprism projected into 3D approximates a torus, hexagonal both in plan and in section.
The p-q duoprisms are identical to the q-p duoprisms, but look different in these projections because they are projected in the center of different cells.
Schlegel diagrams
3-3
3-4
3-5
3-6
3-7
3-8
4-3
4-4
4-5
4-6
4-7
4-8
5-3
5-4
5-5
5-6
5-7
5-8
6-3
6-4
6-5
6-6
6-7
6-8
7-3
7-4
7-5
7-6
7-7
7-8
8-3
8-4
8-5
8-6
8-7
8-8
Orthogonal projections
Vertex-centered orthogonal projections of p-p duoprisms project into [2n] symmetry for odd degrees, and [n] for even degrees. There are n vertices projected into the center. For 4,4, it represents the A3 Coxeter plane of the tesseract. The 5,5 projection is identical to the 3D rhombic triacontahedron.
Orthogonal projection wireframes of p-p duoprisms
Odd
3-3 5-5 7-7 9-9
[3] [6] [5] [10] [7] [14] [9] [18]
Even
4-4 (tesseract) 6-6 8-8 10-10
[4] [8] [6] [12] [8] [16] [10] [20]
Related polytopes
The regular skew polyhedron, {4,4|n}, exists in 4-space as the n2 square faces of a n-n duoprism, using all 2n2 edges and n2 vertices. The 2n n-gonal faces can be seen as removed. (skew polyhedra can be seen in the same way by a n-m duoprism, but these are not regular.)
Duoantiprism
Like the antiprisms as alternated prisms, there is a set of 4-dimensional duoantiprisms: 4-polytopes that can be created by an alternation operation applied to a duoprism. The alternated vertices create nonregular tetrahedral cells, except for the special case, the 4-4 duoprism (tesseract) which creates the uniform (and regular) 16-cell. The 16-cell is the only convex uniform duoantiprism.
The duoprisms , t0,1,2,3{p,2,q}, can be alternated into , ht0,1,2,3{p,2,q}, the "duoantiprisms", which cannot be made uniform in general. The only convex uniform solution is the trivial case of p=q=2, which is a lower symmetry construction of the tesseract , t0,1,2,3{2,2,2}, with its alternation as the 16-cell, , s{2}s{2}.
The only nonconvex uniform solution is p=5, q=5/3, ht0,1,2,3{5,2,5/3}, , constructed from 10 pentagonal antiprisms, 10 pentagrammic crossed-antiprisms, and 50 tetrahedra, known as the great duoantiprism (gudap).[2][3]
Ditetragoltriates
Also related are the ditetragoltriates or octagoltriates, formed by taking the octagon (considered to be a ditetragon or a truncated square) to a p-gon. The octagon of a p-gon can be clearly defined if one assumes that the octagon is the convex hull of two perpendicular rectangles; then the p-gonal ditetragoltriate is the convex hull of two p-p duoprisms (where the p-gons are similar but not congruent, having different sizes) in perpendicular orientations. The resulting polychoron is isogonal and has 2p p-gonal prisms and p2 rectangular trapezoprisms (a cube with D2d symmetry) but cannot be made uniform. The vertex figure is a triangular bipyramid.
Double antiprismoids
Like the duoantiprisms as alternated duoprisms, there is a set of p-gonal double antiprismoids created by alternating the 2p-gonal ditetragoltriates, creating p-gonal antiprisms and tetrahedra while reinterpreting the non-corealmic triangular bipyramidal spaces as two tetrahedra. The resulting figure is generally not uniform except for two cases: the grand antiprism and its conjugate, the pentagrammic double antiprismoid (with p = 5 and 5/3 respectively), represented as the alternation of a decagonal or decagrammic ditetragoltriate. The vertex figure is a variant of the sphenocorona.
k_22 polytopes
The 3-3 duoprism, -122, is first in a dimensional series of uniform polytopes, expressed by Coxeter as k22 series. The 3-3 duoprism is the vertex figure for the second, the birectified 5-simplex. The fourth figure is a Euclidean honeycomb, 222, and the final is a paracompact hyperbolic honeycomb, 322, with Coxeter group [32,2,3], ${\bar {T}}_{7}$. Each progressive uniform polytope is constructed from the previous as its vertex figure.
k22 figures in n dimensions
Space Finite Euclidean Hyperbolic
n 4 5 6 7 8
Coxeter
group
A2A2 E6 ${\tilde {E}}_{6}$=E6+ ${\bar {T}}_{7}$=E6++
Coxeter
diagram
Symmetry [[32,2,-1]] [[32,2,0]] [[32,2,1]] [[32,2,2]] [[32,2,3]]
Order 72 1440 103,680 ∞
Graph ∞ ∞
Name −122 022 122 222 322
See also
• Polytope and 4-polytope
• Convex regular 4-polytope
• Duocylinder
• Tesseract
Notes
1. The Fourth Dimension Simply Explained, Henry P. Manning, Munn & Company, 1910, New York. Available from the University of Virginia library. Also accessible online: The Fourth Dimension Simply Explained—contains a description of duoprisms (double prisms) and duocylinders (double cylinders). Googlebook
2. Jonathan Bowers - Miscellaneous Uniform Polychora 965. Gudap
3. http://www.polychora.com/12GudapsMovie.gif Animation of cross sections
References
• Regular Polytopes, H. S. M. Coxeter, Dover Publications, Inc., 1973, New York, p. 124.
• Coxeter, The Beauty of Geometry: Twelve Essays, Dover Publications, 1999, ISBN 0-486-40919-8 (Chapter 5: Regular Skew Polyhedra in three and four dimensions and their topological analogues)
• Coxeter, H. S. M. Regular Skew Polyhedra in Three and Four Dimensions. Proc. London Math. Soc. 43, 33-62, 1937.
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 26)
• N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
| Wikipedia |
6-demicube
In geometry, a 6-demicube or demihexeract is a uniform 6-polytope, constructed from a 6-cube (hexeract) with alternated vertices removed. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes.
Demihexeract
(6-demicube)
Petrie polygon projection
Type Uniform 6-polytope
Family demihypercube
Schläfli symbol {3,33,1} = h{4,34}
s{21,1,1,1,1}
Coxeter diagrams =
=
Coxeter symbol 131
5-faces4412 {31,2,1}
32 {34}
4-faces25260 {31,1,1}
192 {33}
Cells640160 {31,0,1}
480 {3,3}
Faces640{3}
Edges240
Vertices32
Vertex figure Rectified 5-simplex
Symmetry group D6, [33,1,1] = [1+,4,34]
[25]+
Petrie polygon decagon
Properties convex
E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as HM6 for a 6-dimensional half measure polytope.
Coxeter named this polytope as 131 from its Coxeter diagram, with a ring on one of the 1-length branches, . It can named similarly by a 3-dimensional exponential Schläfli symbol $\left\{3{\begin{array}{l}3,3,3\\3\end{array}}\right\}$ or {3,33,1}.
Cartesian coordinates
Cartesian coordinates for the vertices of a demihexeract centered at the origin are alternate halves of the hexeract:
(±1,±1,±1,±1,±1,±1)
with an odd number of plus signs.
As a configuration
This configuration matrix represents the 6-demicube. The rows and columns correspond to vertices, edges, faces, cells, 4-faces and 5-faces. The diagonal numbers say how many of each element occur in the whole 6-demicube. The nondiagonal numbers say how many of the column's element occur in or at the row's element.[1][2]
The diagonal f-vector numbers are derived through the Wythoff construction, dividing the full group order of a subgroup order by removing one mirror at a time.[3]
D6k-facefkf0f1f2f3f4f5k-figurenotes
A4( ) f0 3215602060153066r{3,3,3,3}D6/A4 = 32*6!/5! = 32
A3A1A1{ } f1 224084126842{}x{3,3}D6/A3A1A1 = 32*6!/4!/2/2 = 240
A3A2{3} f2 33640133331{3}v( )D6/A3A2 = 32*6!/4!/3! = 640
A3A1h{4,3} f3 464160*3030{3}D6/A3A1 = 32*6!/4!/2 = 160
A3A2{3,3} 464*4801221{}v( )D6/A3A2 = 32*6!/4!/3! = 480
D4A1h{4,3,3} f4 824328860*20{ }D6/D4A1 = 32*6!/8/4!/2 = 60
A4{3,3,3} 5101005*19211D6/A4 = 32*6!/5! = 192
D5h{4,3,3,3} f5 16801604080101612*( )D6/D5 = 32*6!/16/5! = 12
A5{3,3,3,3} 6152001506*32D6/A5 = 32*6!/6! = 32
Images
orthographic projections
Coxeter plane B6
Graph
Dihedral symmetry [12/2]
Coxeter plane D6 D5
Graph
Dihedral symmetry [10] [8]
Coxeter plane D4 D3
Graph
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]
Related polytopes
There are 47 uniform polytopes with D6 symmetry, 31 are shared by the B6 symmetry, and 16 are unique:
D6 polytopes
h{4,34}
h2{4,34}
h3{4,34}
h4{4,34}
h5{4,34}
h2,3{4,34}
h2,4{4,34}
h2,5{4,34}
h3,4{4,34}
h3,5{4,34}
h4,5{4,34}
h2,3,4{4,34}
h2,3,5{4,34}
h2,4,5{4,34}
h3,4,5{4,34}
h2,3,4,5{4,34}
The 6-demicube, 131 is third in a dimensional series of uniform polytopes, expressed by Coxeter as k31 series. The fifth figure is a Euclidean honeycomb, 331, and the final is a noncompact hyperbolic honeycomb, 431. Each progressive uniform polytope is constructed from the previous as its vertex figure.
k31 dimensional figures
n 4 5 6 7 8 9
Coxeter
group
A3A1 A5 D6 E7 ${\tilde {E}}_{7}$ = E7+ ${\bar {T}}_{8}$=E7++
Coxeter
diagram
Symmetry [3−1,3,1] [30,3,1] [31,3,1] [32,3,1] [33,3,1] [34,3,1]
Order 48 720 23,040 2,903,040 ∞
Graph - -
Name −131 031 131 231 331 431
It is also the second in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 13k series. The fourth figure is the Euclidean honeycomb 133 and the final is a noncompact hyperbolic honeycomb, 134.
13k dimensional figures
Space Finite Euclidean Hyperbolic
n 4 5 6 7 8 9
Coxeter
group
A3A1 A5 D6 E7 ${\tilde {E}}_{7}$=E7+ ${\bar {T}}_{8}$=E7++
Coxeter
diagram
Symmetry [3−1,3,1] [30,3,1] [31,3,1] [32,3,1] [[33,3,1]] [34,3,1]
Order 48 720 23,040 2,903,040 ∞
Graph - -
Name 13,-1 130 131 132 133 134
Skew icosahedron
Coxeter identified a subset of 12 vertices that form a regular skew icosahedron {3, 5} with the same symmetries as the icosahedron itself, but at different angles. He dubbed this the regular skew icosahedron.[4][5]
References
1. Coxeter, Regular Polytopes, sec 1.8 Configurations
2. Coxeter, Complex Regular Polytopes, p.117
3. Klitzing, Richard. "x3o3o *b3o3o3o - hax".
4. Coxeter, H. S. M. The beauty of geometry : twelve essays (Dover ed.). Dover Publications. pp. 450–451. ISBN 9780486409191.
5. Deza, Michael; Shtogrin, Mikhael (2000). "Embedding the graphs of regular tilings and star-honeycombs into the graphs of hypercubes and cubic lattices". Advanced Studies in Pure Mathematics: 77. doi:10.2969/aspm/02710073. Retrieved 4 April 2020.
• H.S.M. Coxeter:
• Coxeter, Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8, p.296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
• H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973, p.296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
• Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6
• (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
• (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
• (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 26. pp. 409: Hemicubes: 1n1)
• Klitzing, Richard. "6D uniform polytopes (polypeta) x3o3o *b3o3o3o – hax".
External links
• Olshevsky, George. "Demihexeract". Glossary for Hyperspace. Archived from the original on 4 February 2007.
• Multi-dimensional Glossary
Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron
Uniform polychoron Pentachoron 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell
Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221
Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321
Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421
Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds
| Wikipedia |
6-demicubic honeycomb
The 6-demicubic honeycomb or demihexeractic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 6-space. It is constructed as an alternation of the regular 6-cube honeycomb.
6-demicubic honeycomb
(No image)
TypeUniform 6-honeycomb
FamilyAlternated hypercube honeycomb
Schläfli symbolh{4,3,3,3,3,4}
h{4,3,3,3,31,1}
ht0,6{4,3,3,3,3,4}
Coxeter diagram =
=
Facets{3,3,3,3,4}
h{4,3,3,3,3}
Vertex figurer{3,3,3,3,4}
Coxeter group${\tilde {B}}_{6}$ [4,3,3,3,31,1]
${\tilde {D}}_{6}$ [31,1,3,3,31,1]
It is composed of two different types of facets. The 6-cubes become alternated into 6-demicubes h{4,3,3,3,3} and the alternated vertices create 6-orthoplex {3,3,3,3,4} facets.
D6 lattice
The vertex arrangement of the 6-demicubic honeycomb is the D6 lattice.[1] The 60 vertices of the rectified 6-orthoplex vertex figure of the 6-demicubic honeycomb reflect the kissing number 60 of this lattice.[2] The best known is 72, from the E6 lattice and the 222 honeycomb.
The D+
6
lattice (also called D2
6
) can be constructed by the union of two D6 lattices. This packing is only a lattice for even dimensions. The kissing number is 25=32 (2n-1 for n<8, 240 for n=8, and 2n(n-1) for n>8).[3]
∪
The D*
6
lattice (also called D4
6
and C2
6
) can be constructed by the union of all four 6-demicubic lattices:[4] It is also the 6-dimensional body centered cubic, the union of two 6-cube honeycombs in dual positions.
∪ ∪ ∪ = ∪ .
The kissing number of the D6* lattice is 12 (2n for n≥5).[5] and its Voronoi tessellation is a trirectified 6-cubic honeycomb, , containing all birectified 6-orthoplex Voronoi cell, .[6]
Symmetry constructions
There are three uniform construction symmetries of this tessellation. Each symmetry can be represented by arrangements of different colors on the 64 6-demicube facets around each vertex.
Coxeter group Schläfli symbol Coxeter-Dynkin diagram Vertex figure
Symmetry
Facets/verf
${\tilde {B}}_{6}$ = [31,1,3,3,3,4]
= [1+,4,3,3,3,3,4]
h{4,3,3,3,3,4} =
[3,3,3,4]
64: 6-demicube
12: 6-orthoplex
${\tilde {D}}_{6}$ = [31,1,3,31,1]
= [1+,4,3,3,31,1]
h{4,3,3,3,31,1} =
[33,1,1]
32+32: 6-demicube
12: 6-orthoplex
½${\tilde {C}}_{6}$ = [[(4,3,3,3,4,2+)]]ht0,6{4,3,3,3,3,4} 32+16+16: 6-demicube
12: 6-orthoplex
Related honeycombs
This honeycomb is one of 41 uniform honeycombs constructed by the ${\tilde {D}}_{6}$ Coxeter group, all but 6 repeated in other families by extended symmetry, seen in the graph symmetry of rings in the Coxeter–Dynkin diagrams. The 41 permutations are listed with its highest extended symmetry, and related ${\tilde {B}}_{6}$ and ${\tilde {C}}_{6}$ constructions:
D6 honeycombs
Extended
symmetry
Extended
diagram
Order Honeycombs
[31,1,3,3,31,1] ×1 ,
[[31,1,3,3,31,1]] ×2 , , ,
<[31,1,3,3,31,1]>
↔ [31,1,3,3,3,4]
↔
×2 , , , , , , , ,
, , , , , , ,
<2[31,1,3,3,31,1]>
↔ [4,3,3,3,3,4]
↔
×4 ,,
,,
, , , , , , ,
[<2[31,1,3,3,31,1]>]
↔ [[4,3,3,3,3,4]]
↔
×8 , , ,
, , ,
See also
• 6-cubic honeycomb
Notes
1. "The Lattice D6".
2. Sphere packings, lattices, and groups, by John Horton Conway, Neil James Alexander Sloane, Eiichi Bannai
3. Conway (1998), p. 119
4. "The Lattice D6".
5. Conway (1998), p. 120
6. Conway (1998), p. 466
External links
• Kaleidoscopes: Selected Writings of H. S. M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6
• (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
• Conway JH, Sloane NJH (1998). Sphere Packings, Lattices and Groups (3rd ed.). ISBN 0-387-98585-9.
Fundamental convex regular and uniform honeycombs in dimensions 2–9
Space Family ${\tilde {A}}_{n-1}$ ${\tilde {C}}_{n-1}$ ${\tilde {B}}_{n-1}$ ${\tilde {D}}_{n-1}$ ${\tilde {G}}_{2}$ / ${\tilde {F}}_{4}$ / ${\tilde {E}}_{n-1}$
E2 Uniform tiling {3[3]} δ3 hδ3 qδ3 Hexagonal
E3 Uniform convex honeycomb {3[4]} δ4 hδ4 qδ4
E4 Uniform 4-honeycomb {3[5]} δ5 hδ5 qδ5 24-cell honeycomb
E5 Uniform 5-honeycomb {3[6]} δ6 hδ6 qδ6
E6 Uniform 6-honeycomb {3[7]} δ7 hδ7 qδ7 222
E7 Uniform 7-honeycomb {3[8]} δ8 hδ8 qδ8 133 • 331
E8 Uniform 8-honeycomb {3[9]} δ9 hδ9 qδ9 152 • 251 • 521
E9 Uniform 9-honeycomb {3[10]} δ10 hδ10 qδ10
E10 Uniform 10-honeycomb {3[11]} δ11 hδ11 qδ11
En-1 Uniform (n-1)-honeycomb {3[n]} δn hδn qδn 1k2 • 2k1 • k21
| Wikipedia |
6-cube
In geometry, a 6-cube is a six-dimensional hypercube with 64 vertices, 192 edges, 240 square faces, 160 cubic cells, 60 tesseract 4-faces, and 12 5-cube 5-faces.
6-cube
Hexeract
Orthogonal projection
inside Petrie polygon
Orange vertices are doubled, and the center yellow has 4 vertices
TypeRegular 6-polytope
Familyhypercube
Schläfli symbol{4,34}
Coxeter diagram
5-faces12 {4,3,3,3}
4-faces60 {4,3,3}
Cells160 {4,3}
Faces240 {4}
Edges192
Vertices64
Vertex figure5-simplex
Petrie polygondodecagon
Coxeter groupB6, [34,4]
Dual6-orthoplex
Propertiesconvex, Hanner polytope
It has Schläfli symbol {4,34}, being composed of 3 5-cubes around each 4-face. It can be called a hexeract, a portmanteau of tesseract (the 4-cube) with hex for six (dimensions) in Greek. It can also be called a regular dodeca-6-tope or dodecapeton, being a 6-dimensional polytope constructed from 12 regular facets.
Related polytopes
It is a part of an infinite family of polytopes, called hypercubes. The dual of a 6-cube can be called a 6-orthoplex, and is a part of the infinite family of cross-polytopes.
Applying an alternation operation, deleting alternating vertices of the 6-cube, creates another uniform polytope, called a 6-demicube, (part of an infinite family called demihypercubes), which has 12 5-demicube and 32 5-simplex facets.
As a configuration
This configuration matrix represents the 6-cube. The rows and columns correspond to vertices, edges, faces, cells, 4-faces and 5-faces. The diagonal numbers say how many of each element occur in the whole 6-cube. The nondiagonal numbers say how many of the column's element occur in or at the row's element.[1][2]
${\begin{bmatrix}{\begin{matrix}64&6&15&20&15&6\\2&192&5&10&10&5\\4&4&240&4&6&4\\8&12&6&160&3&3\\16&32&24&8&60&2\\32&80&80&40&10&12\end{matrix}}\end{bmatrix}}$
Cartesian coordinates
Cartesian coordinates for the vertices of a 6-cube centered at the origin and edge length 2 are
(±1,±1,±1,±1,±1,±1)
while the interior of the same consists of all points (x0, x1, x2, x3, x4, x5) with −1 < xi < 1.
Construction
There are three Coxeter groups associated with the 6-cube, one regular, with the C6 or [4,3,3,3,3] Coxeter group, and a half symmetry (D6) or [33,1,1] Coxeter group. The lowest symmetry construction is based on hyperrectangles or proprisms, cartesian products of lower dimensional hypercubes.
Name Coxeter Schläfli Symmetry Order
Regular 6-cube
{4,3,3,3,3} [4,3,3,3,3]46080
Quasiregular 6-cube [3,3,3,31,1]23040
hyperrectangle {4,3,3,3}×{}[4,3,3,3,2]7680
{4,3,3}×{4}[4,3,3,2,4]3072
{4,3}2[4,3,2,4,3]2304
{4,3,3}×{}2[4,3,3,2,2]1536
{4,3}×{4}×{}[4,3,2,4,2]768
{4}3[4,2,4,2,4]512
{4,3}×{}3[4,3,2,2,2]384
{4}2×{}2[4,2,4,2,2]256
{4}×{}4[4,2,2,2,2]128
{}6 [2,2,2,2,2]64
Projections
orthographic projections
Coxeter plane B6 B5 B4
Graph
Dihedral symmetry [12] [10] [8]
Coxeter plane Other B3 B2
Graph
Dihedral symmetry [2] [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]
3D Projections
6-cube 6D simple rotation through 2Pi with 6D perspective projection to 3D.
6-cube quasicrystal structure orthographically projected
to 3D using the golden ratio.
Related polytopes
The 64 vertices of a 6-cube also represent a regular skew 4-polytope {4,3,4 | 4}. Its net can be seen as a 4×4×4 matrix of 64 cubes, a periodic subset of the cubic honeycomb, {4,3,4}, in 3-dimensions. It has 192 edges, and 192 square faces. Opposite faces fold together into a 4-cycle. Each fold direction adds 1 dimension, raising it into 6-space.
The 6-cube is 6th in a series of hypercube:
Petrie polygon orthographic projections
Line segment Square Cube 4-cube 5-cube 6-cube 7-cube 8-cube
This polytope is one of 63 uniform 6-polytopes generated from the B6 Coxeter plane, including the regular 6-cube or 6-orthoplex.
B6 polytopes
β6
t1β6
t2β6
t2γ6
t1γ6
γ6
t0,1β6
t0,2β6
t1,2β6
t0,3β6
t1,3β6
t2,3γ6
t0,4β6
t1,4γ6
t1,3γ6
t1,2γ6
t0,5γ6
t0,4γ6
t0,3γ6
t0,2γ6
t0,1γ6
t0,1,2β6
t0,1,3β6
t0,2,3β6
t1,2,3β6
t0,1,4β6
t0,2,4β6
t1,2,4β6
t0,3,4β6
t1,2,4γ6
t1,2,3γ6
t0,1,5β6
t0,2,5β6
t0,3,4γ6
t0,2,5γ6
t0,2,4γ6
t0,2,3γ6
t0,1,5γ6
t0,1,4γ6
t0,1,3γ6
t0,1,2γ6
t0,1,2,3β6
t0,1,2,4β6
t0,1,3,4β6
t0,2,3,4β6
t1,2,3,4γ6
t0,1,2,5β6
t0,1,3,5β6
t0,2,3,5γ6
t0,2,3,4γ6
t0,1,4,5γ6
t0,1,3,5γ6
t0,1,3,4γ6
t0,1,2,5γ6
t0,1,2,4γ6
t0,1,2,3γ6
t0,1,2,3,4β6
t0,1,2,3,5β6
t0,1,2,4,5β6
t0,1,2,4,5γ6
t0,1,2,3,5γ6
t0,1,2,3,4γ6
t0,1,2,3,4,5γ6
References
1. Coxeter, Regular Polytopes, sec 1.8 Configurations
2. Coxeter, Complex Regular Polytopes, p.117
• Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8 p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n>=5)
• Klitzing, Richard. "6D uniform polytopes (polypeta) o3o3o3o3o4x - ax".
External links
• Weisstein, Eric W. "Hypercube". MathWorld.
• Olshevsky, George. "Measure polytope". Glossary for Hyperspace. Archived from the original on 4 February 2007.
• Multi-dimensional Glossary: hypercube Garrett Jones
Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron
Uniform polychoron Pentachoron 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell
Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221
Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321
Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421
Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds
| Wikipedia |
6-simplex
In geometry, a 6-simplex is a self-dual regular 6-polytope. It has 7 vertices, 21 edges, 35 triangle faces, 35 tetrahedral cells, 21 5-cell 4-faces, and 7 5-simplex 5-faces. Its dihedral angle is cos−1(1/6), or approximately 80.41°.
6-simplex
Typeuniform polypeton
Schläfli symbol{35}
Coxeter diagrams
Elements
f5 = 7, f4 = 21, C = 35, F = 35, E = 21, V = 7
(χ=0)
Coxeter groupA6, [35], order 5040
Bowers name
and (acronym)
Heptapeton
(hop)
Vertex figure5-simplex
Circumradius${\sqrt {\tfrac {3}{7}}}$
0.654654[1]
Propertiesconvex, isogonal self-dual
Alternate names
It can also be called a heptapeton, or hepta-6-tope, as a 7-facetted polytope in 6-dimensions. The name heptapeton is derived from hepta for seven facets in Greek and -peta for having five-dimensional facets, and -on. Jonathan Bowers gives a heptapeton the acronym hop.[2]
As a configuration
This configuration matrix represents the 6-simplex. The rows and columns correspond to vertices, edges, faces, cells, 4-faces and 5-faces. The diagonal numbers say how many of each element occur in the whole 6-simplex. The nondiagonal numbers say how many of the column's element occur in or at the row's element. This self-dual simplex's matrix is identical to its 180 degree rotation.[3][4]
${\begin{bmatrix}{\begin{matrix}7&6&15&20&15&6\\2&21&5&10&10&5\\3&3&35&4&6&4\\4&6&4&35&3&3\\5&10&10&5&21&2\\6&15&20&15&6&7\end{matrix}}\end{bmatrix}}$
Coordinates
The Cartesian coordinates for an origin-centered regular heptapeton having edge length 2 are:
$\left({\sqrt {1/21}},\ {\sqrt {1/15}},\ {\sqrt {1/10}},\ {\sqrt {1/6}},\ {\sqrt {1/3}},\ \pm 1\right)$
$\left({\sqrt {1/21}},\ {\sqrt {1/15}},\ {\sqrt {1/10}},\ {\sqrt {1/6}},\ -2{\sqrt {1/3}},\ 0\right)$
$\left({\sqrt {1/21}},\ {\sqrt {1/15}},\ {\sqrt {1/10}},\ -{\sqrt {3/2}},\ 0,\ 0\right)$
$\left({\sqrt {1/21}},\ {\sqrt {1/15}},\ -2{\sqrt {2/5}},\ 0,\ 0,\ 0\right)$
$\left({\sqrt {1/21}},\ -{\sqrt {5/3}},\ 0,\ 0,\ 0,\ 0\right)$
$\left(-{\sqrt {12/7}},\ 0,\ 0,\ 0,\ 0,\ 0\right)$
The vertices of the 6-simplex can be more simply positioned in 7-space as permutations of:
(0,0,0,0,0,0,1)
This construction is based on facets of the 7-orthoplex.
Images
orthographic projections
Ak Coxeter plane A6 A5 A4
Graph
Dihedral symmetry [7] [6] [5]
Ak Coxeter plane A3 A2
Graph
Dihedral symmetry [4] [3]
Related uniform 6-polytopes
The regular 6-simplex is one of 35 uniform 6-polytopes based on the [3,3,3,3,3] Coxeter group, all shown here in A6 Coxeter plane orthographic projections.
A6 polytopes
t0
t1
t2
t0,1
t0,2
t1,2
t0,3
t1,3
t2,3
t0,4
t1,4
t0,5
t0,1,2
t0,1,3
t0,2,3
t1,2,3
t0,1,4
t0,2,4
t1,2,4
t0,3,4
t0,1,5
t0,2,5
t0,1,2,3
t0,1,2,4
t0,1,3,4
t0,2,3,4
t1,2,3,4
t0,1,2,5
t0,1,3,5
t0,2,3,5
t0,1,4,5
t0,1,2,3,4
t0,1,2,3,5
t0,1,2,4,5
t0,1,2,3,4,5
Notes
1. Klitzing, Richard. "heptapeton". bendwavy.org.
2. Klitzing, Richard. "6D uniform polytopes (polypeta) x3o3o3o3o3o — hop".
3. Coxeter 1973, §1.8 Configurations
4. Coxeter, H.S.M. (1991). Regular Complex Polytopes (2nd ed.). Cambridge University Press. p. 117. ISBN 9780521394901.
References
• Coxeter, H.S.M.:
• — (1973). "Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)". Regular Polytopes (3rd ed.). Dover. p. 296. ISBN 0-486-61480-8.
• Sherk, F. Arthur; McMullen, Peter; Thompson, Anthony C.; Weiss, Asia Ivic, eds. (1995). Kaleidoscopes: Selected Writings of H.S.M. Coxeter. Wiley. ISBN 978-0-471-01003-6.
• (Paper 22) — (1940). "Regular and Semi Regular Polytopes I". Math. Zeit. 46: 380–407. doi:10.1007/BF01181449. S2CID 186237114.
• (Paper 23) — (1985). "Regular and Semi-Regular Polytopes II". Math. Zeit. 188 (4): 559–591. doi:10.1007/BF01161657. S2CID 120429557.
• (Paper 24) — (1988). "Regular and Semi-Regular Polytopes III". Math. Zeit. 200: 3–45. doi:10.1007/BF01161745. S2CID 186237142.
• Conway, John H.; Burgiel, Heidi; Goodman-Strass, Chaim (2008). "26. Hemicubes: 1n1". The Symmetries of Things. p. 409. ISBN 978-1-56881-220-5.
• Johnson, Norman (1991). "Uniform Polytopes" (Manuscript). {{cite document}}: Cite document requires |publisher= (help)
• Johnson, N.W. (1966). The Theory of Uniform Polytopes and Honeycombs (PhD). University of Toronto. OCLC 258527038.
External links
• Olshevsky, George. "Simplex". Glossary for Hyperspace. Archived from the original on 4 February 2007.
• Polytopes of Various Dimensions
• Multi-dimensional Glossary
Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron
Uniform polychoron Pentachoron 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell
Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221
Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321
Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421
Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds
| Wikipedia |
600-cell
In geometry, the 600-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol {3,3,5}. It is also known as the C600, hexacosichoron[1] and hexacosihedroid.[2] It is also called a tetraplex (abbreviated from "tetrahedral complex") and a polytetrahedron, being bounded by tetrahedral cells.
600-cell
Schlegel diagram, vertex-centered
(vertices and edges)
TypeConvex regular 4-polytope
Schläfli symbol{3,3,5}
Coxeter diagram
Cells600 ({3,3})
Faces1200 {3}
Edges720
Vertices120
Vertex figure
icosahedron
Petrie polygon30-gon
Coxeter groupH4, [3,3,5], order 14400
Dual120-cell
Propertiesconvex, isogonal, isotoxal, isohedral
Uniform index35
The 600-cell's boundary is composed of 600 tetrahedral cells with 20 meeting at each vertex.[lower-alpha 1] Together they form 1200 triangular faces, 720 edges, and 120 vertices. It is the 4-dimensional analogue of the icosahedron, since it has five tetrahedra meeting at every edge, just as the icosahedron has five triangles meeting at every vertex.[lower-alpha 2] Its dual polytope is the 120-cell.
Geometry
The 600-cell is the fifth in the sequence of 6 convex regular 4-polytopes (in order of size and complexity).[lower-alpha 3] It can be deconstructed into twenty-five overlapping instances of its immediate predecessor the 24-cell,[4] as the 24-cell can be deconstructed into three overlapping instances of its predecessor the tesseract (8-cell), and the 8-cell can be deconstructed into two overlapping instances of its predecessor the 16-cell.[5]
The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length.[lower-alpha 4] The 24-cell's edge length equals its radius, but the 600-cell's edge length is ~0.618 times its radius. The 600-cell's radius and edge length are in the golden ratio.
Regular convex 4-polytopes
Symmetry group A4 B4 F4 H4
Name 5-cell
Hyper-tetrahedron
5-point
16-cell
Hyper-octahedron
8-point
8-cell
Hyper-cube
16-point
24-cell
24-point
600-cell
Hyper-icosahedron
120-point
120-cell
Hyper-dodecahedron
600-point
Schläfli symbol {3, 3, 3} {3, 3, 4} {4, 3, 3} {3, 4, 3} {3, 3, 5} {5, 3, 3}
Coxeter mirrors
Mirror dihedrals 𝝅/3 𝝅/3 𝝅/3 𝝅/2 𝝅/2 𝝅/2 𝝅/3 𝝅/3 𝝅/4 𝝅/2 𝝅/2 𝝅/2 𝝅/4 𝝅/3 𝝅/3 𝝅/2 𝝅/2 𝝅/2 𝝅/3 𝝅/4 𝝅/3 𝝅/2 𝝅/2 𝝅/2 𝝅/3 𝝅/3 𝝅/5 𝝅/2 𝝅/2 𝝅/2 𝝅/5 𝝅/3 𝝅/3 𝝅/2 𝝅/2 𝝅/2
Graph
Vertices 5 tetrahedral 8 octahedral 16 tetrahedral 24 cubical 120 icosahedral 600 tetrahedral
Edges 10 triangular 24 square 32 triangular 96 triangular 720 pentagonal 1200 triangular
Faces 10 triangles 32 triangles 24 squares 96 triangles 1200 triangles 720 pentagons
Cells 5 tetrahedra 16 tetrahedra 8 cubes 24 octahedra 600 tetrahedra 120 dodecahedra
Tori 1 5-tetrahedron 2 8-tetrahedron 2 4-cube 4 6-octahedron 20 30-tetrahedron 12 10-dodecahedron
Inscribed 120 in 120-cell 675 in 120-cell 2 16-cells 3 8-cells 25 24-cells 10 600-cells
Great polygons 2 squares x 3 4 rectangles x 4 4 hexagons x 4 12 decagons x 6 100 irregular hexagons x 4
Petrie polygons 1 pentagon 1 octagon 2 octagons 2 dodecagons 4 30-gons 20 30-gons
Long radius $1$ $1$ $1$ $1$ $1$ $1$
Edge length ${\sqrt {\tfrac {5}{2}}}\approx 1.581$ ${\sqrt {2}}\approx 1.414$ $1$ $1$ ${\tfrac {1}{\phi }}\approx 0.618$ ${\tfrac {1}{\phi ^{2}{\sqrt {2}}}}\approx 0.270$
Short radius ${\tfrac {1}{4}}$ ${\tfrac {1}{2}}$ ${\tfrac {1}{2}}$ ${\sqrt {\tfrac {1}{2}}}\approx 0.707$ ${\sqrt {\tfrac {\phi ^{4}}{8}}}\approx 0.926$ ${\sqrt {\tfrac {\phi ^{4}}{8}}}\approx 0.926$
Area $10\left({\tfrac {5{\sqrt {3}}}{8}}\right)\approx 10.825$ $32\left({\sqrt {\tfrac {3}{4}}}\right)\approx 27.713$ $24$ $96\left({\sqrt {\tfrac {3}{16}}}\right)\approx 41.569$ $1200\left({\tfrac {\sqrt {3}}{4\phi ^{2}}}\right)\approx 198.48$ $720\left({\tfrac {\sqrt {25+10{\sqrt {5}}}}{8\phi ^{4}}}\right)\approx 90.366$
Volume $5\left({\tfrac {5{\sqrt {5}}}{24}}\right)\approx 2.329$ $16\left({\tfrac {1}{3}}\right)\approx 5.333$ $8$ $24\left({\tfrac {\sqrt {2}}{3}}\right)\approx 11.314$ $600\left({\tfrac {\sqrt {2}}{12\phi ^{3}}}\right)\approx 16.693$ $120\left({\tfrac {15+7{\sqrt {5}}}{4\phi ^{6}{\sqrt {8}}}}\right)\approx 18.118$
4-Content ${\tfrac {\sqrt {5}}{24}}\left({\tfrac {\sqrt {5}}{2}}\right)^{4}\approx 0.146$ ${\tfrac {2}{3}}\approx 0.667$ $1$ $2$ ${\tfrac {{\text{Short}}\times {\text{Vol}}}{4}}\approx 3.863$ ${\tfrac {{\text{Short}}\times {\text{Vol}}}{4}}\approx 4.193$
Unit radius Cartesian coordinates
The vertices of a 600-cell of unit radius centered at the origin of 4-space, with edges of length 1/φ ≈ 0.618 (where φ = 1 + √5/2 ≈ 1.618 is the golden ratio), can be given[6] as follows:
8 vertices obtained from
(0, 0, 0, ±1)
by permuting coordinates, and 16 vertices of the form:
(±1/2, ±1/2, ±1/2, ±1/2)
The remaining 96 vertices are obtained by taking even permutations of
(±φ/2, ±1/2, ±φ−1/2, 0)
Note that the first 8 are the vertices of a 16-cell, the second 16 are the vertices of a tesseract, and those 24 vertices together are the vertices of a 24-cell. The remaining 96 vertices are the vertices of a snub 24-cell, which can be found by partitioning each of the 96 edges of another 24-cell (dual to the first) in the golden ratio in a consistent manner.[7]
When interpreted as quaternions,[lower-alpha 5] these are the unit icosians.
In the 24-cell, there are squares, hexagons and triangles that lie on great circles (in central planes through four or six vertices).[lower-alpha 6] In the 600-cell there are twenty-five overlapping inscribed 24-cells, with each square unique to one 24-cell, each hexagon or triangle shared by two 24-cells, and each vertex shared among five 24-cells.[lower-alpha 8] In each 24-cell there are three disjoint 16-cells, so in the 600-cell there are 75 overlapping inscribed 16-cells.[lower-alpha 9] Each 16-cell constitutes a distinct orthonormal basis for the choice of a coordinate reference frame.
The 60 axes and 75 16-cells of the 600-cell constitute a geometric configuration, which in the language of configurations is written as 605754 to indicate that each axis belongs to 5 16-cells, and each 16-cell contains 4 axes.[8] Each axis is orthogonal to exactly 15 others, and these are just its companions in the 5 16-cells in which it occurs.
Hopf spherical coordinates
In the 600-cell there are also great circle pentagons and decagons (in central planes through ten vertices).[lower-alpha 14]
Only the decagon edges are visible elements of the 600-cell (because they are the edges of the 600-cell). The edges of the other great circle polygons are interior chords of the 600-cell, which are not shown in any of the 600-cell renderings in this article (except where shown as dashed lines).
By symmetry, an equal number of polygons of each kind pass through each vertex; so it is possible to account for all 120 vertices as the intersection of a set of central polygons of only one kind: decagons, hexagons, pentagons, squares, or triangles. For example, the 120 vertices can be seen as the vertices of 15 pairs of completely orthogonal[lower-alpha 16] squares which do not share any vertices, or as 100 dual pairs of non-orthogonal hexagons between which all axis pairs are orthogonal, or as 144 non-orthogonal pentagons six of which intersect at each vertex. This latter pentagonal symmetry of the 600-cell is captured by the set of Hopf coordinates[lower-alpha 19] (𝜉i, 𝜂, 𝜉j) given as:
({<10}𝜋/5, {≤5}𝜋/10, {<10}𝜋/5)
where {<10} is the permutation of the ten digits (0 1 2 3 4 5 6 7 8 9) and {≤5} is the permutation of the six digits (0 1 2 3 4 5). The 𝜉i and 𝜉j coordinates range over the 10 vertices of great circle decagons; even and odd digits label the vertices of the two great circle pentagons inscribed in each decagon.[lower-alpha 20]
Polyhedral sections
The mutual distances of the vertices, measured in degrees of arc on the circumscribed hypersphere, only have the values 36° = 𝜋/5, 60° = 𝜋/3, 72° = 2𝜋/5, 90° = 𝜋/2, 108° = 3𝜋/5, 120° = 2𝜋/3, 144° = 4𝜋/5, and 180° = 𝜋. Departing from an arbitrary vertex V one has at 36° and 144° the 12 vertices of an icosahedron,[lower-alpha 1] at 60° and 120° the 20 vertices of a dodecahedron, at 72° and 108° the 12 vertices of a larger icosahedron, at 90° the 30 vertices of an icosidodecahedron, and finally at 180° the antipodal vertex of V.[11][12][13] These can be seen in the H3 Coxeter plane projections with overlapping vertices colored.[14]
These polyhedral sections are solids in the sense that they are 3-dimensional, but of course all of their vertices lie on the surface of the 600-cell (they are hollow, not solid). Each polyhedron lies in Euclidean 4-dimensional space as a parallel cross section through the 600-cell (a hyperplane). In the curved 3-dimensional space of the 600-cell's boundary surface envelope, the polyhedron surrounds the vertex V the way it surrounds its own center. But its own center is in the interior of the 600-cell, not on its surface. V is not actually at the center of the polyhedron, because it is displaced outward from that hyperplane in the fourth dimension, to the surface of the 600-cell. Thus V is the apex of a 4-pyramid based on the polyhedron.
Concentric Hulls
The 600-cell is projected to 3D using an orthonormal basis.
The vertices are sorted and tallied by their 3D norm. Generating the increasingly transparent hull of each set of tallied norms shows:
1) two points at the origin
2) two icosahedra
3) two dodecahedra
4) two larger icosahedra
5) and a single icosidodecahedron
for a total of 120 vertices. This is the view from any origin vertex. The 600-cell contains 60 distinct sets of these concentric hulls, one centered on each pair of antipodal vertices.
Golden chords
See also: 24-cell § Hypercubic chords
The 120 vertices are distributed[15] at eight different chord lengths from each other. These edges and chords of the 600-cell are simply the edges and chords of its five great circle polygons.[16] In ascending order of length, they are √0.𝚫, √1, √1.𝚫, √2, √2.𝚽, √3, √3.𝚽, and √4.[lower-alpha 24]
Notice that the four hypercubic chords of the 24-cell (√1, √2, √3, √4)[lower-alpha 6] alternate with the four new chords of the 600-cell's additional great circles, the decagons and pentagons. The new golden chord lengths are necessarily square roots of fractions, but very special fractions related to the golden ratio[lower-alpha 21] including the two golden sections of √5, as shown in the diagram.[lower-alpha 22]
Boundary envelopes
The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices,[lower-alpha 26] in effect adding twenty-four more overlapping 24-cells inscribed in the 600-cell.[lower-alpha 9] The new surface thus formed is a tessellation of smaller, more numerous cells[lower-alpha 27] and faces: tetrahedra of edge length 1/φ ≈ 0.618 instead of octahedra of edge length 1. It encloses the √1 edges of the 24-cells, which become invisible interior chords in the 600-cell, like the √2 and √3 chords.
Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of 1/φ, the inverse golden ratio), the 600-cell does not have unit edge-length in a unit-radius coordinate system the way the 24-cell and the tesseract do; unlike those two, the 600-cell is not radially equilateral. Like them it is radially triangular in a special way, but one in which golden triangles rather than equilateral triangles meet at the center.[lower-alpha 23]
The boundary envelope of 600 small tetrahedral cells wraps around the twenty-five envelopes of 24 octahedral cells (adding some 4-dimensional space in places between these curved 3-dimensional envelopes). The shape of those interstices must be an octahedral 4-pyramid of some kind, but in the 600-cell it is not regular.[lower-alpha 29]
Geodesics
The vertex chords of the 600-cell are arranged in geodesic great circle polygons of five kinds: decagons, hexagons, pentagons, squares, and triangles.[19]
The √0.𝚫 = 𝚽 edges form 72 flat regular central decagons, 6 of which cross at each vertex.[lower-alpha 1] Just as the icosidodecahedron can be partitioned into 6 central decagons (60 edges = 6 × 10), the 600-cell can be partitioned into 72 decagons (720 edges = 72 × 10). The 720 √0.𝚫 edges divide the surface into 1200 triangular faces and 600 tetrahedral cells: a 600-cell. The 720 edges occur in 360 parallel pairs, √3.𝚽 apart. As in the decagon and the icosidodecahedron, the edges occur in golden triangles[lower-alpha 28] which meet at the center of the polytope.[lower-alpha 23] The 72 great decagons can be divided into 6 sets of 12 non-intersecting Clifford parallel geodesics,[lower-alpha 33] such that only one decagonal great circle in each set passes through each vertex, and the 12 decagons in each set reach all 120 vertices.[21]
The √1 chords form 200 central hexagons (25 sets of 16, with each hexagon in two sets),[lower-alpha 7] 10 of which cross at each vertex[lower-alpha 34] (4 from each of five 24-cells, with each hexagon in two of the 24-cells). Each set of 16 hexagons consists of the 96 edges and 24 vertices of one of the 25 overlapping inscribed 24-cells. The √1 chords join vertices which are two √0.𝚫 edges apart. Each √1 chord is the long diameter of a face-bonded pair of tetrahedral cells (a triangular bipyramid), and passes through the center of the shared face. As there are 1200 faces, there are 1200 √1 chords, in 600 parallel pairs, √3 apart. The hexagonal planes are non-orthogonal (60 degrees apart) but they occur as 100 dual pairs in which all 3 axes of one hexagon are orthogonal to all 3 axes of its dual.[22] The 200 great hexagons can be divided into 10 sets of 20 non-intersecting Clifford parallel geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 20 hexagons in each set reach all 120 vertices.[23]
The √1.𝚫 chords form 144 central pentagons, 6 of which cross at each vertex.[lower-alpha 14] The √1.𝚫 chords run vertex-to-every-second-vertex in the same planes as the 72 decagons: two pentagons are inscribed in each decagon. The √1.𝚫 chords join vertices which are two √0.𝚫 edges apart on a geodesic great circle. The 720 √1.𝚫 chords occur in 360 parallel pairs, √2.𝚽 = φ apart.
The √2 chords form 450 central squares (25 disjoint sets of 18), 15 of which cross at each vertex (3 from each of five 24-cells). Each set of 18 squares consists of the 72 √2 chords and 24 vertices of one of the 25 overlapping inscribed 24-cells. The √2 chords join vertices which are three √0.𝚫 edges apart (and two √1 chords apart). Each √2 chord is the long diameter of an octahedral cell in just one 24-cell. There are 1800 √2 chords, in 900 parallel pairs, √2 apart. The 450 great squares (225 completely orthogonal[lower-alpha 16] pairs) can be divided into 15 sets of 30 non-intersecting Clifford parallel geodesics, such that only one square great circle in each set passes through each vertex, and the 30 squares in each set reach all 120 vertices.[24]
The √2.𝚽 = φ chords form the legs of 720 central isosceles triangles (72 sets of 10 inscribed in each decagon), 6 of which cross at each vertex. The third edge (base) of each isosceles triangle is of length √3.𝚽. The √2.𝚽 chords run vertex-to-every-third-vertex in the same planes as the 72 decagons, joining vertices which are three √0.𝚫 edges apart on a geodesic great circle. There are 720 distinct √2.𝚽 chords, in 360 parallel pairs, √1.𝚫 apart.
The √3 chords form 400 equilateral central triangles (25 sets of 32, with each triangle in two sets), 10 of which cross at each vertex (4 from each of five 24-cells, with each triangle in two of the 24-cells). Each set of 32 triangles consists of the 96 √3 chords and 24 vertices of one of the 25 overlapping inscribed 24-cells. The √3 chords run vertex-to-every-second-vertex in the same planes as the 200 hexagons: two triangles are inscribed in each hexagon. The √3 chords join vertices which are four √0.𝚫 edges apart (and two √1 chords apart on a geodesic great circle). Each √3 chord is the long diameter of two cubic cells in the same 24-cell.[lower-alpha 35] There are 1200 √3 chords, in 600 parallel pairs, √1 apart.
The √3.𝚽 chords (the diagonals of the pentagons) form the legs of 720 central isosceles triangles (144 sets of 5 inscribed in each pentagon), 6 of which cross at each vertex. The third edge (base) of each isosceles triangle is an edge of the pentagon of length √1.𝚫, so these are golden triangles.[lower-alpha 28] The √3.𝚽 chords run vertex-to-every-fourth-vertex in the same planes as the 72 decagons, joining vertices which are four √0.𝚫 edges apart on a geodesic great circle. There are 720 distinct √3.𝚽 chords, in 360 parallel pairs, √0.𝚫 apart.
The √4 chords occur as 60 long diameters (75 sets of 4 orthogonal axes), the 120 long radii of the 600-cell. The √4 chords join opposite vertices which are five √0.𝚫 edges apart on a geodesic great circle. There are 25 distinct but overlapping sets of 12 diameters, each comprising one of the 25 inscribed 24-cells.[lower-alpha 13]
The sum of the squared lengths[lower-alpha 36] of all these distinct chords of the 600-cell is 14,400 = 1202.[lower-alpha 37] These are all the central polygons through vertices, but the 600-cell does have one noteworthy great circle that does not pass through any vertices (a 0-gon).[lower-alpha 41] Moreover, in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all. There are geodesic shortest paths between two 600-cell vertices that are helical rather than simply circular; they correspond to isoclinic (diagonal) rotations rather than simple rotations.[lower-alpha 42]
All the geodesic polygons enumerated above lie in central planes of just three kinds, each characterized by a rotation angle: decagon planes (𝜋/5 apart), hexagon planes (𝜋/3 apart, also in the 25 inscribed 24-cells), and square planes (𝜋/2 apart, also in the 75 inscribed 16-cells and the 24-cells). These central planes of the 600-cell can be divided into 4 central hyperplanes (3-spaces) each forming an icosidodecahedron. There are 450 great squares 90 degrees apart; 200 great hexagons 60 degrees apart; and 72 great decagons 36 degrees apart.[lower-alpha 47] Each great square plane is completely orthogonal[lower-alpha 16] to another great square plane. Each great hexagon plane is completely orthogonal to a plane which intersects only two vertices (one √4 long diameter): a great digon plane.[lower-alpha 48] Each great decagon plane is completely orthogonal to a plane which intersects no vertices: a great 0-gon plane.[lower-alpha 39]
Fibrations of great circle polygons
Each set of similar great circle polygons (squares or hexagons or decagons) can be divided into bundles of non-intersecting Clifford parallel great circles (of 30 squares or 20 hexagons or 12 decagons).[lower-alpha 33] Each fiber bundle of Clifford parallel great circles[lower-alpha 43] is a discrete Hopf fibration which fills the 600-cell, visiting all 120 vertices just once.[29] Each discrete Hopf fibration has its 3-dimensional base which is a distinct polyhedron that acts as a map or scale model of the fibration.[lower-alpha 49] The great circle polygons in each bundle spiral around each other, delineating helical rings of face-bonded cells which nest into each other, pass through each other without intersecting in any cells and exactly fill the 600-cell with their disjoint cell sets. The different fiber bundles with their cell rings each fill the same space (the 600-cell) but their fibers run Clifford parallel in different "directions"; great circle polygons in different fibrations are not Clifford parallel.[30]
Decagons
The fibrations of the 600-cell include 6 fibrations of its 72 great decagons: 6 fiber bundles of 12 great decagons.[lower-alpha 32] Each fiber bundle[lower-alpha 44] delineates 20 helical rings of 30 tetrahedral cells each,[lower-alpha 31] with five rings nesting together around each decagon.[31] The Hopf map of this fibration is the icosahedron, where each of 12 vertices lifts to a great decagon, and each of 20 triangular faces lifts to a 30-cell ring.[lower-alpha 49] Each tetrahedral cell occupies only one cell ring in each of the 6 fibrations. The tetrahedral cell contributes each of its 6 edges to a decagon in a different fibration, but contributes that edge to five distinct cell rings in the fibration.[lower-alpha 30]
The 12 great circles and 30-cell rings of the 600-cell's 6 characteristic Hopf fibrations make the 600-cell a geometric configuration of 30 "points" and 12 "lines" written as 302125. It is called the Schläfli double six configuration after Ludwig Schläfli,[33] the Swiss mathematician who discovered the 600-cell and the complete set of regular polytopes in n dimensions.[34]
Hexagons
The fibrations of the 24-cell include 4 fibrations of its 16 great hexagons: 4 fiber bundles of 4 great hexagons. Each fiber bundle delineates 4 helical rings of 6 octahedral cells each, with three rings nesting together around each hexagon. Each octahedral cell occupies only one cell ring in each of the 4 fibrations. The octahedral cell contributes 3 of its 12 edges to 3 different Clifford parallel hexagons in each fibration, but contributes each edge to three distinct cell rings in the fibration.
The 600-cell contains 25 24-cells, and can be seen (10 different ways) as a compound of 5 disjoint 24-cells.[lower-alpha 14] It has 10 fibrations of its 200 great hexagons: 10 fiber bundles of 20 great hexagons. Each fiber bundle[lower-alpha 45] delineates 20 helical rings of 6 octahedral cells each, with three rings nesting together around each hexagon. The Hopf map of this fibration is the dodecahedron, where the 20 vertices lift to a bundle of great hexagons.[23] Each octahedral cell occupies only one cell ring in each of the 10 fibrations. The 20 6-octahedron rings belong to 5 disjoint 24-cells of 4 6-octahedron rings each; each hexagonal fibration of the 600-cell consists of 5 disjoint 24-cells.
Squares
The fibrations of the 16-cell include 3 fibrations of its 6 great squares: 3 fiber bundles of 2 great squares. Each fiber bundle delineates 2 helical rings of 8 tetrahedral cells each. Each tetrahedral cell occupies only one cell ring in each of the 3 fibrations. The tetrahedral cell contributes each of its 6 edges to a different square (contributing two opposite non-intersecting edges to each of the 3 fibrations), but contributes each edge to both of the two distinct cell rings in the fibration.
The 600-cell contains 75 16-cells, and can be seen (10 different ways) as a compound of 15 disjoint 16-cells. It has 15 fibrations of its 450 great squares: 15 fiber bundles of 30 great squares. Each fiber bundle[lower-alpha 46] delineates 75 cell-disjoint helical rings of 8 tetrahedral cells each.[lower-alpha 51] The Hopf map of this fibration is the icosidodecahedron,[lower-alpha 23] where the 30 vertices lift to a bundle of great squares.[24] Each tetrahedral cell occupies only one of the 75 cell rings in each of the 15 fibrations.
Clifford parallel cell rings
The densely packed helical cell rings[35][36][29] of fibrations are cell-disjoint, but they share vertices, edges and faces. Each fibration of the 600-cell can be seen as a dense packing of cell rings with the corresponding faces of adjacent cell rings face-bonded to each other.[lower-alpha 54] The same fibration can also be seen as a minimal sparse arrangement of fewer completely disjoint cell rings that do not touch at all.[lower-alpha 10]
The fibrations of great decagons can be seen (five different ways) as 4 completely disjoint 30-cell rings with spaces separating them, rather than as 20 face-bonded cell rings, by leaving out all but one cell ring of the five that meet at each decagon.[37] The five different ways you can do this are equivalent, in that all five correspond to the same discrete fibration (in the same sense that the 6 decagonal fibrations are equivalent, in that all 6 cover the same 600-cell). The 4 cell rings still constitute the complete fibration: they include all 12 Clifford parallel decagons, which visit all 120 vertices.[lower-alpha 55] This subset of 4 of 20 cell rings is dimensionally analogous[lower-alpha 2] to the subset of 12 of 72 decagons, in that both are sets of completely disjoint Clifford parallel polytopes which visit all 120 vertices.[lower-alpha 56] The subset of 4 of 20 cell rings is one of 5 fibrations within the fibration of 12 of 72 decagons: a fibration of a fibration. All the fibrations have this two level structure with subfibrations.
The fibrations of the 24-cell's great hexagons can be seen (three different ways) as 2 completely disjoint 6-cell rings with spaces separating them, rather than as 4 face-bonded cell rings, by leaving out all but one cell ring of the three that meet at each hexagon. Therefore each of the 10 fibrations of the 600-cell's great hexagons can be seen as 2 completely disjoint octahedral cell rings.
The fibrations of the 16-cell's great squares can be seen (two different ways) as a single 8-tetrahedral-cell ring with an adjacent cell-ring-sized empty space, rather than as 2 face-bonded cell rings, by leaving out one of the two cell rings that meet at each square. Therefore each of the 15 fibrations of the 600-cell's great squares can be seen as a single tetrahedral cell ring.[lower-alpha 51]
The sparse constructions of the 600-cell's fibrations correspond to lower-symmetry decompositions of the 600-cell, 24-cell or 16-cell with cells of different colors to distinguish the cell rings from the spaces between them.[lower-alpha 57] The particular lower-symmetry form of the 600-cell corresponding to the sparse construction of the great decagon fibrations is dimensionally analogous[lower-alpha 2] to the snub tetrahedron form of the icosahedron (which is the base[lower-alpha 49] of these fibrations on the 2-sphere). Each of the 20 Boerdijk-Coxeter cell rings[lower-alpha 31] is lifted from a corresponding face of the icosahedron.[lower-alpha 60]
Constructions
The 600-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell, the 120-cell, and the polygons {7} and above.[41] Consequently, there are numerous ways to construct or deconstruct the 600-cell, but none of them are trivial. The construction of the 600-cell from its regular predecessor the 24-cell can be difficult to visualize.
Gosset's construction
Thorold Gosset discovered the semiregular 4-polytopes, including the snub 24-cell with 96 vertices, which falls between the 24-cell and the 600-cell in the sequence of convex 4-polytopes of increasing size and complexity in the same radius. Gosset's construction of the 600-cell from the 24-cell is in two steps, using the snub 24-cell as an intermediate form. In the first, more complex step (described elsewhere) the snub 24-cell is constructed by a special snub truncation of a 24-cell at the golden sections of its edges.[7] In the second step the 600-cell is constructed in a straightforward manner by adding 4-pyramids (vertices) to facets of the snub 24-cell.[42]
The snub 24-cell is a diminished 600-cell from which 24 vertices (and the cluster of 20 tetrahedral cells around each) have been truncated,[lower-alpha 26] leaving a "flat" icosahedral cell in place of each removed icosahedral pyramid.[lower-alpha 1] The snub 24-cell thus has 24 icosahedral cells and the remaining 120 tetrahedral cells. The second step of Gosset's construction of the 600-cell is simply the reverse of this diminishing: an icosahedral pyramid of 20 tetrahedral cells is placed on each icosahedral cell.
Constructing the unit-radius 600-cell from its precursor the unit-radius 24-cell by Gosset's method actually requires three steps. The 24-cell precursor to the snub-24 cell is not of the same radius: it is larger, since the snub-24 cell is its truncation. Starting with the unit-radius 24-cell, the first step is to reciprocate it around its midsphere to construct its outer canonical dual: a larger 24-cell, since the 24-cell is self-dual. That larger 24-cell can then be snub truncated into a unit-radius snub 24-cell.
Cell clusters
Since it is so indirect, Gosset's construction may not help us very much to directly visualize how the 600 tetrahedral cells fit together into a curved 3-dimensional surface envelope,[lower-alpha 27] or how they lie on the underlying surface envelope of the 24-cell's octahedral cells. For that it is helpful to build up the 600-cell directly from clusters of tetrahedral cells.
Most of us have difficulty visualizing the 600-cell from the outside in 4-space, or recognizing an outside view of the 600-cell due to our total lack of sensory experience in 4-dimensional spaces,[43] but we should be able to visualize the surface envelope of 600 cells from the inside because that volume is a 3-dimensional space that we could actually "walk around in" and explore.[44] In these exercises of building the 600-cell up from cell clusters, we are entirely within a 3-dimensional space, albeit a strangely small, closed curved space, in which we can go a mere ten edge lengths away in a straight line in any direction and return to our starting point.
Icosahedra
The vertex figure of the 600-cell is the icosahedron.[lower-alpha 1] Twenty tetrahedral cells meet at each vertex, forming an icosahedral pyramid whose apex is the vertex, surrounded by its base icosahedron. The 600-cell has a dihedral angle of 𝜋/3 + arccos(−1/4) ≈ 164.4775°.[46]
An entire 600-cell can be assembled from 24 such icosahedral pyramids (bonded face-to-face at 8 of the 20 faces of the icosahedron, colored yellow in the illustration), plus 24 clusters of 5 tetrahedral cells (four cells face-bonded around one) which fill the voids remaining between the icosahedra. Each icosahedron is face-bonded to each adjacent cluster of 5 cells by two blue faces that share an edge (which is also one of the six edges of the central tetrahedron of the five). Six clusters of 5 cells surround each icosahedron, and six icosahedra surround each cluster of 5 cells. Five tetrahedral cells surround each icosahedron edge: two from inside the icosahedral pyramid, and three from outside it.[lower-alpha 64]
The apexes of the 24 icosahedral pyramids are the vertices of a 24-cell inscribed in the 600-cell. The other 96 vertices (the vertices of the icosahedra) are the vertices of an inscribed snub 24-cell, which has exactly the same structure of icosahedra and tetrahedra described here, except that the icosahedra are not 4-pyramids filled by tetrahedral cells; they are only "flat" 3-dimensional icosahedral cells, because the central apical vertex is missing.
The 24-cell edges joining icosahedral pyramid apex vertices run through the centers of the yellow faces. Coloring the icosahedra with 8 yellow and 12 blue faces can be done in 5 distinct ways.[lower-alpha 65] Thus each icosahedral pyramid's apex vertex is a vertex of 5 distinct 24-cells,[lower-alpha 12] and the 120 vertices comprise 25 (not 5) 24-cells.[lower-alpha 9]
The icosahedra are face-bonded into geodesic "straight lines" by their opposite yellow faces, bent in the fourth dimension into a ring of 6 icosahedral pyramids. Their apexes are the vertices of a great circle hexagon. This hexagonal geodesic traverses a ring of 12 tetrahedral cells, alternately bonded face-to-face and vertex-to-vertex. The long diameter of each face-bonded pair of tetrahedra (each triangular bipyramid) is a hexagon edge (a 24-cell edge). There are 4 rings of 6 icosahedral pyramids intersecting at each apex-vertex, just as there are 4 cell-disjoint interlocking rings of 6 octahedra in the 24-cell (a hexagonal fibration).[lower-alpha 68]
The tetrahedral cells are face-bonded into triple helices, bent in the fourth dimension into rings of 30 tetrahedral cells.[lower-alpha 31] The three helices are geodesic "straight lines" of 10 edges: great circle decagons which run Clifford parallel[lower-alpha 33] to each other. Each tetrahedron, having six edges, participates in six different decagons[lower-alpha 30] and thereby in all 6 of the decagonal fibrations of the 600-cell.
The partitioning of the 600-cell into clusters of 20 cells and clusters of 5 cells is artificial, since all the cells are the same. One can begin by picking out an icosahedral pyramid cluster centered at any arbitrarily chosen vertex, so there are 120 overlapping icosahedra in the 600-cell.[lower-alpha 62] Their 120 apexes are each a vertex of five 24-vertex 24-cells, so there are 5*120/24 = 25 overlapping 24-cells.[lower-alpha 14]
Octahedra
There is another useful way to partition the 600-cell surface, into 24 clusters of 25 tetrahedral cells, which reveals more structure[51] and a direct construction of the 600-cell from its predecessor the 24-cell.
Begin with any one of the clusters of 5 cells (above), and consider its central cell to be the center object of a new larger cluster of tetrahedral cells. The central cell is the first section of the 600-cell beginning with a cell. By surrounding it with more tetrahedral cells, we can reach the deeper sections beginning with a cell.
First, note that a cluster of 5 cells consists of 4 overlapping pairs of face-bonded tetrahedra (triangular dipyramids) whose long diameter is a 24-cell edge (a hexagon edge) of length √1. Six more triangular dipyramids fit into the concavities on the surface of the cluster of 5,[lower-alpha 69] so the exterior chords connecting its 4 apical vertices are also 24-cell edges of length √1. They form a tetrahedron of edge length √1, which is the second section of the 600-cell beginning with a cell.[lower-alpha 70] There are 600 of these √1 tetrahedral sections in the 600-cell.[lower-alpha 71]
With the six triangular dipyamids fit into the concavities, there are 12 new cells and 6 new vertices in addition to the 5 cells and 8 vertices of the original cluster. The 6 new vertices form the third section of the 600-cell beginning with a cell, an octahedron of edge length √1, obviously the cell of a 24-cell. As partially filled so far (by 17 tetrahedral cells), this √1 octahedron has concave faces into which a short triangular pyramid fits; it has the same volume as a regular tetrahedral cell but an irregular tetrahedral shape.[lower-alpha 72] Each octahedron surrounds 1 + 4 + 12 + 8 = 25 tetrahedral cells: 17 regular tetrahedral cells plus 8 volumetrically equivalent tetrahedral cells each consisting of 6 one-sixth fragments from 6 different regular tetrahedral cells that each span three adjacent octahedral cells.[lower-alpha 73]
Thus the unit-radius 600-cell may be constructed directly from its predecessor,[lower-alpha 29] the unit-radius 24-cell, by placing on each of its octahedral facets a truncated[lower-alpha 74] irregular octahedral pyramid of 14 vertices[lower-alpha 75] constructed (in the above manner) from 25 regular tetrahedral cells of edge length 1/φ ≈ 0.618.
Union of two tori
There is yet another useful way to partition the 600-cell surface into clusters of tetrahedral cells, which reveals more structure[52] and the decagonal fibrations of the 600-cell. An entire 600-cell can be assembled around two rings of 5 icosahedral pyramids, bonded vertex-to-vertex into two geodesic "straight lines".
The 120-cell can be decomposed into two disjoint tori. Since it is the dual of the 600-cell, this same dual tori structure exists in the 600-cell, although it is somewhat more complex. The 10-cell geodesic path in the 120-cell corresponds to the 10-vertex decagon path in the 600-cell.[53]
Start by assembling five tetrahedra around a common edge. This structure looks somewhat like an angular "flying saucer". Stack ten of these, vertex to vertex, "pancake" style. Fill in the annular ring between each pair of "flying saucers" with 10 tetrahedra to form an icosahedron. You can view this as five vertex-stacked icosahedral pyramids, with the five extra annular ring gaps also filled in.[lower-alpha 76] The surface is the same as that of ten stacked pentagonal antiprisms: a triangular-faced column with a pentagonal cross-section.[54] Bent into a columnar ring this is a torus consisting of 150 cells, ten edges long, with 100 exposed triangular faces,[lower-alpha 77] 150 exposed edges, and 50 exposed vertices. Stack another tetrahedron on each exposed face. This will give you a somewhat bumpy torus of 250 cells with 50 raised vertices, 50 valley vertices, and 100 valley edges.[lower-alpha 78] The valleys are 10 edge long closed paths and correspond to other instances of the 10-vertex decagon path mentioned above (great circle decagons). These decagons spiral around the center core decagon,[lower-alpha 79] but mathematically they are all equivalent (they all lie in central planes).
Build a second identical torus of 250 cells that interlinks with the first. This accounts for 500 cells. These two tori mate together with the valley vertices touching the raised vertices, leaving 100 tetrahedral voids that are filled with the remaining 100 tetrahedra that mate at the valley edges. This latter set of 100 tetrahedra are on the exact boundary of the duocylinder and form a Clifford torus. They can be "unrolled" into a square 10×10 array. Incidentally this structure forms one tetrahedral layer in the tetrahedral-octahedral honeycomb. There are exactly 50 "egg crate" recesses and peaks on both sides that mate with the 250 cell tori. In this case into each recess, instead of an octahedron as in the honeycomb, fits a triangular bipyramid composed of two tetrahedra.
This decomposition of the 600-cell has symmetry [[10,2+,10]], order 400, the same symmetry as the grand antiprism.[55] The grand antiprism is just the 600-cell with the two above 150-cell tori removed, leaving only the single middle layer of 300 tetrahedra, dimensionally analogous[lower-alpha 2] to the 10-face belt of an icosahedron with the 5 top and 5 bottom faces removed (a pentagonal antiprism).[lower-alpha 80]
The two 150-cell tori each contain 6 Clifford parallel great decagons (five around one), and the two tori are Clifford parallel to each other, so together they constitute a complete fibration of 12 decagons that reaches all 120 vertices, despite filling only half the 600-cell with cells.
Boerdijk–Coxeter helix rings
The 600-cell can also be partitioned into 20 cell-disjoint intertwining rings of 30 cells,[31] each ten edges long, forming a discrete Hopf fibration which fills the entire 600-cell.[56][57] Each ring of 30 face-bonded tetrahedra is a cylindrical Boerdijk–Coxeter helix bent into a ring in the fourth dimension.
A single 30-tetrahedron Boerdijk–Coxeter helix ring within the 600-cell, seen in stereographic projection.[lower-alpha 31]
A 30-tetrahedron ring can be seen along the perimeter of this 30-gonal orthogonal projection of the 600-cell.[lower-alpha 41]
The 30-cell ring as a {30/11} polygram of 30 edges wound into a helix that twists around its axis 11 times. This projection along the axis of the 30-cell cylinder shows the 30 vertices 12° apart around the cylinder's circular cross section, with the edges connecting every 11th vertex on the circle.[lower-alpha 40]
The 30-vertex, 30-tetrahedron Boerdijk–Coxeter helix ring, cut and laid out flat in 3-dimensional space. Three cyan Clifford parallel great decagons bound the ring.[lower-alpha 32] They are bridged by a skew 30-gram helix of 30 magenta edges linking all 30 vertices: the Petrie polygon of the 600-cell.[lower-alpha 81] The 15 orange edges and 15 yellow edges form separate 15-gram helices, the edge-paths of isoclines.
The 30-cell ring is the 3-dimensional space occupied by the 30 vertices of three cyan Clifford parallel great decagons that lie adjacent to each other, 36° = 𝜋/5 = one 600-cell edge length apart at all their vertex pairs.[lower-alpha 82] The 30 magenta edges joining these vertex pairs form a helical triacontagram, a skew 30-gram spiral of 30 edge-bonded triangular faces, that is the Petrie polygon of the 600-cell.[lower-alpha 81] The dual of the 30-cell ring (the skew 30-gon made by connecting its cell centers) is the Petrie polygon of the 120-cell, the 600-cell's dual polytope.[lower-alpha 50] The central axis of the 30-cell ring is a great 30-gon geodesic that passes through the center of 30 faces, but does not intersect any vertices.[lower-alpha 41]
The 15 orange edges and 15 yellow edges form separate 15-gram helices. Each orange or yellow edge crosses between two cyan great decagons. Successive orange or yellow edges of these 15-gram helices do not lie on the same great circle; they lie in different central planes inclined at 36° = 𝝅/5 to each other.[lower-alpha 47] Each 15-gram helix is noteworthy as the edge-path of an isocline, the geodesic path of an isoclinic rotation.[lower-alpha 42] The isocline is a circular curve which intersects every second vertex of the 15-gram, missing the vertex in between. A single isocline runs twice around each orange (or yellow) 15-gram through every other vertex, hitting half the vertices on the first loop and the other half of them on the second loop. The two connected loops forms a single Möbius loop, a skew {15/2} pentadecagram. The pentadecagram is not shown in these illustrations (but see below), because its edges are invisible chords between vertices which are two orange (or two yellow) edges apart, and no chords are shown in these illustrations. Although the 30 vertices of the 30-cell ring do not lie in one great 30-gon central plane,[lower-alpha 82] these invisible pentadecagram isoclines are true geodesic circles of a special kind, that wind through all four dimensions rather than lying in a 2-dimensional plane as an ordinary geodesic great circle does.[lower-alpha 83]
Five of these 30-cell helices nest together and spiral around each of the 10-vertex decagon paths, forming the 150-cell torus described in the grand antiprism decomposition above.[55] Thus every great decagon is the center core decagon of a 150-cell torus.[lower-alpha 84] The 600-cell may be decomposed into 20 30-cell rings, or into two 150-cell tori and 10 30-cell rings, but not into four 150-cell tori of this kind.[lower-alpha 85] The 600-cell can be decomposed into four 150-cell tori of a different kind.[lower-alpha 86]
Radial golden triangles
The 600-cell can be constructed radially from 720 golden triangles of edge lengths √0.𝚫 √1 √1 which meet at the center of the 4-polytope, each contributing two √1 radii and a √0.𝚫 edge.[lower-alpha 23] They form 1200 triangular pyramids with their apexes at the center: irregular tetrahedra with equilateral √0.𝚫 bases (the faces of the 600-cell). These form 600 tetrahedral pyramids with their apexes at the center: irregular 5-cells with regular √0.𝚫 tetrahedron bases (the cells of the 600-cell).
Characteristic orthoscheme
Characteristics of the 600-cell[59]
edge[60] arc dihedral[61]
𝒍 ${\tfrac {1}{\phi }}\approx 0.618$ 36° ${\tfrac {\pi }{5}}$ 164°29′ $\pi -2{\text{𝟁}}$
𝟀 ${\sqrt {\tfrac {2}{3\phi ^{2}}}}\approx 0.505$ 22°15′20″ ${\tfrac {\pi }{3}}-{\text{𝜼}}$ 60° ${\tfrac {\pi }{3}}$
𝝓[lower-alpha 87] ${\sqrt {\tfrac {1}{2\phi ^{2}}}}\approx 0.437$ 18° ${\tfrac {\pi }{10}}$ 36° ${\tfrac {\pi }{5}}$
𝟁 ${\sqrt {\tfrac {1}{6\phi ^{2}}}}\approx 0.252$ 17°44′40″ ${\text{𝜼}}-{\tfrac {\pi }{6}}$ 60° ${\tfrac {\pi }{3}}$
$_{0}R^{3}/l$ ${\sqrt {\tfrac {3}{4\phi ^{2}}}}\approx 0.535$ 22°15′20″ ${\tfrac {\pi }{3}}-{\text{𝜼}}$ 90° ${\tfrac {\pi }{2}}$
$_{1}R^{3}/l$ ${\sqrt {\tfrac {1}{4\phi ^{2}}}}\approx 0.309$ 18° ${\tfrac {\pi }{10}}$ 90° ${\tfrac {\pi }{2}}$
$_{2}R^{3}/l$ ${\sqrt {\tfrac {1}{12\phi ^{2}}}}\approx 0.178$ 17°44′40″ ${\text{𝜼}}-{\tfrac {\pi }{6}}$ 90° ${\tfrac {\pi }{2}}$
$_{0}R^{4}/l$ $1$
$_{1}R^{4}/l$ ${\sqrt {\tfrac {5+{\sqrt {5}}}{8}}}\approx 0.951$
$_{2}R^{4}/l$ ${\sqrt {\tfrac {\phi ^{2}}{3}}}\approx 0.934$
$_{3}R^{4}/l$ ${\sqrt {\tfrac {\phi ^{4}}{8}}}\approx 0.926$
${\text{𝜼}}$ 37°44′40″ ${\tfrac {{\text{arc sec }}4}{2}}$
Every regular 4-polytope has its characteristic 4-orthoscheme, an irregular 5-cell.[lower-alpha 88] The characteristic 5-cell of the regular 600-cell is represented by the Coxeter-Dynkin diagram , which can be read as a list of the dihedral angles between its mirror facets. It is an irregular tetrahedral pyramid based on the characteristic tetrahedron of the regular tetrahedron. The regular 600-cell is subdivided by its symmetry hyperplanes into 14400 instances of its characteristic 5-cell that all meet at its center.[lower-alpha 52]
The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 600-cell).[lower-alpha 89] If the regular 600-cell has unit radius and edge length ${\text{𝒍}}={\tfrac {1}{\phi }}\approx 0.618$, its characteristic 5-cell's ten edges have lengths ${\sqrt {\tfrac {2}{3\phi ^{2}}}}$, ${\sqrt {\tfrac {1}{2\phi ^{2}}}}$, ${\sqrt {\tfrac {1}{6\phi ^{2}}}}$ (the exterior right triangle face, the characteristic triangle 𝟀, 𝝓, 𝟁), plus ${\sqrt {\tfrac {3}{4\phi ^{2}}}}$, ${\sqrt {\tfrac {1}{4\phi ^{2}}}}$, ${\sqrt {\tfrac {1}{12\phi ^{2}}}}$ (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the characteristic radii of the regular tetrahedron), plus $1$, ${\sqrt {\tfrac {5+{\sqrt {5}}}{8}}}$, ${\sqrt {\tfrac {\phi ^{2}}{3}}}$, ${\sqrt {\tfrac {\phi ^{4}}{8}}}$ (edges which are the characteristic radii of the 600-cell). The 4-edge path along orthogonal edges of the orthoscheme is ${\sqrt {\tfrac {1}{2\phi ^{2}}}}$, ${\sqrt {\tfrac {1}{6\phi ^{2}}}}$, ${\sqrt {\tfrac {1}{4\phi ^{2}}}}$, ${\sqrt {\tfrac {\phi ^{4}}{8}}}$, first from a 600-cell vertex to a 600-cell edge center, then turning 90° to a 600-cell face center, then turning 90° to a 600-cell tetrahedral cell center, then turning 90° to the 600-cell center.
Reflections
The 600-cell can be constructed by the reflections of its characteristic 5-cell in its own facets (its tetrahedral mirror walls).[lower-alpha 90] Reflections and rotations are related: a reflection in an even number of intersecting mirrors is a rotation.[63][64] For example, a full isoclinic rotation of the 600-cell in decagonal invariant planes takes each of the 120 vertices through 15 vertices and back to itself, on a skew pentadecagram2 geodesic isocline of circumference 5𝝅 that winds around the 3-sphere, as each great decagon rotates (like a wheel) and also tilts sideways (like a coin flipping) in the completely orthogonal plane.[lower-alpha 91] Any set of four orthogonal pairs of antipodal vertices (the 8 vertices of one of the 75 inscribed 16-cells)[lower-alpha 55] performing such an orbit visits 15 * 8 = 120 distinct vertices and generates the 600-cell sequentially in one full isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 120 vertices simultaneously by reflection.[lower-alpha 67]
Weyl orbits
Another construction method uses quaternions and the Icosahedral symmetry of Weyl group orbits $O(\Lambda )=W(H_{4})=I$ of order 120.[66] The following are the orbits of weights of D4 under the Weyl group W(D4):
O(0100) : T = {±1,±e1,±e2,±e3,(±1±e1±e2±e3)/2}
O(1000) : V1
O(0010) : V2
O(0001) : V3
With quaternions $(p,q)$ where ${\bar {p}}$ is the conjugate of $p$ and $[p,q]:r\rightarrow r'=prq$ and $[p,q]^{*}:r\rightarrow r''=p{\bar {r}}q$, then the Coxeter group $W(H_{4})=\lbrace [p,{\bar {p}}]\oplus [p,{\bar {p}}]^{*}\rbrace $ is the symmetry group of the 600-cell and the 120-cell of order 14400.
Given $p\in T$ such that ${\bar {p}}=\pm p^{4},{\bar {p}}^{2}=\pm p^{3},{\bar {p}}^{3}=\pm p^{2},{\bar {p}}^{4}=\pm p$ and $p^{\dagger }$ as an exchange of $-1/\varphi \leftrightarrow \varphi $ within $p$, we can construct:
• the snub 24-cell $S=\sum _{i=1}^{4}\oplus p^{i}T$
• the 600-cell $I=T+S=\sum _{i=0}^{4}\oplus p^{i}T$
• the 120-cell $J=\sum _{i,j=0}^{4}\oplus p^{i}{\bar {p}}^{\dagger j}T'$
Rotations
The regular convex 4-polytopes are an expression of their underlying symmetry which is known as SO(4), the group of rotations[67] about a fixed point in 4-dimensional Euclidean space.[lower-alpha 100]
The 600-cell is generated by isoclinic rotations[lower-alpha 42] of the 24-cell by 36° = 𝜋/5 (the arc of one 600-cell edge length).[lower-alpha 102]
Twenty-five 24-cells
There are 25 inscribed 24-cells in the 600-cell. Therefore there are also 25 inscribed snub 24-cells, 75 inscribed tesseracts and 75 inscribed 16-cells.[lower-alpha 9]
The 8-vertex 16-cell has 4 long diameters inclined at 90° = 𝜋/2 to each other, often taken as the 4 orthogonal axes or basis of the coordinate system.
The 24-vertex 24-cell has 12 long diameters inclined at 60° = 𝜋/3 to each other: 3 disjoint sets of 4 orthogonal axes, each set comprising the diameters of one of 3 inscribed 16-cells, isoclinically rotated by 𝜋/3 with respect to each other.[lower-alpha 103]
The 120-vertex 600-cell has 60 long diameters: not just 5 disjoint sets of 12 diameters, each comprising one of 5 inscribed 24-cells (as we might suspect by analogy), but 25 distinct but overlapping sets of 12 diameters, each comprising one of 25 inscribed 24-cells.[71] There are 5 disjoint 24-cells in the 600-cell, but not just 5: there are 10 different ways to partition the 600-cell into 5 disjoint 24-cells.[lower-alpha 13]
Like the 16-cells and 8-cells inscribed in the 24-cell, the 25 24-cells inscribed in the 600-cell are mutually isoclinic polytopes. The rotational distance between inscribed 24-cells is always an equal-angled rotation of 𝜋/5 in each pair of completely orthogonal invariant planes of rotation.[lower-alpha 101]
Five 24-cells are disjoint because they are Clifford parallel: their corresponding vertices are 𝜋/5 apart on two non-intersecting Clifford parallel[lower-alpha 33] decagonal great circles (as well as 𝜋/5 apart on the same decagonal great circle).[lower-alpha 32] An isoclinic rotation of decagonal planes by 𝜋/5 takes each 24-cell to a disjoint 24-cell (just as an isoclinic rotation of hexagonal planes by 𝜋/3 takes each 16-cell to a disjoint 16-cell).[lower-alpha 104] Each isoclinic rotation occurs in two chiral forms: there are 4 disjoint 24-cells to the left of each 24-cell, and another 4 disjoint 24-cells to its right.[lower-alpha 106] The left and right rotations reach different 24-cells; therefore each 24-cell belongs to two different sets of five disjoint 24-cells.
All Clifford parallel polytopes are isoclinic, but not all isoclinic polytopes are Clifford parallels (completely disjoint objects).[lower-alpha 107] Each 24-cell is isoclinic and Clifford parallel to 8 others, and isoclinic but not Clifford parallel to 16 others.[lower-alpha 7] With each of the 16 it shares 6 vertices: a hexagonal central plane.[lower-alpha 12] Non-disjoint 24-cells are related by a simple rotation by 𝜋/5 in an invariant plane intersecting only two vertices of the 600-cell, a rotation in which the completely orthogonal fixed plane is their common hexagonal central plane. They are also related by an isoclinic rotation in which both planes rotate by 𝜋/5.[lower-alpha 109]
There are two kinds of 𝜋/5 isoclinic rotations which take each 24-cell to another 24-cell.[lower-alpha 104] Disjoint 24-cells are related by a 𝜋/5 isoclinic rotation of an entire fibration of 12 Clifford parallel decagonal invariant planes. (There are 6 such sets of fibers, and a right or a left isoclinic rotation possible with each set, so there are 12 such distinct rotations.)[lower-alpha 106] Non-disjoint 24-cells are related by a 𝜋/5 isoclinic rotation of an entire fibration of 20 Clifford parallel hexagonal invariant planes.[lower-alpha 111] (There are 10 such sets of fibers, so there are 20 such distinct rotations.)[lower-alpha 108]
On the other hand, each of the 10 sets of five disjoint 24-cells is Clifford parallel because its corresponding great hexagons are Clifford parallel. (24-cells do not have great decagons.) The 16 great hexagons in each 24-cell can be divided into 4 sets of 4 non-intersecting Clifford parallel geodesics, each set of which covers all 24 vertices of the 24-cell. The 200 great hexagons in the 600-cell can be divided into 10 sets of 20 non-intersecting Clifford parallel geodesics, each set of which covers all 120 vertices and constitutes a discrete hexagonal fibration. Each of the 10 sets of 20 disjoint hexagons can be divided into five sets of 4 disjoint hexagons, each set of 4 covering a disjoint 24-cell. Similarly, the corresponding great squares of disjoint 24-cells are Clifford parallel.
Rotations on polygram isoclines
The regular convex 4-polytopes each have their characteristic kind of right (and left) isoclinic rotation, corresponding to their characteristic kind of discrete Hopf fibration of great circles.[lower-alpha 59] For example, the 600-cell can be fibrated six different ways into a set of Clifford parallel great decagons, so the 600-cell has six distinct right (and left) isoclinic rotations in which those great decagon planes are invariant planes of rotation. We say these isoclinic rotations are characteristic of the 600-cell because they only emerge in the 600-cell, although they are also found in larger regular polytopes (the 120-cell) which contain inscribed instances of the 600-cell.
Just as the geodesic polygons (decagons or hexagons or squares) in the 600-cell's central planes form fiber bundles of Clifford parallel great circles, the corresponding geodesic skew polygrams (which trace the paths on the Clifford torus of vertices under isoclinic rotation)[75] form fiber bundles of Clifford parallel isoclines: helical circles which wind through all four dimensions.[lower-alpha 42] Since isoclinic rotations are chiral, occurring in left-handed and right-handed forms, each polygon fiber bundle has corresponding left and right polygram fiber bundles.[76] All the fiber bundles are aspects of the same discrete Hopf fibration, because the fibration is the various expressions of the same distinct left-right pair of isoclinic rotations.
Cell rings are another expression of the Hopf fibration. Each discrete fibration has a set of cell-disjoint cell rings that tesselates the 4-polytope.[lower-alpha 54] The isoclines in each chiral bundle spiral around each other: they are axial geodesics of the rings of face-bonded cells. The Clifford parallel cell rings of the fibration nest into each other, pass through each other without intersecting in any cells, and exactly fill the 600-cell with their disjoint cell sets.
Isoclinic rotations rotate a rigid object's vertices along parallel paths, each vertex circling within two completely orthogonal planes, the way a loom weaves a piece of fabric from two orthogonal sets of parallel fibers. A bundle of Clifford parallel great circle polygons and a corresponding bundle of Clifford parallel skew polygram isoclines are the warp and woof of the same distinct left or right isoclinic rotation, which takes Clifford parallel great circle polygons to each other, flipping them like coins and rotating them through a Clifford parallel set of central planes. Meanwhile, because the polygons are also rotating individually like wheels, vertices are displaced along helical Clifford parallel isoclines (the chords of which form the skew polygram), through vertices which lie in successive Clifford parallel polygons.[lower-alpha 58]
In the 600-cell, each family of isoclinic skew polygrams (moving vertex paths in the decagon {10}, hexagon {6}, or square {4} great polygon rotations) can be divided into bundles of non-intersecting Clifford parallel polygram isoclines.[77] The isocline bundles occur in pairs of left and right chirality; the isoclines in each rotation act as chiral objects, as does each distinct isoclinic rotation itself.[lower-alpha 53] Each fibration contains an equal number of left and right isoclines, in two disjoint bundles, which trace the paths of the 600-cell's vertices during the fibration's left or right isoclinic rotation respectively. Each left or right fiber bundle of isoclines by itself constitutes a discrete Hopf fibration which fills the entire 600-cell, visiting all 120 vertices just once. It is a different bundle of fibers than the bundle of Clifford parallel polygon great circles, but the two fiber bundles describe the same discrete fibration because they enumerate those 120 vertices together in the same distinct right (or left) isoclinic rotation, by their intersection as a fabric of cross-woven parallel fibers.
Each isoclinic rotation involves pairs of completely orthogonal invariant central planes of rotation, which both rotate through the same angle. There are two ways they can do this: by both rotating in the "same" direction, or by rotating in "opposite" directions (according to the right hand rule by which we conventionally say which way is "up" on each of the 4 coordinate axes). The right polygram and right isoclinic rotation conventionally correspond to invariant pairs rotating in the same direction; the left polygram and left isoclinic rotation correspond to pairs rotating in opposite directions.[73] Left and right isoclines are different paths that go to different places. In addition, each distinct isoclinic rotation (left or right) can be performed in a positive or negative direction along the circular parallel fibers.
A fiber bundle of Clifford parallel isoclines is the set of helical vertex circles described by a distinct left or right isoclinic rotation. Each moving vertex travels along an isocline contained within a (moving) cell ring. While the left and right isoclinic rotations each double-rotate the same set of Clifford parallel invariant planes of rotation, they step through different sets of great circle polygons because left and right isoclinic rotations hit alternate vertices of the great circle {2p} polygon (where p is a prime ≤ 5).[lower-alpha 114] The left and right rotation share the same Hopf bundle of {2p} polygon fibers, which is both a left and right bundle, but they have different bundles of {p} polygons[78] because the discrete fibers are opposing left and right {p} polygons inscribed in the {2p} polygon.[lower-alpha 115]
A simple rotation is direct and local, taking some vertices to adjacent vertices along great circles, and some central planes to other central planes within the same hyperplane.[lower-alpha 116] In a simple rotation, there is just a single pair of completely orthogonal invariant central planes of rotation; it does not constitute a fibration.
An isoclinic rotation is diagonal and global, taking all the vertices to non-adjacent vertices (two or more edge-lengths away)[lower-alpha 93] along diagonal isoclines, and all the central plane polygons to Clifford parallel polygons (of the same kind). A left-right pair of isoclinic rotations constitutes a discrete fibration. All the Clifford parallel central planes of the fibration are invariant planes of rotation, separated by two equal angles and lying in different hyperplanes.[lower-alpha 47] The diagonal isocline[lower-alpha 94] is a shorter route between the non-adjacent vertices than the multiple simple routes between them available along edges: it is the shortest route on the 3-sphere, the geodesic.
Decagons and pentadecagrams
The fibrations of the 600-cell include 6 fibrations of its 72 great decagons: 6 fiber bundles of 12 great decagons,[lower-alpha 32] each delineating 20 chiral cell rings of 30 tetrahedral cells each,[lower-alpha 31] with three great decagons bounding each cell ring, and five cell rings nesting together around each decagon. The bundle of 12 Clifford parallel decagon fibers is divided into a bundle of 12 left pentagon fibers and a bundle of 12 right pentagon fibers, with each left-right pair of pentagons inscribed in a decagon.[79]
12 great decagons comprise a fiber bundle covering all 120 vertices in a discrete Hopf fibration. There are 20 cell-disjoint 30-cell rings in the fibration, but only 4 completely disjoint 30-cell rings.[lower-alpha 10] The 600-cell has six such discrete decagonal fibrations, and each is the domain (container) of a unique left-right pair of isoclinic rotations (left and right fiber bundles of 12 great pentagons).[lower-alpha 117] Each great decagon belongs to just one fibration,[78] but each 30-cell ring belongs to 5 of the six fibrations (and is completely disjoint from 1 other fibration). The 600-cell contains 72 great decagons, divided among six fibrations, each of which is a set of 20 cell-disjoint 30-cell rings (4 completely disjoint 30-cell rings), but the 600-cell has only 20 distinct 30-cell rings altogether. Each 30-cell ring contains 3 of the 12 Clifford parallel decagons in each of 5 fibrations, and 30 of the 120 vertices.
In these decagonal isoclinic rotations, vertices travel along isoclines which follow the edges of hexagons,[23] advancing a pythagorean distance of one hexagon edge in each double 36°×36° rotational unit.[lower-alpha 111] In an isoclinic rotation, each successive hexagon edge travelled lies in a different great hexagon, so the isocline describes a skew polygram, not a polygon. In a 60°×60° isoclinic rotation (as in the 24-cell's characteristic hexagonal rotation, and below in the hexagonal rotations of the 600-cell) this polygram is a hexagram: the isoclinic rotation follows a 6-edge circular path, just as a simple hexagonal rotation does, although it takes two revolutions to enumerate all the vertices in it, because the isocline is a double loop through every other vertex, and its chords are √3 chords of the hexagon instead of √1 hexagon edges.[lower-alpha 119] But in the 600-cell's 36°×36° characteristic decagonal rotation, successive great hexagons are closer together and more numerous, and the isocline polygram formed by their 15 hexagon edges is a pentadecagram (15-gram).[lower-alpha 91] It is not only not the same period as the hexagon or the simple decagonal rotation, it is not even an integer multiple of the period of the hexagon, or the decagon, or either's simple rotation. Only the compound {30/4}=2{15/2} triacontagram (30-gram), which is two 15-grams rotating in parallel (a black and a white), is a multiple of them all, and so constitutes the rotational unit of the decagonal isoclinic rotation.[lower-alpha 114]
In the 30-cell ring, the non-adjacent vertices linked by isoclinic rotations are two edge-lengths apart, with three other vertices of the ring lying between them.[lower-alpha 121] The two non-adjacent vertices are linked by a √1 chord of the isocline which is a great hexagon edge (a 24-cell edge). The √1 chords of the 30-cell ring (without the √0.𝚫 600-cell edges) form a skew triacontagram{30/4}=2{15/2} which contains 2 disjoint {15/2} Möbius double loops, a left-right pair of pentadecagram2 isoclines. Each left (or right) bundle of 12 pentagon fibers is crossed by a left (or right) bundle of 8 Clifford parallel pentadecagram fibers. Each distinct 30-cell ring has 2 double-loop pentadecagram isoclines running through its even or odd (black or white) vertices respectively.[lower-alpha 97] The pentadecagram helices have no inherent chirality, but each acts as either a left or right isocline in any distinct isoclinic rotation.[lower-alpha 113] The 2 pentadecagram fibers belong to the left and right fiber bundles of 5 different fibrations.
At each vertex, there are six great decagons and six pentadecagram isoclines (six black or six white) that cross at the vertex.[lower-alpha 125] Eight pentadecagram isoclines (four black and four white) comprise a unique right (or left) fiber bundle of isoclines covering all 120 vertices in the distinct right (or left) isoclinic rotation. Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of 12 pentagons and 8 pentadecagram isoclines. There are only 20 distinct black isoclines and 20 distinct white isoclines in the 600-cell. Each distinct isocline belongs to 5 fiber bundles.
Three sets of 30-cell ring chords from the same orthogonal projection viewpoint
Pentadecagram {15/2} Triacontagram {30/4}=2{15/2} Triacontagram {30/6}=6{5}
All edges are pentadecagram isocline chords of length √1, which are also great hexagon edges of 24-cells inscribed in the 600-cell. Only great pentagon edges of length √1.𝚫 ≈ 1.176.
A single black (or white) isocline is a Möbius double loop skew pentadecagram {15/2} of circumference 5𝝅.[lower-alpha 91] The √1 chords are 24-cell edges (hexagon edges) from different inscribed 24-cells. These chords are invisible (not shown) in the 30-cell ring illustration, where they join opposite vertices of two face-bonded tetrahedral cells that are two orange edges apart or two yellow edges apart. The 30-cell ring as a skew compound of two disjoint pentadecagram {15/2} isoclines (a black-white pair, shown here as orange-yellow).[lower-alpha 97] The √1 chords of the isoclines link every 4th vertex of the 30-cell ring in a straight chord under two orange edges or two yellow edges. The doubly-curved isocline is the geodesic (shortest path) between those vertices; they are also two edges apart by three different angled paths along the edges of the face-bonded tetrahedra. Each pentadecagram isocline (at left) intersects all six great pentagons (above) in two or three vertices. The pentagons lie on flat 2𝝅 great circles in the decagon invariant planes of rotation. The pentadecagrams are not flat: they are helical 5𝝅 isocline circles whose 15 chords lie in successive great hexagon planes inclined at 𝝅/5 = 36° to each other. The isocline circle is said to be twisting either left or right with the rotation, but all such pentadecagrams are directly congruent, each acting as a left or right isocline in different fibrations.
No 600-cell edges appear in these illustrations, only invisible interior chords of the 600-cell. In this article, they should all properly be drawn as dashed lines.
Two 15-gram double-loop isoclines are axial to each 30-cell ring. The 30-cell rings are chiral; each fibration contains 10 right (clockwise-spiraling) rings and 10 left (counterclockwise spiraling) rings, but the two isoclines in each 3-cell ring are directly congruent.[lower-alpha 98] Each acts as a left (or right) isocline a left (or right) rotation, but has no inherent chirality.[lower-alpha 113] The fibration's 20 left and 20 right 15-grams altogether contain 120 disjoint open pentagrams (60 left and 60 right), the open ends of which are adjacent 600-cell vertices (one √0.𝚫 edge-length apart). The 30 chords joining the isocline's 30 vertices are √1 hexagon edges (24-cell edges), connecting 600-cell vertices which are two 600-cell √0.𝚫 edges apart on a decagon great circle. [lower-alpha 95] These isocline chords are both hexagon edges and pentagram edges.
The 20 Clifford parallel isoclines (30-cell ring axes) of each left (or right) isocline bundle do not intersect each other. Either distinct decagonal isoclinic rotation (left or right) rotates all 120 vertices (and all 600 cells), but pentadecagram isoclines and pentagons are connected such that vertices alternate as 60 black and 60 white vertices (and 300 black and 300 white cells), like the black and white squares of the chessboard.[lower-alpha 124] In the course of the rotation, the vertices on a left (or right) isocline rotate within the same 15-vertex black (or white) isocline, and the cells rotate within the same black (or white) 30-cell ring.
Hexagons and hexagrams
The fibrations of the 600-cell include 10 fibrations of its 200 great hexagons: 10 fiber bundles of 20 great hexagons. Each fiber bundle[lower-alpha 45] delineates 20 directly congruent cell rings of 6 octahedral cells each, with three cell rings nesting together around each hexagon. The bundle of 20 Clifford parallel hexagon fibers is divided into a bundle of 20 black √3 great triangle fibers and a bundle of 20 white great triangle fibers, with 6 black and 6 white fibers in each 6-octahedron ring and a black and a white triangle inscribed in each hexagon. The black or white fibers are joined into black or white isoclines, each of which is a different kind of great circle of 6 vertices: a skew hexagram helix consisting of 6 √3 chords from 6 different great hexagons, joined end-to-end in a helical loop, and winding twice around the 600-cell through all four dimensions rather than lying flat in a central plane like a √1 great circle hexagon.[lower-alpha 119]
There is a connection between these 10 fibrations of hexagons and hexagrams and the 6 fibrations of decagons and pentadecagrams described above: the √1 edges of the great hexagons are the isocline chords, the edges of the pentadecagrams. Each fiber bundle of 20 hexagons is distributed among the 600-cell's 20 chiral 30-cell rings of 30 tetrahedral cells,[lower-alpha 31] in each of which 5 great hexagons from 5 different fiber bundles provide their 6 √1 edges as chords of a pentadecagram isocline of 15 chords running through the 30-cell ring, with 2 pentadecagrams running through each 30-cell ring, and each hexagon providing 3 non-adjacent edges to each pentadecagram. The bundle of 20 Clifford parallel hexagon fibers is divided into a bundle of 10 left great hexagon fibers and a bundle of 10 right great hexagon fibers, with 5 right (or left) great hexagons contained within each right (or left) 30-cell ring. Conversely, the √3 isocline chords of the 200 great hexagons (the hexagram edges) join every 4th vertex of the 24-cell's {12/5} Petrie dodecagram, the 24-cell's only feature with a 5 in its Schläfli symbol,[83] while the 600-cell's √1 isocline chords (the pentadecagram edges) join every 3rd vertex of an inscribed 24-cell's Petrie dodecagram.
Squares and octagrams
The fibrations of the 600-cell include 15 fibrations of its 450 great squares: 15 fiber bundles of 30 great squares. Each fiber bundle[lower-alpha 46] delineates 30 chiral cell rings of 8 tetrahedral cells each,[lower-alpha 51] with a left and a right cell ring nesting together to fill each of the 15 disjoint 16-cells inscribed in the 600-cell. The bundle of 30 Clifford parallel square fibers is divided into a bundle of 30 black √4 great digon fibers and a bundle of 30 white great digon fibers, with 3 black and 3 white fibers in each 8-tetrahedron ring and a black and a white digon inscribed in each great square (as its diagonals). The black or white fibers are joined into black or white isoclines, each of which is a different kind of great circle of 8 vertices: a skew octagram helix consisting of 8 √4 chords from 4 orthogonal great digons (each used twice), joined end-to-end in a helical loop, and winding twice around the 600-cell through all four dimensions rather than lying flat in a central plane like a √2 great circle square.
There is a connection between these 15 fibrations of squares and octagrams and the 6 fibrations of decagons and pentadecagrams described above: the √4 chords of the 450 great squares are edges of pentadecagrams.[lower-alpha 55] Each fiber bundle of 30 squares is distributed among the 600-cell's 20 chiral 30-cell rings of 30 tetrahedral cells,[lower-alpha 31] in each of which 15 great squares from 15 different fiber bundles provide their 2 √4 chords as edges of 2 regular {15/4} pentadecagrams which wind 4 times around the 600-cell through every 4th vertex of the 30-cell ring. These {15/4} pentadecagrams connect every 4th vertex of one of the two disjoint 15-gons contained in the 30-cell ring, just as the {15/2} pentadecagrams connect every 2nd vertex of the same two 15-gons. The bundle of 30 Clifford parallel square fibers is divided into a bundle of 30 black great digon fibers and a bundle of 30 white great digon fibers, with 15 black (and 15 white) great digons comprising a black (and a white) pentadecagon contained within each right (or left) 30-cell ring.
There is also a connection between these 15 fibrations of squares and octagrams and the 10 fibrations of hexagons and hexagrams described above: the √2 edges of the 450 great squares join every 3rd vertex of the 24-cell's Petrie dodecagon, forming a {12/3}=3{4} dodecagram. The √3 chords of the 200 great hexagons (the hexagram edges) join every 4th vertex of the same Petrie polygon, forming a {12/4}=4{3} dodecagram.
As a configuration
This configuration matrix[84] represents the 600-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 600-cell. The non-diagonal numbers say how many of the column's element occur in or at the row's element.
${\begin{bmatrix}{\begin{matrix}120&12&30&20\\2&720&5&5\\3&3&1200&2\\4&6&4&600\end{matrix}}\end{bmatrix}}$
Here is the configuration expanded with k-face elements and k-figures. The diagonal element counts are the ratio of the full Coxeter group order, 14400, divided by the order of the subgroup with mirror removal.
H4 k-facefkf0f1f2f3k-fig Notes
H3( ) f0 120123020{3,5}H4/H3 = 14400/120 = 120
A1H2{ } f1 272055{5}H4/H2A1 = 14400/10/2 = 720
A2A1{3} f2 3312002{ }H4/A2A1 = 14400/6/2 = 1200
A3{3,3} f3 464600( )H4/A3 = 14400/24 = 600
Symmetries
The icosians are a specific set of Hamiltonian quaternions with the same symmetry as the 600-cell.[85] The icosians lie in the golden field, (a + b√5) + (c + d√5)i + (e + f√5)j + (g + h√5)k, where the eight variables are rational numbers.[86] The finite sums of the 120 unit icosians are called the icosian ring.
When interpreted as quaternions,[lower-alpha 5] the 120 vertices of the 600-cell form a group under quaternionic multiplication. This group is often called the binary icosahedral group and denoted by 2I as it is the double cover of the ordinary icosahedral group I.[88] It occurs twice in the rotational symmetry group RSG of the 600-cell as an invariant subgroup, namely as the subgroup 2IL of quaternion left-multiplications and as the subgroup 2IR of quaternion right-multiplications. Each rotational symmetry of the 600-cell is generated by specific elements of 2IL and 2IR; the pair of opposite elements generate the same element of RSG. The centre of RSG consists of the non-rotation Id and the central inversion −Id. We have the isomorphism RSG ≅ (2IL × 2IR) / {Id, -Id}. The order of RSG equals 120 × 120/2 = 7200.
The binary icosahedral group is isomorphic to SL(2,5).
The full symmetry group of the 600-cell is the Weyl group of H4.[89] This is a group of order 14400. It consists of 7200 rotations and 7200 rotation-reflections. The rotations form an invariant subgroup of the full symmetry group. The rotational symmetry group was first described by S.L. van Oss.[90] The H4 group and its Clifford algebra construction from 3-dimensional symmetry groups by induction is described by Dechant.[91]
Visualization
The symmetries of the 3-D surface of the 600-cell are somewhat difficult to visualize due to both the large number of tetrahedral cells,[lower-alpha 27] and the fact that the tetrahedron has no opposing faces or vertices.[lower-alpha 53] One can start by realizing the 600-cell is the dual of the 120-cell. One may also notice that the 600-cell also contains the vertices of a dodecahedron,[41] which with some effort can be seen in most of the below perspective projections.
2D projections
The H3 decagonal projection shows the plane of the van Oss polygon.
Orthographic projections by Coxeter planes[14]
H4 - F4
[30]
(Red=1)
[20]
(Red=1)
[12]
(Red=1)
H3 A2 / B3 / D4 A3 / B2
[10]
(Red=1,orange=5,yellow=10)
[6]
(Red=1,orange=3,yellow=6)
[4]
(Red=1,orange=2,yellow=4)
3D projections
A three-dimensional model of the 600-cell, in the collection of the Institut Henri Poincaré, was photographed in 1934–1935 by Man Ray, and formed part of two of his later "Shakesperean Equation" paintings.[92]
Vertex-first projection
This image shows a vertex-first perspective projection of the 600-cell into 3D. The 600-cell is scaled to a vertex-center radius of 1, and the 4D viewpoint is placed 5 units away. Then the following enhancements are applied:
• The 20 tetrahedra meeting at the vertex closest to the 4D viewpoint are rendered in solid color. Their icosahedral arrangement is clearly shown.
• The tetrahedra immediately adjoining these 20 cells are rendered in transparent yellow.
• The remaining cells are rendered in edge-outline.
• Cells facing away from the 4D viewpoint (those lying on the "far side" of the 600-cell) have been culled, to reduce visual clutter in the final image.
Cell-first projection
This image shows the 600-cell in cell-first perspective projection into 3D. Again, the 600-cell to a vertex-center radius of 1 and the 4D viewpoint is placed 5 units away. The following enhancements are then applied:
• The nearest cell to the 4d viewpoint is rendered in solid color, lying at the center of the projection image.
• The cells surrounding it (sharing at least 1 vertex) are rendered in transparent yellow.
• The remaining cells are rendered in edge-outline.
• Cells facing away from the 4D viewpoint have been culled for clarity.
This particular viewpoint shows a nice outline of 5 tetrahedra sharing an edge, towards the front of the 3D image.
Frame synchronized orthogonal isometric (left) and perspective (right) projections
Diminished 600-cells
The snub 24-cell may be obtained from the 600-cell by removing the vertices of an inscribed 24-cell and taking the convex hull of the remaining vertices.[93] This process is a diminishing of the 600-cell.
The grand antiprism may be obtained by another diminishing of the 600-cell: removing 20 vertices that lie on two mutually orthogonal rings and taking the convex hull of the remaining vertices.[55]
A bi-24-diminished 600-cell, with all tridiminished icosahedron cells has 48 vertices removed, leaving 72 of 120 vertices of the 600-cell. The dual of a bi-24-diminished 600-cell, is a tri-24-diminished 600-cell, with 48 vertices and 72 hexahedron cells.
There are a total of 314,248,344 diminishings of the 600-cell by non-adjacent vertices. All of these consist of regular tetrahedral and icosahedral cells.[94]
Diminished 600-cells
Name Tri-24-diminished 600-cell Bi-24-diminished 600-cell Snub 24-cell
(24-diminished 600-cell)
Grand antiprism
(20-diminished 600-cell)
600-cell
Vertices 48 72 96 100 120
Vertex figure
(Symmetry)
dual of tridiminished icosahedron
([3], order 6)
tetragonal antiwedge
([2]+, order 2)
tridiminished icosahedron
([3], order 6)
bidiminished icosahedron
([2], order 4)
Icosahedron
([5,3], order 120)
Symmetry Order 144 (48×3 or 72×2) [3+,4,3]
Order 576 (96×6)
[[10,2+,10]]
Order 400 (100×4)
[5,3,3]
Order 14400 (120×120)
Net
Ortho
H4 plane
Ortho
F4 plane
Related polytopes and honeycombs
The 600-cell is one of 15 regular and uniform polytopes with the same H4 symmetry [3,3,5]:[95]
H4 family polytopes
120-cell rectified
120-cell
truncated
120-cell
cantellated
120-cell
runcinated
120-cell
cantitruncated
120-cell
runcitruncated
120-cell
omnitruncated
120-cell
{5,3,3} r{5,3,3} t{5,3,3} rr{5,3,3} t0,3{5,3,3} tr{5,3,3} t0,1,3{5,3,3} t0,1,2,3{5,3,3}
600-cell rectified
600-cell
truncated
600-cell
cantellated
600-cell
bitruncated
600-cell
cantitruncated
600-cell
runcitruncated
600-cell
omnitruncated
600-cell
{3,3,5} r{3,3,5} t{3,3,5} rr{3,3,5} 2t{3,3,5} tr{3,3,5} t0,1,3{3,3,5} t0,1,2,3{3,3,5}
It is similar to three regular 4-polytopes: the 5-cell {3,3,3}, 16-cell {3,3,4} of Euclidean 4-space, and the order-6 tetrahedral honeycomb {3,3,6} of hyperbolic space. All of these have tetrahedral cells.
{3,3,p} polytopes
Space S3 H3
Form Finite Paracompact Noncompact
Name {3,3,3}
{3,3,4}
{3,3,5}
{3,3,6}
{3,3,7}
{3,3,8}
... {3,3,∞}
Image
Vertex
figure
{3,3}
{3,4}
{3,5}
{3,6}
{3,7}
{3,8}
{3,∞}
This 4-polytope is a part of a sequence of 4-polytope and honeycombs with icosahedron vertex figures:
{p,3,5} polytopes
Space S3 H3
Form Finite Compact Paracompact Noncompact
Name {3,3,5}
{4,3,5}
{5,3,5}
{6,3,5}
{7,3,5}
{8,3,5}
... {∞,3,5}
Image
Cells
{3,3}
{4,3}
{5,3}
{6,3}
{7,3}
{8,3}
{∞,3}
The regular complex polygons 3{5}3, and 5{3}5, , in $\mathbb {C} ^{2}$ have a real representation as 600-cell in 4-dimensional space. Both have 120 vertices, and 120 edges. The first has Complex reflection group 3[5]3, order 360, and the second has symmetry 5[3]5, order 600.[96]
Regular complex polytope in orthogonal projection of H4 Coxeter plane[14]
{3,3,5}
Order 14400
3{5}3
Order 360
5{3}5
Order 600
See also
• 24-cell, the predecessor 4-polytope on which the 600-cell is based
• 120-cell, the dual 4-polytope to the 600-cell, and its successor
• Uniform 4-polytope family with [5,3,3] symmetry
• Regular 4-polytope
• Polytope
Notes
1. In the curved 3-dimensional space of the 600-cell's boundary surface, at each vertex one finds the twelve nearest other vertices surrounding the vertex the way an icosahedron's vertices surround its center. Twelve 600-cell edges converge at the icosahedron's center, where they appear to form six straight lines which cross there. However, the center is actually displaced in the 4th dimension (radially outward from the center of the 600-cell), out of the hyperplane defined by the icosahedron's vertices. Thus the vertex icosahedron is actually a canonical icosahedral pyramid,[lower-alpha 62] composed of 20 regular tetrahedra on a regular icosahedron base, and the vertex is its apex.[lower-alpha 63]
2. One might ask whether dimensional analogy "always works", or if it is perhaps "just guesswork" that might sometimes be incapable of producing a correct dimensionally analogous figure, especially when reasoning from a lower to a higher dimension. Apparently dimensional analogy in both directions has firm mathematical foundations. Dechant[38] derived the 4D symmetry groups from their 3D symmetry group counterparts by induction, demonstrating that there is nothing in 4D symmetry that is not already inherent in 3D symmetry. He showed that neither 4D symmetry nor 3D symmetry is more fundamental than the other, as either can be derived from the other. This is true whether dimensional analogies are computed using Coxeter group theory, or Clifford geometric algebra. These two rather different kinds of mathematics contribute complementary geometric insights.
3. The convex regular 4-polytopes can be ordered by size as a measure of 4-dimensional content (hypervolume) for the same radius. Each greater polytope in the sequence is rounder than its predecessor, enclosing more content[3] within the same radius. The 4-simplex (5-cell) is the limit smallest case, and the 120-cell is the largest. Complexity (as measured by comparing configuration matrices or simply the number of vertices) follows the same ordering. This provides an alternative numerical naming scheme for regular polytopes in which the 600-cell is the 120-point 4-polytope: fifth in the ascending sequence that runs from 5-point 4-polytope to 600-point 4-polytope.
4. The edge length will always be different unless predecessor and successor are both radially equilateral, i.e. their edge length is the same as their radius (so both are preserved). Since radially equilateral polytopes are rare, the only such construction (in any dimension) is from the 8-cell to the 24-cell.
5. In four-dimensional Euclidean geometry, a quaternion is simply a (w, x, y, z) Cartesian coordinate. Hamilton did not see them as such when he discovered the quaternions. Schläfli would be the first to consider four-dimensional Euclidean space, publishing his discovery of the regular polyschemes in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space.[87] Although he described a quaternion as an ordered four-element multiple of real numbers, the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.
6. The 600-cell geometry is based on the 24-cell. The 600-cell rounds out the 24-cell with 2 more great circle polygons (exterior decagon and interior pentagon), adding 4 more chord lengths which alternate with the 24-cell's 4 chord lengths.
7. A 24-cell contains 16 hexagons. In the 600-cell, with 25 24-cells, each 24-cell is disjoint from 8 24-cells and intersects each of the other 16 24-cells in six vertices that form a hexagon.[9] A 600-cell contains 25・16/2 = 200 such hexagons.
8. In cases where inscribed 4-polytopes of the same kind occupy disjoint sets of vertices (such as the two 16-cells inscribed in the tesseract, or the three 16-cells inscribed in the 24-cell), their sets of vertex chords and central polygons must likewise be disjoint. In the cases where they share vertices (such as the three tesseracts inscribed in the 24-cell, or the 25 24-cells inscribed in the 600-cell), they also share some vertex chords and central polygons.[lower-alpha 7]
9. The 600-cell contains exactly 25 24-cells, 75 16-cells and 75 8-cells, with each 16-cell and each 8-cell lying in just one 24-cell.[18]
10. Polytopes are completely disjoint if all their element sets are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.
11. Each of the 25 24-cells of the 600-cell contains exactly one vertex of each great pentagon.[9] Six pentagons intersect at each 600-cell vertex, so each 24-cell intersects all 144 great pentagons.
12. Five 24-cells meet at each icosahedral pyramid apex[lower-alpha 1] of the 600-cell. Each 24-cell shares not just one vertex but 6 vertices (one of its four hexagonal central planes) with each of the other four 24-cells.[lower-alpha 7]
13. Schoute was the first to state (a century ago) that there are exactly ten ways to partition the 120 vertices of the 600-cell into five disjoint 24-cells. The 25 24-cells can be placed in a 5 x 5 array such that each row and each column of the array partitions the 120 vertices of the 600-cell into five disjoint 24-cells. The rows and columns of the array are the only ten such partitions of the 600-cell.[18]
14. The 600-cell contains 25 distinct 24-cells, bound to each other by pentagonal rings. Each pentagon links five completely disjoint[lower-alpha 10] 24-cells together, the collective vertices of which are the 120 vertices of the 600-cell. Each 24-point 24-cell contains one fifth of all the vertices in the 120-point 600-cell, and is linked to the other 96 vertices (which comprise a snub 24-cell) by the 600-cell's 144 pentagons. Each of the 25 24-cells intersects each of the 144 great pentagons at just one vertex.[lower-alpha 11] Five 24-cells meet at each 600-cell vertex,[lower-alpha 12] so all 25 24-cells are linked by each great pentagon. The 600-cell can be partitioned into five disjoint 24-cells (10 different ways),[lower-alpha 13] and also into 24 disjoint pentagons (inscribed in the 12 Clifford parallel great decagons of one of the 6 decagonal fibrations) by choosing a pentagon from the same fibration at each 24-cell vertex.
15. In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is completely orthogonal to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.
16. Two flat planes A and B of a Euclidean space of four dimensions are called completely orthogonal if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.[lower-alpha 15]
17. The angles 𝜉i and 𝜉j are angles of rotation in the two completely orthogonal[lower-alpha 16] invariant planes which characterize rotations in 4-dimensional Euclidean space. The angle 𝜂 is the inclination of both these planes from the north-south pole axis, where 𝜂 ranges from 0 to 𝜋/2. The (𝜉i, 0, 𝜉j) coordinates describe the great circles which intersect at the north and south pole ("lines of longitude"). The (𝜉i, 𝜋/2, 𝜉j) coordinates describe the great circles orthogonal to longitude ("equators"); there is more than one "equator" great circle in a 4-polytope, as the equator of a 3-sphere is a whole 2-sphere of great circles. The other Hopf coordinates (𝜉i, 0 < 𝜂 < 𝜋/2, 𝜉j) describe the great circles (not "lines of latitude") which cross an equator but do not pass through the north or south pole.
18. The conversion from Hopf coordinates (𝜉i, 𝜂, 𝜉j) to unit-radius Cartesian coordinates (w, x, y, z) is:
w = cos 𝜉i sin 𝜂
x = cos 𝜉j cos 𝜂
y = sin 𝜉j cos 𝜂
z = sin 𝜉i sin 𝜂
The "Hopf north pole" (0, 0, 0) is Cartesian (0, 1, 0, 0).
Cartesian (1, 0, 0, 0) is Hopf (0, 𝜋/2, 0).
19. The Hopf coordinates[10] are triples of three angles:
(𝜉i, 𝜂, 𝜉j)
that parameterize the 3-sphere by numbering points along its great circles. A Hopf coordinate describes a point as a rotation from the "north pole" (0, 0, 0).[lower-alpha 17] Hopf coordinates are a natural alternative to Cartesian coordinates[lower-alpha 18] for framing regular convex 4-polytopes, because the group of 4-dimensional rotations, denoted SO(4), generates those polytopes.
20. There are 600 permutations of these coordinates, but there are only 120 vertices in the 600-cell. These are actually the Hopf coordinates of the vertices of the 120-cell, which has 600 vertices and can be seen (two different ways) as a compound of 5 disjoint 600-cells.
21. The fractional-root golden chords exemplify that the golden ratio φ is a circle ratio related to 𝜋:[17]
𝜋/5 = arccos (φ/2)
is one decagon edge, the 𝚽 = √0.𝚫 = √0.382~ ≈ 0.618 chord. Reciprocally, in this function discovered by Robert Everest expressing ϕ as a function of 𝜋 and the numbers 1, 2, 3 and 5 of the Fibonacci series:
φ = 1 – 2 cos (3𝜋/5)
3𝜋/5 is the arc length of the φ = √2.𝚽 = √2.618~ ≈ 1.618 chord.
22. The 600-cell edges are decagon edges of length √0.𝚫, which is 𝚽, the smaller golden section of √5; the edges are in the inverse golden ratio 1/φ to the √1 hexagon chords (the 24-cell edges). The other fractional-root chords exhibit golden relationships as well. The chord of length √1.𝚫 is a pentagon edge. The next fractional-root chord is a decagon diagonal of length √2.𝚽 which is φ, the larger golden section of √5; it is in the golden ratio[lower-alpha 21] to the √1 chord (and the radius).[lower-alpha 25] The last fractional-root chord is the pentagon diagonal of length √3.𝚽. The diagonal of a regular pentagon is always in the golden ratio to its edge, and indeed φ√1.𝚫 is √3.𝚽.
23. The long radius (center to vertex) of the 600-cell is in the golden ratio to its edge length; thus its radius is φ if its edge length is 1, and its edge length is 1/φ if its radius is 1. Only a few uniform polytopes have this property, including the four-dimensional 600-cell, the three-dimensional icosidodecahedron, and the two-dimensional decagon. (The icosidodecahedron is the equatorial cross section of the 600-cell, and the decagon is the equatorial cross section of the icosidodecahedron.) Radially golden polytopes are those which can be constructed, with their radii, from golden triangles[lower-alpha 28] which meet at the center, each contributing two radii and an edge.
24. The fractional square roots are given as decimal fractions where:
𝚽 ≈ 0.618 is the inverse golden ratio 1/φ
𝚫 = 1 - 𝚽 = 𝚽2 ≈ 0.382
For example:
𝚽 = √0.𝚫 = √0.382~ ≈ 0.618
25. Notice in the diagram how the φ chord (the larger golden section) sums with the adjacent 𝚽 edge (the smaller golden section) to √5, as if together they were a √5 chord bent to fit inside the √4 diameter.
26. Consider one of the 24-vertex 24-cells inscribed in the 120-vertex 600-cell. The other 96 vertices constitute a snub 24-cell. Removing any one 24-cell from the 600-cell produces a snub 24-cell.
27. Each tetrahedral cell touches, in some manner, 56 other cells. One cell contacts each of the four faces; two cells contact each of the six edges, but not a face; and ten cells contact each of the four vertices, but not a face or edge.
28. A golden triangle is an isosceles triangle in which the duplicated side a is in the golden ratio to the distinct side b:
a/b = φ = 1 + √5/2 ≈ 1.618
It can be found in a regular decagon by connecting any two adjacent vertices to the center, and in the regular pentagon by connecting any two adjacent vertices to the vertex opposite them.
The vertex angle is:
𝛉 = arccos(φ/2) = 𝜋/5 = 36°
so the base angles are each 2𝜋/5 = 72°. The golden triangle is uniquely identified as the only triangle to have its three angles in 2:2:1 proportions.
29. Beginning with the 16-cell, every regular convex 4-polytope in the unit-radius sequence is inscribed in its successor.[5] Therefore the successor may be constructed by placing 4-pyramids of some kind on the cells of its predecessor. Between the 16-cell and the tesseract, we have 16 right tetrahedral pyramids, with their apexes filling the corners of the tesseract. Between the tesseract and the 24-cell, we have 8 canonical cubic pyramids. But if we place 24 canonical octahedral pyramids on the 24-cell, we only get another tesseract (of twice the radius and edge length), not the successor 600-cell. Between the 24-cell and the 600-cell there must be 24 smaller, irregular 4-pyramids on a regular octahedral base.
30. The six great decagons which pass by each tetrahedral cell along its edges do not all intersect with each other, because the 6 edges of the tetrahedron do not all share a vertex. Each decagon intersects four of the others (at 60 degrees), but just misses one of the others as they run past each other (at 90 degrees) along the opposite and perpendicular skew edges of the tetrahedron. Each tetrahedron links three pairs of decagons which do not intersect at a vertex of the tetrahedron. However, none of the six decagons are Clifford parallel;[lower-alpha 33] each belongs to a different Hopf fiber bundle of 12. Only one of the tetrahedron's six edges may be part of a helix in any one Boerdijk–Coxeter triple helix ring.[lower-alpha 31] Incidentally, this footnote is one of a tetrahedron of four footnotes about Clifford parallel decagons[lower-alpha 32] that all reference each other.
31. Since tetrahedra[lower-alpha 30] do not have opposing faces, the only way they can be stacked face-to-face in a straight line is in the form of a twisted chain called a Boerdijk-Coxeter helix. This is a Clifford parallel[lower-alpha 33] triple helix as shown in the illustration. In the 600-cell we find them bent in the fourth dimension into geodesic rings. Each ring has 30 cells and touches 30 vertices. The cells are each face-bonded to two adjacent cells, but one of the six edges of each tetrahedron belongs only to that cell, and these 30 edges form 3 Clifford parallel great decagons which spiral around each other.[lower-alpha 32] 5 30-cell rings meet at and spiral around each decagon (as 5 tetrahedra meet at each edge). A bundle of 20 such cell-disjoint rings fills the entire 600-cell, thus constituting a discrete Hopf fibration. There are 6 distinct such Hopf fibrations, covering the same space but running in different "directions".
32. Two Clifford parallel[lower-alpha 33] great decagons don't intersect, but their corresponding vertices are linked by one edge of another decagon. The two parallel decagons and the ten linking edges form a double helix ring. Three decagons can also be parallel (decagons come in parallel fiber bundles of 12) and three of them may form a triple helix ring. If the ring is cut and laid out flat in 3-space, it is a Boerdijk–Coxeter helix[lower-alpha 31] 30 tetrahedra[lower-alpha 30] long. The three Clifford parallel decagons can be seen as the cyan edges in the triple helix illustration. Each magenta edge is one edge of another decagon linking two parallel decagons.
33. Clifford parallels are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point. A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the 3-sphere.[20] Whereas in 3-dimensional space, any two geodesic great circles on the 2-sphere will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. They spiral around each other in Hopf fiber bundles which, in the 600-cell, visit all 120 vertices just once. For example, each of the 600 tetrahedra participates in 6 great decagons[lower-alpha 30] belonging to 6 discrete Hopf fibrations, each filling the whole 600-cell. Each fibration is a bundle of 12 Clifford parallel decagons which form 20 cell-disjoint intertwining rings of 30 tetrahedral cells,[lower-alpha 31] each bounded by three of the 12 great decagons.[lower-alpha 32]
34. The 10 hexagons which cross at each vertex lie along the 20 short radii of the icosahedral vertex figure.[lower-alpha 1]
35. The 25 inscribed 24-cells each have 3 inscribed tesseracts, which each have 8 √1 cubic cells. The 1200 √3 chords are the 4 long diameters of these 600 cubes; the 3 tesseracts overlap and each chord is the long diameter of a cube in two different tesseracts.
36. The sum of 0.𝚫・720 + 1・1200 + 1.𝚫・720 + 2・1800 + 2.𝚽・720 + 3・1200 + 3.𝚽・720 + 4・60 is 14,400.
37. The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.[25]
38. A triacontagon or 30-gon is a thirty-sided polygon. The triacontagon is the largest regular polygon whose interior angle is the sum of the interior angles of smaller polygons: 168° is the sum of the interior angles of the equilateral triangle (60°) and the regular pentagon (108°).
39. The 600-cell has 72 great 30-gons: 6 sets of 12 Clifford parallel 30-gon central planes, each completely orthogonal to a decagon central plane. Unlike the great circles of the unit-radius 600-cell that pass through its vertices, this 30-gon is not actually a great circle of the unit-radius 3-sphere. Because it passes through face centers rather than vertices, it has a shorter radius and lies on a smaller 3-sphere. Of course, there is also a unit-radius great circle in this central plane completely orthogonal to a decagon central plane, but as a great circle polygon it is a 0-gon, not a 30-gon, because it intersects none of the points of the 600-cell. In the 600-cell, the great circle polygon completely orthogonal to each great decagon is a 0-gon.
40. The 30 vertices and 30 edges of the 30-cell ring lie on a skew {30/11} star polygon with a winding number of 11 called a triacontagram11, a continuous tight corkscrew helix bent into a loop of 30 edges (the magenta edges in the triple helix illustration), which winds 11 times around itself in the course of a single revolution around the 600-cell, accompanied by a single 360 degree twist of the 30-cell ring.[31] The same 30-cell ring can also be characterized as the Petrie polygon of the 600-cell.[lower-alpha 81]
41. Each great decagon central plane is completely orthogonal[lower-alpha 16] to a great 30-gon[lower-alpha 38] central plane which does not intersect any vertices of the 600-cell. The 72 30-gons are each the center axis of a 30-cell Boerdijk–Coxeter triple helix ring,[lower-alpha 31] with each segment of the 30-gon passing through a tetrahedron similarly. The 30-gon great circle resides completely in the curved 3-dimensional surface of its 3-sphere;[lower-alpha 39] its curved segments are not chords. It does not touch any edges or vertices, but it does hit faces. It is the central axis of a spiral skew 30-gram, the Petrie polygon of the 600-cell which links all 30 vertices of the 30-cell Boerdijk–Coxeter helix, with three of its edges in each cell.[lower-alpha 40]
42. A point under isoclinic rotation traverses the diagonal[lower-alpha 94] straight line of a single isoclinic geodesic, reaching its destination directly, instead of the bent line of two successive simple geodesics. A geodesic is the shortest path through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do not lie in a single plane; they are 4-dimensional spirals rather than simple 2-dimensional circles.[lower-alpha 58] But they are not like 3-dimensional screw threads either, because they form a closed loop like any circle.[lower-alpha 95] Isoclinic geodesics are 4-dimensional great circles, and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.[lower-alpha 83] They are true circles,[lower-alpha 91] and even form fibrations like ordinary 2-dimensional great circles. These isoclines are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere[lower-alpha 96] they always occur in chiral pairs as Villarceau circles on the Clifford torus,[lower-alpha 99] the geodesic paths traversed by vertices in an isoclinic rotation. They are helices bent into a Möbius loop in the fourth dimension, taking a diagonal winding route around the 3-sphere through the non-adjacent vertices of a 4-polytope's skew Clifford polygon.[lower-alpha 98]
43. In 4-space no more than 4 great circles may be Clifford parallel[lower-alpha 33] and all the same angular distance apart.[27] Such central planes are mutually isoclinic: each pair of planes is separated by two equal angles, and an isoclinic rotation by that angle will bring them together. Where three or four such planes are all separated by the same angle, they are called equi-isoclinic.
44. The decagonal planes in the 600-cell occur in equi-isoclinic[lower-alpha 43] groups of 3, everywhere 3 Clifford parallel decagons 36° (𝝅/5) apart form a 30-cell Boerdijk–Coxeter triple helix ring.[lower-alpha 31] Also Clifford parallel to those 3 decagons are 3 equi-isoclinic decagons 72° (2𝝅/5) apart, 3 108° (3𝝅/5) apart, and 3 144° (4𝝅/5) apart, for a total of 12 Clifford parallel decagons (120 vertices) that comprise a discrete Hopf fibration. Because the great decagons lie in isoclinic planes separated by two equal angles, their corresponding vertices are separated by a combined vector relative to both angles. Vectors in 4-space may be combined by quaternionic multiplication, discovered by Hamilton.[28] The corresponding vertices of two great polygons which are 36° (𝝅/5) apart by isoclinic rotation are 60° (𝝅/3) apart in 4-space. The corresponding vertices of two great polygons which are 108° (3𝝅/5) apart by isoclinic rotation are also 60° (𝝅/3) apart in 4-space. The corresponding vertices of two great polygons which are 72° (2𝝅/5) apart by isoclinic rotation are 120° (2𝝅/3) apart in 4-space, and the corresponding vertices of two great polygons which are 144° (4𝝅/5) apart by isoclinic rotation are also 120° (2𝝅/3) apart in 4-space.
45. The hexagonal planes in the 600-cell occur in equi-isoclinic[lower-alpha 43] groups of 4, everywhere 4 Clifford parallel hexagons 60° (𝝅/3) apart form a 24-cell. Also Clifford parallel to those 4 hexagons are 4 equi-isoclinic hexagons 36° (𝝅/5) apart, 4 72° (2𝝅/5) apart, 4 108° (3𝝅/5) apart, and 4 144° (4𝝅/5) apart, for a total of 20 Clifford parallel hexagons (120 vertices) that comprise a discrete Hopf fibration.
46. The square planes in the 600-cell occur in equi-isoclinic[lower-alpha 43] groups of 2, everywhere 2 Clifford parallel squares 90° (𝝅/2) apart form a 16-cell. Also Clifford parallel to those 2 squares are 4 equi-isoclinic groups of 4, where 3 Clifford parallel 16-cells 60° (𝝅/3) apart form a 24-cell. Also Clifford parallel are 4 equi-isoclinic groups of 3: 3 36° (𝝅/5) apart, 3 72° (2𝝅/5) apart, 3 108° (3𝝅/5) apart, and 3 144° (4𝝅/5) apart, for a total of 30 Clifford parallel squares (120 vertices) that comprise a discrete Hopf fibration.
47. Two angles are required to fix the relative positions of two planes in 4-space.[26] Since all planes in the same hyperplane are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great decagons are a multiple (from 0 to 4) of 36° (𝝅/5) apart in each angle, and may be the same angle apart in both angles.[lower-alpha 44] Great hexagons may be 60° (𝝅/3) apart in one or both angles, and may be a multiple (from 0 to 4) of 36° (𝝅/5) apart in one or both angles.[lower-alpha 45] Great squares may be 90° (𝝅/2) apart in one or both angles, may be 60° (𝝅/3) apart in one or both angles, and may be a multiple (from 0 to 4) of 36° (𝝅/5) apart in one or both angles.[lower-alpha 46] Planes which are separated by two equal angles are called isoclinic.[lower-alpha 43] Planes which are isoclinic have Clifford parallel great circles.[lower-alpha 33] A great hexagon and a great decagon are neither isoclinic nor Clifford parallel; they are separated by a 𝝅/3 (60°) angle and a multiple (from 1 to 4) of 𝝅/5 (36°) angle.
48. In the 24-cell each great square plane is completely orthogonal[lower-alpha 16] to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two antipodal vertices: a great digon plane.
49. Each Hopf fibration of the 3-sphere into Clifford parallel great circle fibers has a map (called its base) which is an ordinary 2-sphere.[39] On this map each great circle fiber appears as a single point. The base of a great decagon fibration of the 600-cell is the icosahedron, in which each vertex represents one of the 12 great decagons.[21] To a toplogist the base is not necessarily any part of the thing it maps: the base icosahedron is not expected to be a cell or interior feature of the 600-cell, it is merely the dimensionally analogous sphere,[lower-alpha 2] useful for reasoning about the fibration. But in fact the 600-cell does have icosahedra in it: 120 icosahedral vertex figures,[lower-alpha 1] any of which can be seen as its base: a 3-dimensional 1:10 scale model of the whole 4-dimensional 600-cell. Each 3-dimensional vertex icosahedron is lifted to the 4-dimensional 600-cell by a 720 degree isoclinic rotation,[lower-alpha 42] which takes each of its 4 disjoint triangular faces in a circuit around one of 4 disjoint 30-vertex rings of 30 tetrahedral cells (each braided of 3 Clifford parallel great decagons), and so visits all 120 vertices of the 600-cell. Since the 12 decagonal great circles (of the 4 rings) are Clifford parallel decagons of the same fibration, we can see geometrically how the icosahedron works as a map of a Hopf fibration of the entire 600-cell, and how the Hopf fibration is an expression of an isoclinic symmetry.[40]
50. The regular skew 30-gon is the Petrie polygon of the 600-cell and its dual the 120-cell. The Petrie polygons of the 120-cell occur in the 600-cell as duals of the 30-cell Boerdijk–Coxeter helix rings: connecting their 30 cell centers together produces the Petrie polygons of the dual 120-cell, as noticed by Rolfdieter Frank (circa 2001). Thus he discovered that the vertex set of the 120-cell partitions into 20 non-intersecting Petrie polygons. This set of 20 disjoint Clifford parallel skew polygons is a discrete Hopf fibration of the 120-cell (just as their 20 dual 30-cell rings are a discrete fibration of the 600-cell).
51. These are the √2 tetrahedral cells of the 75 inscribed 16-cells, not the √0.𝚫 tetrahedral cells of the 600-cell.
52. ‟The Petrie polygons of the Platonic solid $\{p,q\}$ correspond to equatorial polygons of the truncation $\{{\tfrac {p}{q}}\}$ and to equators of the simplicially subdivided spherical tessellation $\{p,q\}$. This "simplicial subdivision" is the arrangement of $g=g_{p,q}$ right-angled spherical triangles into which the sphere is decomposed by the planes of symmetry of the solid. The great circles that lie in these planes were formerly called "lines of symmetry", but perhaps a more vivid name is reflecting circles. The analogous simplicial subdivision of the spherical honeycomb $\{p,q,r\}$ consists of the $g=g_{p,q,r}$ tetrahedra 0123 into which a hypersphere (in Euclidean 4-space) is decomposed by the hyperplanes of symmetry of the polytope $\{p,q,r\}$. The great spheres which lie in these hyperplanes are naturally called reflecting spheres. Since the orthoscheme has no obtuse angles, it entirely contains the arc that measures the absolutely shortest distance 𝝅/h [between the] 2h tetrahedra [that] are strung like beads on a necklace, or like a "rotating ring of tetrahedra" ... whose opposite edges are generators of a helicoid. The two opposite edges of each tetrahedron are related by a screw-displacement.[lower-alpha 67] Hence the total number of spheres is 2h.”[62]
53. The fibration's Clifford parallel cell rings may or may not be chiral objects, depending upon whether the 4-polytope's cells have opposing faces or not. The characteristic cell rings of the 16-cell and 600-cell (with tetrahedral cells) are chiral: they twist either clockwise or counterclockwise. Isoclines acting with either left or right chirality (not both) run through cell rings of this kind, though each fibration contains both left and right cell rings.[lower-alpha 113] The characteristic cell rings of the tesseract, 24-cell and 120-cell (with cubical, octahedral, and dodecahedral cells respectively) are directly congruent, not chiral: there is only one kind of characteristic cell ring in each of these 4-polytopes, and it is not twisted (it has no torsion). Pairs of left-handed and right-handed isoclines run through cell rings of this kind. Note that all these 4-polytopes (except the 16-cell) contain fibrations of their inscribed predecessors' characteristic cell rings in addition to their own characteristic fibrations, so the 600-cell contains both chiral and directly congruent cell rings.
54. The choice of a partitioning of a regular 4-polytope into cell rings is arbitrary, because all of its cells are identical. No particular fibration is distinguished, unless the 4-polytope is rotating. In isoclinic rotations, one set of cell rings (one fibration) is distinguished as the unique container of that distinct left-right pair of rotations and its isoclines.
55. The only way to partition the 120 vertices of the 600-cell into 4 completely disjoint 30-vertex, 30-cell rings[lower-alpha 31] is by partitioning each of 15 completely disjoint 16-cells similarly into 4 symmetric parts: 4 antipodal vertex pairs lying on the 4 orthogonal axes of the 16-cell. The 600-cell contains 75 distinct 16-cells which can be partitioned into sets of 15 completely disjoint 16-cells. In any set of 4 completely disjoint 30-cell rings, there is a set of 15 completely disjoint 16-cells, with one axis of each 16-cell in each 30-cell ring.
56. Unlike their bounding decagons, the 20 cell rings themselves are not all Clifford parallel to each other, because only completely disjoint polytopes are Clifford parallel.[lower-alpha 10] The 20 cell rings have 5 different subsets of 4 Clifford parallel cell rings. Each cell ring is bounded by 3 Clifford parallel great decagons, so each subset of 4 Clifford parallel cell rings is bounded by a total of 12 Clifford parallel great decagons (a discrete Hopf fibration). In fact each of the 5 different subsets of 4 cell rings is bounded by the same 12 Clifford parallel great decagons (the same Hopf fibration); there are 5 different ways to see the same 12 decagons as a set of 4 cell rings (and equivalently, just one way to see them as a single set of 20 cell rings).
57. Note that the differently colored helices of cells are different cell rings (or ring-shaped holes) in the same fibration, not the different fibrations of the 4-polytope. Each fibration is the entire 4-polytope.
58. In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be invariant because the points in each stay in their places in the plane as the plane moves, rotating and tilting sideways by the angle that the other plane rotates.
59. The invariant axis of a rotating 2-sphere is dimensionally analogous to the pair of invariant planes of a rotating 3-sphere. The poles of the rotating 2-sphere are dimensionally analogous to linked great circles on the 3-sphere. By dimensional analogy, each 1D point in 3D lifts to a 2D line in 4D, in this case a circle.[lower-alpha 49] The two antipodal rotation poles lift to a pair of circular Hopf fibers which are not merely Clifford parallel and interlinked,[lower-alpha 33] but also completely orthogonal.[lower-alpha 16] The invariant great circles of the 4D rotation are its poles. In the case of an isoclinic rotation, there is not merely one such pair of 2D poles (completely orthogonal Hopf great circle fibers), there are many such pairs: a finite number of circle-pairs if the 3-sphere fibration is discrete (e.g. a regular polytope with a finite number of vertices), or else an infinite number of orthogonal circle-pairs, entirely filling the 3-sphere. Every point in the curved 3-space of the 3-sphere lies on one such circle (never on two, since the completely orthogonal circles, like all the Clifford parallel Hopf great circle fibers, do not intersect). Where a 2D rotation has one pole, and a 3D rotation of a 2-sphere has 2 poles, an isoclinic 4D rotation of a 3-sphere has nothing but poles, an infinite number of them. In a discrete 4-polytope, all the Clifford parallel invariant great polygons of the rotation are poles, and they fill the 4-polytope, passing through every vertex just once. In one full revolution of such a rotation, every point in the space loops exactly once through its pole-circle.[lower-alpha 112] The circles are arranged with a surprising symmetry, so that each pole-circle links with every other pole-circle, like a maximally dense fabric of 4D chain mail in which all the circles are linked through each other, but no two circles ever intersect.
60. The 4 red faces of the snub tetrahedron correspond to the 4 completely disjoint cell rings of the sparse construction of the fibration (its subfibration). The red faces are centered on the vertices of an inscribed tetrahedron, and lie in the center of the larger faces of an inscribing tetrahedron.
61. Because the octahedron can be snub truncated yielding an icosahedron,[45] another name for the icosahedron is snub octahedron. This term refers specifically to a lower symmetry arrangement of the icosahedron's faces (with 8 faces of one color and 12 of another).
62. The 120-point 600-cell has 120 overlapping icosahedral pyramids.[lower-alpha 1]
63. The icosahedron is not radially equilateral in Euclidean 3-space, but an icosahedral pyramid is radially equilateral in the curved 3-space of the 600-cell's surface (the 3-sphere). In 4-space the 12 edges radiating from its apex are not actually its radii: the apex of the icosahedral pyramid is not actually its center, just one of its vertices. But in curved 3-space the edges radiating symmetrically from the apex are radii, so the icosahedron is radially equilateral in that curved 3-space. In Euclidean 4-space 24 edges radiating symmetrically from a central point make the radially equilateral 24-cell, and a symmetrical subset of 16 of those edges make the radially equilateral tesseract.
64. An icosahedron edge between two blue faces is surrounded by two blue-faced icosahedral pyramid cells and 3 cells from an adjacent cluster of 5 cells (one of which is the central tetrahedron of the five)
65. The pentagonal pyramids around each vertex of the "snub octahedron" icosahedron all look the same, with two yellow and three blue faces. Each pentagon has five distinct rotational orientations. Rotating any pentagonal pyramid rotates all of them, so the five rotational positions are the only five different ways to arrange the colors.
66. Notice that the contraction is chiral, since there are two choices of diagonal on which to begin folding the square faces.
67. Let Q denote a rotation, R a reflection, T a translation, and let Qq Rr T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q2 is a double rotation (in four dimensions). Every orthogonal transformation is expressible as
Qq Rr
where 2q + r ≤ n, the number of dimensions.[65]
68. There is a vertex icosahedron[lower-alpha 1] inside each 24-cell octahedral central section (not inside a √1 octahedral cell, but in the larger √2 octahedron that lies in a central hyperplane), and a larger icosahedron inside each 24-cell cuboctahedron. The two different-sized icosahedra are the second and fourth sections of the 600-cell (beginning with a vertex). The octahedron and the cuboctahedron are the central sections of the 24-cell (beginning with a vertex and beginning with a cell, respectively).[47] The cuboctahedron, large icosahedron, octahedron, and small icosahedron nest like Russian dolls and are related by a helical contraction.[48] The contraction begins with the square faces of the cuboctahedron folding inward along their diagonals to form pairs of triangles.[lower-alpha 66] The 12 vertices of the cuboctahedron move toward each other to the points where they form a regular icosahedron (the large icosahedron); they move slightly closer together until they form a Jessen's icosahedron; they continue to spiral toward each other until they merge into the 8 vertices of the octahedron;[49] and they continue moving along the same helical paths, separating again into the 12 vertices of the snub octahedron (the small icosahedron).[lower-alpha 61] The geometry of this sequence of transformations[lower-alpha 67] in S3 is similar to the kinematics of the cuboctahedron and the tensegrity icosahedron in R3. The twisting, expansive-contractive transformations between these polyhedra were named Jitterbug transformations by Buckminster Fuller.[50]
69. These 12 cells are edge-bonded to the central cell, face-bonded to the exterior faces of the cluster of 5, and face-bonded to each other in pairs. They are blue-faced cells in the 6 different icosahedral pyramids surrounding the cluster of 5.
70. The √1 tetrahedron has a volume of 9 √0.𝚫 tetrahedral cells. In the curved 3-dimensional volume of the 600 cells, it encloses the cluster of 5 cells, which do not entirely fill it. The 6 dipyramids (12 cells) which fit into the concavities of the cluster of 5 cells overfill it: only one third of each dipyramid lies within the √1 tetrahedron. The dipyramids contribute one-third of each of 12 cells to it, a volume equivalent to 4 cells.
71. We also find √1 tetrahedra as the cells of the unit-radius 5-cell, and radially around the center of the 24-cell (one behind each of the 96 faces). Those radial √1 tetrahedra also occur in the 600-cell (in the 25 inscribed 24-cells), but note that those are not the same tetrahedra as the 600 √1 tetrahedral sections.
72. Each √1 edge of the octahedral cell is the long diameter of another tetrahedral dipyramid (two more face-bonded tetrahedral cells). In the 24-cell, three octahedral cells surround each edge, so one third of the dipyramid lies inside each octahedron, split between two adjacent concave faces. Each concave face is filled by one-sixth of each of the three dipyramids that surround its three edges, so it has the same volume as one tetrahedral cell.
73. A √1 octahedral cell (of any 24-cell inscribed in the 600-cell) has six vertices which all lie in the same hyperplane: they bound an octahedral section (a flat three-dimensional slice) of the 600-cell. The same √1 octahedron filled by 25 tetrahedral cells has a total of 14 vertices lying in three parallel three-dimensional sections of the 600-cell: the 6-point √1 octahedral section, a 4-point √1 tetrahedral section, and a 4-point √0.𝚫 tetrahedral section. In the curved three-dimensional space of the 600-cell's surface, the √1 octahedron surrounds the √1 tetrahedron which surrounds the √0.𝚫 tetrahedron, as three concentric hulls. This 14-vertex 4-polytope is a 4-pyramid with a regular octahedron base: not a canonical octahedral pyramid with one apex (which has only 7 vertices) but an irregular truncated octahedral pyramid. Because its base is a regular octahedron which is a 24-cell octahedral cell, this 4-pyramid lies on the surface of the 24-cell.
74. The apex of a canonical √1 octahedral pyramid has been truncated into a regular tetrahedral cell with shorter √0.𝚫 edges, replacing the apex with four vertices. The truncation has also created another four vertices (arranged as a √1 tetrahedron in a hyperplane between the octahedral base and the apex tetrahedral cell), and linked these eight new vertices with √0.𝚫 edges. The truncated pyramid thus has eight 'apex' vertices above the hyperplane of its octahedral base, rather than just one apex: 14 vertices in all. The original pyramid had flat sides: the five geodesic routes from any base vertex to the opposite base vertex ran along two √1 edges (and just one of those routes ran through the single apex). The truncated pyramid has rounded sides: five geodesic routes from any base vertex to the opposite base vertex run along three √0.𝚫 edges (and pass through two 'apexes').
75. The uniform 4-polytopes which this 14-vertex, 25-cell irregular 4-polytope most closely resembles may be the 10-vertex, 10-cell rectified 5-cell and its dual (it has characteristics of both).
76. The annular ring gaps between icosahedra are filled by a ring of 10 face-bonded tetrahedra that all meet at the vertex where the two icosahedra meet. This 10-cell ring is shaped like a pentagonal antiprism which has been hollowed out like a bowl on both its top and bottom sides, so that it has zero thickness at its center. This center vertex, like all the other vertices of the 600-cell, is itself the apex of an icosahedral pyramid where 20 tetrahedra meet.[lower-alpha 62] Therefore the annular ring of 10 tetrahedra is itself an equatorial ring of an icosahedral pyramid, containing 10 of the 20 cells of its icosahedral pyramid.
77. The 100-face surface of the triangular-faced 150-cell column could be scissors-cut lengthwise along a 10 edge path and peeled and laid flat as a 10×10 parallelogram of triangles.
78. Because the 100-face surface of the 150-cell torus is alternately convex and concave, 100 tetrahedra stack on it in face-bonded pairs, as 50 triangular bipyramids which share one raised vertex and bury one formerly exposed valley edge. The triangular bipyramids are vertex-bonded to each other in 5 parallel lines of 5 bipyramids (10 tetrahedra) each, which run straight up and down the outside surface of the 150-cell column.
79. 5 decagons spiral clockwise and 5 spiral counterclockwise, intersecting each other at the 50 valley vertices.
80. The same 10-face belt of an icosahedral pyramid is an annular ring of 10 tetrahedra around the apex.[lower-alpha 76]
81. The 600-cell's Petrie polygon is a skew triacontagon {30}. It can be seen in orthogonal projection as the circumference of a triacontagram {30/3}=3{10} helix which zig-zags 60° left and right, bridging the space between the 3 Clifford parallel great decagons of the 30-cell ring. In the completely orthogonal plane it projects to the regular triacontagram {30/11}.[58]
82. The 30 vertices of the Boerdijk–Coxeter triple-helix ring lie in 3 decagonal central planes which intersect only at one point (the center of the 600-cell), even though they are not completely orthogonal or orthogonal at all: they are π/5 apart.[lower-alpha 47] Their decagonal great circles are Clifford parallel: one 600-cell edge-length apart at every point.[lower-alpha 33] They are ordinary 2-dimensional great circles, not helices, but they are linked Clifford parallel circles.
83. Isoclinic geodesics are 4-dimensional great circles in the sense that they are 1-dimensional geodesic lines that curve in 4-space in two completely orthogonal planes at once. They should not be confused with great 2-spheres,[69] which are the 4-space analogues[lower-alpha 2] of 2-dimensional great circles in 3-space (great 1-spheres).
84. The 20 30-cell rings are chiral objects; they either spiral clockwise (right) or counterclockwise (left). The 150-cell torus (formed by five cell-disjoint 30-cell rings of the same chirality surrounding a great decagon) is not itself a chiral object, since it can be decomposed into either five parallel left-handed rings or five parallel right-handed rings. Unlike the 20-cell rings, the 150-cell tori are directly congruent with no torsion, like the octahedral 6-cell rings of the 24-cell. Each great decagon has five left-handed 30-cell rings surrounding it, and also five right-handed 30-cell rings surrounding it; but left-handed and right-handed 30-cell rings are not cell-disjoint and belong to different distinct rotations: the left and right rotations of the same fibration. In either distinct isoclinic rotation (left or right), the vertices of the 600-cell move along the axial 15-gram isoclines of 20 left 30-cell rings or 20 right 30-cell rings. Thus the great decagons, the 30-cell rings, and the 150-cell tori all occur as sets of Clifford parallel interlinked circles,[lower-alpha 33] although the exact way they nest together, avoid intersecting each other, and pass through each other to form a Hopf link is not identical for these three different kinds of Clifford parallel polytopes, in part because the linked pairs are variously of no inherent chirality (the decagons), the same chirality (the 30-cell rings), or no net torsion and both left and right interior organization (the 150-cell tori) but tracing the same chirality of interior organization in any distinct left or right rotation.
85. A point on the icosahedron Hopf map[lower-alpha 49] of the 600-cell's decagonal fibration lifts to a great decagon; a triangular face lifts to a 30-cell ring; and a pentagonal pyramid of 5 faces lifts to a 150-cell torus.[54] In the grand antiprism decomposition, two completely disjoint 150-cell tori are lifted from antipodal pentagons, leaving an equatorial ring of 10 icosahedron faces between them: a Petrie decagon of 10 triangles, which lift to 10 30-cell rings. But to get a decomposition of the 600-cell into four 150-cell tori of this kind, the icosahedral map would have to be decomposed into four pentagons, centered at the vertices of an inscribed tetrahedron, and the icosahedron cannot be decomposed that way.
86. Sadoc describes the decomposition of the 600-cell into four tori.[37] It is the same fibration of 12 great decagons and 20 30-cell rings, seen as a fibration of four completely disjoint 30-cell rings[lower-alpha 10] with spaces between them, which still encompasses all 12 decagons and all 120 vertices. If we look closely at the spaces between the four disjoint 30-cell rings, we can discern four 150-cell rings of 5 30-cell rings each. But these 150-cell rings do not have 5 30-cell rings around a common decagon axis. Their axis is a 30-cell ring, not a decagon. To construct them, on each of the four completely disjoint 30-cell rings, face-bond three more 30-cell rings to the exterior faces, making four stellated ("bumpy") rings containing four 30-cell rings (120 cells) each. Collectively they contain 16 of the 20 30-cell rings: there are still four 30-cell ring "holes" left to fill in the 600-cell. To do that, fill the surface concavities of each 120-tetrahedron ring by wrapping a fifth 30-cell ring around its circumference, completely orthogonal to the axial 30-cell ring you started with. The result is four 150-cell tori, of 5 30-cell rings each, each having two completely orthogonal 30-cell ring axes, either of which can be seen as either an axis or a circumference: it is both. On the icosahedron Hopf map,[lower-alpha 49] the four 30-cell rings lift from a star of four icosahedron faces (three faces edge-bonded around one). The fifth 30-cell ring lifts from a fifth face edge-bonded to the star, a sort of "extra flap" like the sixth square flap of the net of a cube before you fold it up into a cube. It does not matter which of the six possible adjacent faces you choose as the flap, but the choice determines the choice for all four 150-cell rings. There are six choices because there are six decagonal fibrations; this is when you fix which fibration you are taking. Thus every 30-cell ring is the center core of a 150-cell ring.
87. The greek letter phi 𝝓 as used by Coxeter to represent this characteristic angle is not to be confused with the more common use of phi 𝝓 to represent the golden ratio constant ≈ 1.618, as in some values in this table.
88. An orthoscheme is a chiral irregular simplex with right triangle faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own facets (its mirror walls). Every regular polytope can be dissected radially into instances of its characteristic orthoscheme surrounding its center. The characteristic orthoscheme has the shape described by the same Coxeter-Dynkin diagram as the regular polytope without the generating point ring.
89. The four edges of each 4-orthoscheme which meet at the center of a regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.
90. The reflecting surface of a (3-dimensional) polyhedron consists of 2-dimensional faces; the reflecting surface of a (4-dimensional) polychoron consists of 3-dimensional cells.
91. An isoclinic rotation by 36° is two simple rotations by 36° at the same time.[lower-alpha 126] It moves all the vertices 60° at the same time, in various different directions. Fifteen successive diagonal rotational increments, of 36°×36° each, move each vertex 900° through 15 vertices on a Möbius double loop of circumference 5𝝅 called an isocline, winding around the 600-cell and back to its point of origin, in one-and-one-half the time (15 rotational increments) that it would take a simple rotation to take the vertex once around the 600-cell on an ordinary {10} great circle (in 10 rotational increments).[lower-alpha 95] The helical double loop 5𝝅 isocline is just a special kind of single full circle, of 1.5 the period (15 chords instead of 10) as the simple great circle. The isocline is one true circle, as perfectly round and geodesic as the simple great circle, even through its chords are φ longer, its circumference is 5𝝅 instead of 2𝝅, it circles through four dimensions instead of two, and it acts in two chiral forms (left and right) even though all such circles of the same circumference are directly congruent. Nevertheless, to avoid confusion we always refer to it as an isocline and reserve the term great circle for a geodesic circle in the plane.[lower-alpha 42]
92. Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations a and b: the left double rotation as a then b, and the right double rotation as b then a. Simple rotations are not commutative; left and right rotations (in general) reach different destinations. The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: each moving vertex reaches its destination directly without passing through the intermediate point touched by a then b, or the other intermediate point touched by b then a, by rotating on a single helical geodesic (so it is the shortest path).[lower-alpha 58] Conversely, any simple rotation can be seen as the composition of two equal-angled double rotations (a left isoclinic rotation and a right isoclinic rotation), as discovered by Cayley; perhaps surprisingly, this composition is commutative, and is possible for any double rotation as well.[68]
93. Isoclinic rotations take each vertex to a non-adjacent vertex at least two edge-lengths away. In the characteristic isoclinic rotations of the 5-cell, 16-cell, 24-cell and 600-cell, the non-adjacent vertex is exactly two edge-lengths away along one of several great circle geodesic routes: the opposite vertex of a neighboring cell. In the 8-cell it is three zig-zag edge-lengths away in the same cell: the diagonally opposite vertex of a cube.
94. In an isoclinic rotation, each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a 4-dimensional diagonal.[lower-alpha 42] The point is displaced a total Pythagorean distance equal to the square root of four times the square of that distance. All vertices are displaced to a vertex at least two edge-lengths away.[lower-alpha 93] For example, when the unit-radius 600-cell rotates isoclinically 36 degrees in a decagon invariant plane and 36 degrees in its completely orthogonal invariant plane,[lower-alpha 41] each vertex is displaced to another vertex √1 (60°) distant, moving √1/4 = 1/2 unit radius in four orthogonal directions.
95. Because the 600-cell's helical pentadecagram2 geodesic is bent into a twisted ring in the fourth dimension like a Möbius strip, its screw thread doubles back across itself after each revolution, without ever reversing its direction of rotation (left or right). The 30-vertex isoclinic path follows a Möbius double loop, forming a single continuous 15-vertex loop traversed in two revolutions. The Möbius helix is a geodesic "straight line" or isocline. The isocline connects the vertices of a lower frequency (longer wavelength) skew polygram than the Petrie polygon. The Petrie triacontagon has √0.𝚫 edges; the isoclinic pentadecagram2 has √1 edges which join vertices which are two √0.𝚫 edges apart. Each √1 edge belongs to a different great hexagon, and successive √1 edges belong to different 24-cells, as the isoclinic rotation takes hexagons to Clifford parallel hexagons and passes through successive Clifford parallel 24-cells.
96. All isoclines are geodesics, and isoclines on the 3-sphere are circles (curving equally in all four dimensions), but not all isoclines on 3-manifolds in 4-space are circles.
97. Isoclinic rotations[lower-alpha 42] partition the 600 cells (and the 120 vertices) of the 600-cell into two disjoint subsets of 300 cells (and 60 vertices), even and odd (or black and white), which shift places among themselves on black or white isoclines, in a manner dimensionally analogous[lower-alpha 2] to the way the bishops' diagonal moves restrict them to the white or the black squares of the chessboard.[lower-alpha 124] The black and white subsets are also divided among black and white invariant great circle polygons of the isoclinic rotation. In a discrete rotation (as of a 4-polytope with a finite number of vertices) the black and white subsets correspond to sets of inscribed great polygons {p} in invariant great circle polygons {2p}. For example, in the 600-cell a black and a white great pentagon {5} are inscribed in an invariant great decagon {10} of the characteristic decagonal isoclinic rotation. Importantly, a black and white pair of polygons {p} of the same distinct isoclinic rotation are never inscribed in the same {2p} polygon; there is always a black and a white {p} polygon inscribed in each invariant {2p} polygon, but they belong to distinct isoclinic rotations: the left and right rotation of the same fibraton, which share the same set of invariant planes. Black (white) isoclines intersect only black (white) great {p} polygons, so each vertex is either black or white.
98. The chord-path of an isocline may be called the 4-polytope's Clifford polygon, as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic Clifford displacement.[82] The isocline is a helical Möbius double loop which reverses its chirality twice in the course of a full double circuit. The two loops are both entirely contained within the same cell ring, where they both follow chords connecting even (odd) vertices: typically opposite vertices of adjacent cells, two edge lengths apart.[lower-alpha 97] Both "halves" of the double loop pass through each cell in the cell ring, but intersect only two even (odd) vertices in each even (odd) cell. Each pair of intersected vertices in an even (odd) cell lie opposite each other on the Möbius strip, exactly one edge length apart. Thus each cell has two helices passing through it, which are Clifford parallels[lower-alpha 33] of opposite chirality at each pair of parallel points. Globally these two helices are a single connected circle of both chiralities,[lower-alpha 91] with no net torsion. An isocline acts as a left (or right) isocline when traversed by a left (or right) rotation (of different fibrations).
99. Isoclines on the 3-sphere occur in non-intersecting pairs of even/odd coordinate parity.[lower-alpha 97] A single black or white isocline forms a Möbius loop called the {1,1} torus knot or Villarceau circle[70] in which each of two "circles" linked in a Möbius "figure eight" loop traverses through all four dimensions.[lower-alpha 98] The double loop is a true circle in four dimensions.[lower-alpha 91] Even and odd isoclines are also linked, not in a Möbius loop but as a Hopf link of two non-intersecting circles,[lower-alpha 33] as are all the Clifford parallel isoclines of a Hopf fiber bundle.
100. A rotation in 4-space is completely characterized by choosing an invariant plane and an angle and direction (left or right) through which it rotates, and another angle and direction through which its one completely orthogonal[lower-alpha 16] invariant plane rotates. Two rotational displacements are identical if they have the same pair of invariant planes of rotation, through the same angles in the same directions (and hence also the same chiral pairing of directions). Thus the general rotation in 4-space is a double rotation, characterized by two angles. A simple rotation is a special case in which one rotational angle is 0.[lower-alpha 92] An isoclinic rotation is a different special case, similar but not identical to two simple rotations through the same angle.[lower-alpha 42]
101. There are an infinite number of pairs of completely orthogonal[lower-alpha 16] invariant planes in each isoclinic rotation, all rotating through the same angle;[lower-alpha 59] nonetheless, not all central planes are invariant planes of rotation. The invariant planes of an isoclinic rotation constitute a fibration of the entire 4-polytope.[72] In every isoclinic rotation of the 600-cell taking vertices to vertices either 12 Clifford parallel great decagons, or 20 Clifford parallel great hexagons or 30 Clifford parallel great squares are invariant planes of rotation.
102. In a Clifford displacement, also known as an isoclinic rotation, all the Clifford parallel[lower-alpha 33] invariant planes[lower-alpha 101] are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted sideways by that same angle. A Clifford displacement is 4-dimensionally diagonal.[lower-alpha 94] Every plane that is Clifford parallel to one of the completely orthogonal planes is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane rotates sideways. All central polygons (of every kind) rotate by the same angle (though not all do so invariantly), and are also displaced sideways by the same angle to a Clifford parallel polygon (of the same kind).
103. The three 16-cells in the 24-cell are rotated by 60° (𝜋/3) isoclinically with respect to each other. Because an isoclinic rotation is a rotation in two completely orthogonal planes at the same time, this means their corresponding vertices are 120° (2𝜋/3) apart. In a unit-radius 4-polytope, vertices 120° apart are joined by a √3 chord.
104. Any isoclinic rotation by 𝜋/5 in decagonal invariant planes[lower-alpha 110] takes every central polygon, geodesic cell ring or inscribed 4-polytope[lower-alpha 9] in the 600-cell to a Clifford parallel polytope 𝜋/5 away.
105. Five 24-cells meet at each vertex of the 600-cell,[lower-alpha 12] so there are four different directions in which the vertices can move to rotate the 24-cell (or all the 24-cells at once in an isoclinic rotation[lower-alpha 104]) directly toward an adjacent 24-cell.
106. A disjoint 24-cell reached by an isoclinic rotation is not any of the four adjacent 24-cells; the double rotation[lower-alpha 100] takes it past (not through) the adjacent 24-cell it rotates toward,[lower-alpha 105] and left or right to a more distant 24-cell from which it is completely disjoint.[lower-alpha 10] The four directions reach 8 different 24-cells[lower-alpha 7] because in an isoclinic rotation each vertex moves in a spiral along two completely orthogonal great circles at once. Four paths are right-hand threaded (like most screws and bolts), moving along the circles in the "same" directions, and four are left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are "opposite" directions (according to the right hand rule by which we conventionally say which way is "up" on each of the 4 coordinate axes).[73]
107. All isoclinic polygons are Clifford parallels (completely disjoint).[lower-alpha 10] Polyhedra (3-polytopes) and polychora (4-polytopes) may be isoclinic and not disjoint, if all of their corresponding central polygons are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same object, shared). For example, the 24-cell, 600-cell and 120-cell contain pairs of inscribed tesseracts (8-cells) which are isoclinically rotated by 𝜋/3 with respect to each other, yet are not disjoint: they share a 16-cell (8 vertices, 6 great squares and 4 octahedral central hyperplanes), and some corresponding pairs of their great squares are cocellular (intersecting) rather than Clifford parallel (disjoint).
108. At each vertex, a 600-cell has four adjacent (non-disjoint)[lower-alpha 10] 24-cells that can each be reached by a simple rotation in that direction.[lower-alpha 105] Each 24-cell has 4 great hexagons crossing at each of its vertices, one of which it shares with each of the adjacent 24-cells; in a simple rotation that hexagonal plane remains fixed (its vertices do not move) as the 600-cell rotates around the common hexagonal plane. The 24-cell has 16 great hexagons altogether, so it is adjacent (non-disjoint) to 16 other 24-cells.[lower-alpha 7] In addition to being reachable by a simple rotation, each of the 16 can also be reached by an isoclinic rotation in which the shared hexagonal plane is not fixed: it rotates (non-invariantly) through 𝜋/5. The double rotation reaches an adjacent 24-cell directly as if indirectly by two successive simple rotations:[lower-alpha 92] first to one of the other adjacent 24-cells, and then to the destination 24-cell (adjacent to both of them).
109. In the 600-cell, there is a simple rotation which will take any vertex directly to any other vertex, also moving most or all of the other vertices but leaving at most 6 other vertices fixed (the vertices that the fixed central plane intersects). The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great decagon, a great hexagon, a great square or a great digon,[lower-alpha 48] and the completely orthogonal fixed plane intersects 0 vertices (a 30-gon),[lower-alpha 41] 2 vertices (a digon), 4 vertices (a square) or 6 vertices (a hexagon) respectively. Two non-disjoint 24-cells are related by a simple rotation through 𝜋/5 of the digon central plane completely orthogonal to their common hexagonal central plane. In this simple rotation, the hexagon does not move. The two non-disjoint 24-cells are also related by an isoclinic rotation in which the shared hexagonal plane does move.[lower-alpha 108]
110. Any isoclinic rotation in a decagonal invariant plane is an isoclinic rotation in 24 invariant planes: 12 Clifford parallel decagonal planes,[lower-alpha 101] and the 12 Clifford parallel 30-gon planes completely orthogonal to each of those decagonal planes.[lower-alpha 41] As the invariant planes rotate in two completely orthogonal directions at once,[lower-alpha 58] all points in the planes move with them (stay in their planes and rotate with them), describing helical isoclines[lower-alpha 42] through 4-space. Note however that in a discrete decagonal fibration of the 600-cell (where 120 vertices are the only points considered), the 12 30-gon planes contain no points.
111. Notice the apparent incongruity of rotating hexagons by 𝜋/5, since only their opposite vertices are an integral multiple of 𝜋/5 apart. However, recall that 600-cell vertices which are one hexagon edge apart are exactly two decagon edges and two tetrahedral cells (one triangular dipyramid) apart. The hexagons have their own 10 discrete fibrations and cell rings, not Clifford parallel to the decagonal fibrations but also by fives[lower-alpha 14] in that five 24-cells meet at each vertex, each pair sharing a hexagon.[lower-alpha 12] Each hexagon rotates non-invariantly by 𝜋/5 in a hexagonal isoclinic rotation between non-disjoint 24-cells.[lower-alpha 108] Conversely, in all 𝜋/5 isoclinic rotations in decagonal invariant planes, all the vertices travel along isoclines[lower-alpha 42] which follow the edges of hexagons.
112. Consider the statement: In one full revolution of an isoclinic rotation, every point in the space loops exactly once through its great circle Hopf fiber. It can be found in the literature, expressed in the mathematical language of the Hopf fibration,[74] but as a plain language statement of Euclidean geometry, how exactly should we visualize it? It paints a clear picture of all the great circles of a Hopf fibration rotating as rigid wheels, in parallel. That is a correct visualization, except for the fact that points moving under isoclinic rotation traverse an invariant great circle only in the sense that they stay on that circle as the whole circle itself is tilting in the completely orthogonal plane.[lower-alpha 42] With respect to the stationary reference frame, the points move diagonally on a helical isocline, they do not move on a planar great circle.[lower-alpha 58] Each helical isocline is itself a kind of circle, but it is not a planar great circle of the Hopf fibration: it is a special kind of geodesic circle whose circumference is greater than 2𝝅r, and it is not pictured explicitly at all by the plain statement we are trying to visualize. We cannot easily visualize this statement about the Hopf great circles in a stationary reference frame. The statement does not simply mean that in an isoclinic rotation every point on a stationary Hopf great circle loops through its stationary great circle. Rather, it means that every point on every Hopf great circle loops through its great circle as every great circle itself is moving orthogonally, flipping like a coin in the plane completely orthogonal to its own plane (at any instant, because of course the completely orthogonal plane is moving too). This simultaneous twisting rotation in two completely orthogonal planes is a double rotation; if the angle of rotation in the two completely orthogonal planes is exactly the same, it is isoclinic. An isoclinic rotation takes each rigid planar Hopf great circle to the stationary position of another Hopf great circle, while simultaneously each Hopf great circle also rotates like a wheel. This fibration of doubly rotating rigid wheels is undoubtably hard to visualize. In any graphical animation (whether actually rendered or merely imagined) it will be difficult to track the motions of the different rotating wheels, because Clifford parallel circles are not parallel in the ordinary sense, and every great circle is moving in a different direction at any one instant. There is one more way in which this simple statement belies the full complexity of the isoclinic motion. While it is true that every point loops through its Hopf great circle exactly once in a full isoclinic revolution, every vertex moves more than 360 degrees, as measured in the stationary reference frame. In any distinct isoclinic rotation, all the vertices move the same angular distance in the stationary reference frame in one full revolution, but each distinct pair of left-right isoclinic rotations corresponds to a unique Hopf fibration,[72] and the characteristic distance moved is different for each kind of Hopf fibration. For example, in the isoclinic rotation of a great hexagon fibration of the 24-cell, each vertex moves 720 degrees in the stationary reference frame (2 times the distance it moves within its moving Hopf great circle); but in the isoclinic rotation of a great decagon fibration of the 600-cell, each vertex moves 900 degrees in the stationary reference frame (2.5 times its great circle distance).
113. Each isocline has no inherent chirality but can act as either a left or right isocline; it is shared by a distinct left rotation and a distinct right rotation of different fibrations.
114. The analogous relationships among three kinds of {2p} isoclinic rotations, in Clifford parallel bundles of {4}, {6} or {10} great polygon invariant planes respectively, are at the heart of the complex nested relationship among the regular convex 4-polytopes.[lower-alpha 3] In the √1 hexagon {6} rotations characteristic of the 24-cell, the isocline chords (polygram edges) are simply √3 chords of the great hexagon, so the simple {6} hexagon rotation and the isoclinic {6/2} hexagram rotation both rotate circles of 6 vertices. The hexagram isocline, a special kind of great circle, has a circumference of 4𝝅 compared to the hexagon 2𝝅 great circle.[lower-alpha 119] The invariant central plane completely orthogonal to each {6} great hexagon is a {2} great digon,[lower-alpha 48] so an isoclinic {6} rotation of hexagrams is also a {2} rotation of axes.[lower-alpha 109] In the √2 square {4} rotations characteristic of the 16-cell, the isocline chords are √4 digon edges (axes), and the isocline polygram is an octagram, so the isocline has a circumference of 8𝝅. The isoclinic {8/2} octagram rotation rotates a circle of twice as many vertices as the simple {4} square rotation in the same time (number of rotational increments). The invariant central plane completely orthogonal to each {4} great square is another {4} great square, so a right {4} rotation of squares is also a left {4} rotation of squares. The 16-cell's dual polytope the 8-cell tesseract inherits the same simple {4} and isoclinic {8/2} rotations, but its characteristic isoclinic rotation takes place in completely orthogonal invariant planes which contain a {4} great rectangle or a {2} great digon (from its successor the 24-cell). In the 8-cell this is a rotation of √1 × √3 great rectangles, and also a rotation of √4 axes, but it is the same isoclinic rotation as the 24-cell's characteristic rotation of great hexagons (in which the great rectangles are inscribed), as a consequence of the unique circumstance that the 8-cell and 24-cell have the same edge length. In the √0.𝚫 decagon {10} rotations characteristic of the 600-cell, the isocline chords are √1 hexagon edges, the isocline polygram is a pentadecagram, and the isocline has a circumference of 5𝝅.[lower-alpha 91] The isoclinic {15/2} pentadecagram rotation rotates a circle of {15} vertices in the same time as the simple decagon rotation of {10} vertices. The invariant central plane completely orthogonal to each (10} great decagon is a {0} great 0-gon,[lower-alpha 39] so a {10} rotation of decagons is also a {0} rotation of planes containing no vertices. The 600-cell's dual polytope the 120-cell inherits the same simple {10} and isoclinic {15/2} rotations, but its characteristic isoclinic rotation takes place in completely orthogonal invariant planes which contain {2} great digons (from its successor the 5-cell).[lower-alpha 120] This is a {15/4} rotation of irregular great hexagons {6} of two alternating edge lengths (analogous to the tesseract's great rectangles), where the two different-length edges are three 120-cell edges and three 5-cell edges.
115. Each discrete fibration of a regular convex 4-polytope is characterized by a unique left-right pair of isoclinic rotations and a unique bundle of great circle {2p} polygons (0 ≤ p ≤ 5) in the invariant planes of that pair of rotations. Each distinct rotation has a unique bundle of left (or right) {p} polygons inscribed in the {2p} polygons, and a unique bundle of skew {2p} polygrams which are its discrete left (or right) isoclines. The {p} polygons weave the {2p} polygrams into a bundle, and vice versa.
116. The 600-cell has four orthogonal central hyperplanes, each of which is an icosidodecahedron.[lower-alpha 23]
117. There are six congruent decagonal fibrations of the 600-cell. Choosing one decagonal fibration means choosing a bundle of 12 directly congruent Clifford parallel decagonal great circles, and a cell-disjoint set of 20 directly congruent 30-cell rings which tesselate the 600-cell. The fibration and its great circles are not chiral, but it has distinct left and right expressions in a left-right pair of isoclinic rotations. In the right (left) rotation the vertices move along a right (left) Hopf fiber bundle of Clifford parallel isoclines and intersect a right (left) Hopf fiber bundle of Clifford parallel great pentagons. The 30-cell rings are the only chiral objects, other than the bundles of isoclines or pentagons.[78] A right (left) pentagon bundle contains 12 great pentagons, inscribed in the 12 Clifford parallel great decagons. A right (left) isocline bundle contains 20 Clifford parallel pentadecagrams, one in each 30-cell ring.
118. The composition of two simple 60° rotations in a pair of completely orthogonal invariant planes is a 60° isoclinic rotation in four pairs of completely orthogonal invariant planes.[lower-alpha 92] Thus the isoclinic rotation is the compound of four simple rotations, and all 24 vertices rotate in invariant hexagon planes, versus just 6 vertices in a simple rotation.
119. An isoclinic rotation by 60° is two simple rotations by 60° at the same time.[lower-alpha 118] It moves all the vertices 120° at the same time, in various different directions. Six successive diagonal rotational increments, of 60°x60° each, move each vertex through 720° on a Möbius double loop called an isocline, twice around the 24-cell and back to its point of origin, in the same time (six rotational units) that it would take a simple rotation to take the vertex once around the 24-cell on an ordinary great circle. The helical double loop 4𝝅 isocline is just another kind of single full circle, of the same time interval and period (6 chords) as the simple great circle. The isocline is one true circle,[lower-alpha 83] as perfectly round and geodesic as the simple great circle, even through its chords are √3 longer, its circumference is 4𝝅 instead of 2𝝅,[lower-alpha 99] it circles through four dimensions instead of two, and it acts in two chiral forms (left and right) even though all such circles of the same circumference are directly congruent.[lower-alpha 98] Nevertheless, to avoid confusion we always refer to it as an isocline and reserve the term great circle for a geodesic circle in the plane.
120. 120 regular 5-cells are inscribed in the 120-cell. The 5-cell has digon central planes, no two of which are orthogonal. It has 10 digon central planes, where each vertex pair is an edge, not an axis. The 5-cell is self-dual, so by reciprocation the 120-cell can be inscribed in a regular 5-cell of larger radius. Therefore the finite sequence of 6 regular 4-polytopes[lower-alpha 3] nested like Russian dolls can also be seen as an infinite sequence.
121. In the 30-cell ring, each isocline runs from a vertex to a non-adjacent vertex in the third shell of vertices surrounding it. Three other vertices between these two vertices can be seen in the 30-cell ring, two adjacent in the first surrounding shell, and one in the second surrounding shell.
122. Since it is difficult to color points and lines white, we sometimes use black and red instead of black and white. In particular, isocline chords are sometimes shown as black or red dashed lines.
123. Chirality and even/odd parity are distinct flavors. Things which have even/odd coordinate parity are black or white: the squares of the chessboard,[lower-alpha 122] cells, vertices and the isoclines which connect them by isoclinic rotation.[lower-alpha 42] Everything else is black and white: e.g. adjacent face-bonded cell pairs, or edges and chords which are black at one end and white at the other. Things which have chirality come in right or left enantiomorphous forms: isoclinic rotations and chiral objects which include characteristic orthoschemes, pairs of Clifford parallel great polygon planes,[81] fiber bundles of Clifford parallel circles (whether or not the circles themselves are chiral), and the chiral cell rings found in the 16-cell and 600-cell. Things which have neither an even/odd parity nor a chirality include all edges and faces (shared by black and white cells), great circle polygons and their fibrations, and non-chiral cell rings such as the 24-cell's cell rings of octahedra. Some things have both an even/odd parity and a chirality: isoclines are black or white because they connect vertices which are all of the same color, and they act as left or right chiral objects when they are vertex paths in a left or right rotation, although they have no inherent chirality themselves.[lower-alpha 113] Each left (or right) rotation traverses an equal number of black and white isoclines.[lower-alpha 98]
124. Left and right isoclinic rotations partition the 600 cells (and 120 vertices) into black and white in the same way.[14] The rotations of all fibrations of the same kind of great polygon use the same chessboard, which is a convention of the coordinate system based on even and odd coordinates.[80] Left and right are not colors: in either a left (or right) rotation half the moving vertices are black, running along black isoclines through black vertices, and the other half are white vertices rotating among themselves.[lower-alpha 123]
125. Each axis of the 600-cell touches a left isocline of each fibration at one end and a right isocline of the fibration at the other end. Each 30-cell ring's axial isocline passes through only one of the two antipodal vertices of each of the 30 (of 60) 600-cell axes that the isocline's 30-vertex, 30-cell ring touches (at only one end).
126. The composition of two simple 36° rotations in a pair of completely orthogonal invariant planes is a 36° isoclinic rotation in twelve pairs of completely orthogonal invariant planes.[lower-alpha 92] Thus the isoclinic rotation is the compound of twelve simple rotations, and all 120 vertices rotate in invariant decagon planes, versus just 10 vertices in a simple rotation.
Citations
1. N.W. Johnson: Geometries and Transformations, (2018) ISBN 978-1-107-10340-5 Chapter 11: Finite Symmetry Groups, 11.5 Spherical Coxeter groups, p.249
2. Matila Ghyka, The Geometry of Art and Life (1977), p.68
3. Coxeter 1973, pp. 292–293, Table I(ii): The sixteen regular polytopes {p,q,r} in four dimensions; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of unit radius.
4. Coxeter 1973, p. 153, §8.51; "In fact, the vertices of {3, 3, 5}, each taken 5 times, are the vertices of 25 {3, 4, 3}'s."
5. Coxeter 1973, p. 305, Table VII: Regular Compounds in Four Dimensions.
6. Coxeter 1973, pp. 156–157, §8.7 Cartesian coordinates.
7. Coxeter 1973, pp. 151–153, §8.4 The snub {3,4,3}.
8. Waegell & Aravind 2009, pp. 3–4, §3.2 The 75 bases of the 600-cell; In the 600-cell the configuration's "points" and "lines" are axes ("rays") and 16-cells ("bases"), respectively.
9. Denney et al. 2020, p. 438.
10. Zamboj 2021, pp. 10–11, §Hopf coordinates.
11. Coxeter 1973, p. 298, Table V: The Distribution of Vertices of Four-dimensional Polytopes in Parallel Solid Sections (§13.1); (iii) Sections of {3, 3, 5} (edge 2𝜏−1) beginning with a vertex.
12. Oss 1899; van Oss does not mention the arc distances between vertices of the 600-cell.
13. Buekenhout & Parker 1998.
14. Dechant 2021, pp. 18–20, §6. The Coxeter Plane.
15. Coxeter 1973, p. 298, Table V: The Distribution of Vertices of Four-dimensional Polytopes in Parallel Solid Sections (§13.1); (iii) Sections of {3, 3, 5} (edge 2𝜏−1) beginning with a vertex; see column a.
16. Steinbach 1997, p. 23, Figure 3; Steinbach derived a formula relating the diagonals and edge lengths of successive regular polygons, and illustrated it with a "fan of chords" diagram like the one here.
17. Baez, John (7 March 2017). "Pi and the Golden Ratio". Azimuth. Retrieved 10 October 2022.
18. Denney et al. 2020, p. 434.
19. Denney et al. 2020, pp. 437–439, §4 The planes of the 600-cell.
20. Kim & Rote 2016, pp. 8–10, Relations to Clifford Parallelism.
21. Sadoc 2001, p. 576, §2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis.
22. Waegell & Aravind 2009, p. 5, §3.4. The 24-cell: points, lines, and Reye's configuration; Here Reye's "points" and "lines" are axes and hexagons, respectively. The dual hexagon planes are not orthogonal to each other, only their dual axis pairs. Dual hexagon pairs do not occur in individual 24-cells, only between 24-cells in the 600-cell.
23. Sadoc 2001, pp. 576–577, §2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis.
24. Sadoc 2001, p. 577, §2.4 Discretising the fibration for the {3, 3, 5} polytope: the four-fold screw axis.
25. Copher 2019, p. 6, §3.2 Theorem 3.4.
26. Kim & Rote 2016, p. 7, §6 Angles between two Planes in 4-Space; "In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, k angles are defined between k-dimensional subspaces.)"
27. Lemmens & Seidel 1973.
28. Mamone, Pileio & Levitt 2010, p. 1433, §4.1; A Cartesian 4-coordinate point (w,x,y,z) is a vector in 4D space from (0,0,0,0). Four-dimensional real space is a vector space: any two vectors can be added or multiplied by a scalar to give another vector. Quaternions extend the vectorial structure of 4D real space by allowing the multiplication of two 4D vectors $\left(w,x,y,z\right)_{1}$ and $\left(w,x,y,z\right)_{2}$ according to
${\begin{pmatrix}w_{2}\\x_{2}\\y_{2}\\z_{2}\end{pmatrix}}*{\begin{pmatrix}w_{1}\\x_{1}\\y_{1}\\z_{1}\end{pmatrix}}={\begin{pmatrix}{w_{2}w_{1}-x_{2}x_{1}-y_{2}y_{1}-z_{2}z_{1}}\\{w_{2}x_{1}+x_{2}w_{1}+y_{2}z_{1}-z_{2}y_{1}}\\{w_{2}y_{1}-x_{2}z_{1}+y_{2}w_{1}+z_{2}x_{1}}\\{w_{2}z_{1}+x_{2}y_{1}-y_{2}x_{1}+z_{2}w_{1}}\end{pmatrix}}$
29. Sadoc 2001, pp. 575–578, §2 Geometry of the {3,3,5}-polytope in S3; Sadoc studied all the Hopf fibrations of the 600-cell into sets of {4}, {6} or {10} great circle fibers on different screw axes, gave their Hopf maps, and fully illustrated the characteristic decagonal cell rings.
30. Tyrrell & Semple 1971, pp. 6–7, §4. Isoclinic planes in Euclidean space E4.
31. Sadoc 2001, pp. 577–578, §2.5 The 30/11 symmetry: an example of other kind of symmetries.
32. Coxeter 1973, p. 211, §11.x Historical remarks; "The finite group [32, 2, 1] is isomorphic with the group of incidence-preserving permutations of the 27 lines on the general cubic surface. (For the earliest description of these lines, see Schlafli 2.)".
33. Schläfli 1858; this paper of Schläfli's describing the double six configuration was one of the only fragments of his discovery of the regular polytopes in higher dimensions to be published during his lifetime.[32]
34. Coxeter 1973, pp. 141–144, §7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks; "Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassman and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions."
35. Coxeter 1970, studied cell rings in the general case of their geometry and group theory, identifying each cell ring as a polytope in its own right which fills a three-dimensional manifold (such as the 3-sphere) with its corresponding honeycomb.[lower-alpha 52] He found that cell rings follow Petrie polygons and some (but not all) cell rings and their honeycombs are twisted, occurring in left- and right-handed chiral forms. Specifically, he found that the regular 4-polytopes with tetrahedral cells (5-cell, 16-cell, 600-cell) have twisted cell rings, and the others (whose cells have opposing faces) do not.[lower-alpha 53] Separately, he categorized cell rings by whether they form their honeycombs in hyperbolic or Euclidean space, the latter being those found in the 4-polytopes which can tile 4-space by translation to form Euclidean honeycombs (16-cell, 8-cell, 24-cell).
36. Banchoff 2013, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the Clifford torus, showed how the honeycombs correspond to Hopf fibrations, and made decompositions composed of meridian and equatorial cell rings with illustrations.
37. Sadoc 2001, p. 578, §2.6 The {3, 3, 5} polytope: a set of four helices.
38. Dechant 2021, §1. Introduction.
39. Zamboj 2021.
40. Sadoc & Charvolin 2009, §1.2 The curved space approach; studies the helical orientation of molecules in crystal structures and their imperfect packings ("frustrations") in 3-dimensional space. "The frustration, which arises when the molecular orientation is transported along the two [circular] AB paths of figure 1 [helix], is imposed by the very topological nature of the Euclidean space R3. It would not occur if the molecules were embedded in the non-Euclidean space of the 3-sphere S3, or hypersphere. This space with a homogeneous positive curvature can indeed be described by equidistant and uniformly twisted fibers,[lower-alpha 33] along which the molecules can be aligned without any conflict between compactness and torsion.... The fibres of this Hopf fibration are great circles of S3, the whole family of which is also called the Clifford parallels. Two of these fibers are C∞ symmetry axes for the whole fibration; each fibre makes one turn around each axis and regularly rotates when moving from one axis to another.[lower-alpha 58] These fibers build a double twist configuration while staying parallel, i.e. without any frustration, in the whole volume of S3.[lower-alpha 59] They can therefore be used as models to study the condensation of long molecules in the presence of a double twist constraint."
41. Coxeter 1973, p. 303, Table VI (iii): 𝐈𝐈 = {3,3,5}.
42. Coxeter 1973, p. 153, §8.5 Gosset's construction for {3,3,5}.
43. Borovik 2006; "The environment which directed the evolution of our brain never provided our ancestors with four-dimensional experiences.... [Nevertheless] we humans are blessed with a remarkable piece of mathematical software for image processing hardwired into our brains. Coxeter made full use of it, and expected the reader to use it.... Visualization is one of the most powerful interiorization techniques. It anchors mathematical concepts and ideas into one of the most powerful parts of our brain, the visual processing module. Coxeter Theory [of polytopes generated by] finite reflection groups allow[s] an approach to their study based on a systematic reduction of complex geometric configurations to much simpler two- and three-dimensional special cases."
44. Miyazaki 1990; Miyazaki showed that the surface envelope of the 600-cell can be realized architecturally in our ordinary 3-dimensional space as physical buildings (geodesic domes).
45. Coxeter 1973, pp. 50–52, §3.7.
46. Coxeter 1973, p. 293; 164°29'
47. Coxeter 1973, p. 298, Table V: The Distribution of Vertices of Four-dimensional Polytopes in Parallel Solid Sections.
48. Coxeter 1973, pp. 50–52, §3.7: Coordinates for the vertices of the regular and quasi-regular solids.
49. Itoh & Nara 2021, §4. From the 24-cell onto an octahedron; "This article addresses the 24-cell and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller."
50. Verheyen, H. F. (1989). "The complete set of Jitterbug transformers and the analysis of their motion". Computers and Mathematics with Applications. 17 (1–3): 203–250. doi:10.1016/0898-1221(89)90160-0. MR 0994201.
51. Coxeter 1973, p. 299, Table V: (iv) Simplified sections of {3,3,5} ... beginning with a cell.
52. Sadoc 2001, pp. 576–577, §2.4 Discretising the fibration for the {3, 3, 5}; "Let us now proceed to a toroidal decomposition of the {3, 3, 5} polytope."
53. Coxeter 1970, pp. 19–23, §9. The 120-cell and the 600-cell.
54. Sadoc 2001, pp. 576–577, §2.4 Discretising the fibration for the {3, 3, 5}, Fig. 2. A five fold symmetry column; in caption (sic) dodecagons should be decagons.
55. Dechant 2021, pp. 20–22, §7. The Grand Antiprism and H2 × H2.
56. Banchoff 1988.
57. Zamboj 2021, pp. 6–12, §2 Mathematical background.
58. Coxeter 1973, pp. 292–293, Table I(ii); 600-cell h1 h2.
59. Coxeter 1973, pp. 292–293, Table I(ii); "600-cell".
60. Coxeter 1973, p. 139, §7.9 The characteristic simplex.
61. Coxeter 1973, p. 290, Table I(ii); "dihedral angles".
62. Coxeter 1973, pp. 227−233, §12.7 A necklace of tetrahedral beads.
63. Coxeter 1973, pp. 33–38, §3.1 Congruent transformations.
64. Dechant 2017, pp. 410–419, §6. The Coxeter Plane; see p. 416, Table 1. Summary of the factorisations of the Coxeter versors of the 4D root systems; "Coxeter (reflection) groups in the Clifford framework ... afford a uniquely simple prescription for reflections. Via the Cartan-Dieudonné theorem, performing two reflections successively generates a rotation, which in Clifford algebra is described by a spinor that is simply the geometric product of the two vectors generating the reflections."
65. Coxeter 1973, p. 217, §12.2 Congruent transformations.
66. Koca, Al-Ajmi & Ozdes Koca 2011, pp. 986–988, 6. Dual of the snub 24-cell.
67. Mamone, Pileio & Levitt 2010, pp. 1438–1439, §4.5 Regular Convex 4-Polytopes; the 600-cell has 14,400 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝛨4.
68. Perez-Gracia & Thomas 2017.
69. Stillwell 2001, p. 24.
70. Dorst 2019, p. 44, §1. Villarceau Circles; "In mathematics, the path that the (1, 1) knot on the torus traces is also known as a Villarceau circle. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a Hopf fibration.... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut."
71. Waegell & Aravind 2009, pp. 2–5, §3. The 600-cell.
72. Kim & Rote 2016, pp. 13–14, §8.2 Equivalence of an Invariant Family and a Hopf Bundle.
73. Perez-Gracia & Thomas 2017, pp. 12−13, §5. A useful mapping.
74. Kim & Rote 2016, pp. 12–16, 8 The Construction of Hopf Fibrations; see Theorem 13.
75. Perez-Gracia & Thomas 2017, pp. 2−3, §2. Isoclinic rotations.
76. Kim & Rote 2016, p. 12-16, §8 The Construction of Hopf Fibrations; see §8.3.
77. Perez-Gracia & Thomas 2017, §1. Introduction; "This article [will] derive a spectral decomposition of isoclinic rotations and explicit formulas in matrix and Clifford algebra for the computation of Cayley's [isoclinic] factorization."[lower-alpha 92]
78. Kim & Rote 2016, p. 14, §8.3 Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle.
79. Kim & Rote 2016, pp. 14–16, §8.3 Properties of the Hopf Fibration.
80. Coxeter 1973, p. 156: "...the chess-board has an n-dimensional analogue."
81. Kim & Rote 2016, p. 8, Left and Right Pairs of Isoclinic Planes.
82. Tyrrell & Semple 1971, pp. 34–57, Linear Systems of Clifford Parallels.
83. Coxeter 1973, pp. 292–293, Table I(ii); 24-cell h1 is {12}, h2 is {12/5}.
84. Coxeter 1973, p. 12, §1.8. Configurations.
85. van Ittersum 2020, pp. 80–95, §4.3.
86. Steinbach 1997, p. 24.
87. Stillwell 2001, p. 18-21.
88. Stillwell 2001, pp. 22–23, The Poincaré Homology Sphere.
89. Denney et al. 2020, §2 The Labeling of H4.
90. Oss 1899, pp. 1–18.
91. Dechant 2021, Abstract; "[E]very 3D root system allows the construction of a corresponding 4D root system via an 'induction theorem'. In this paper, we look at the icosahedral case of H3 → H4 in detail and perform the calculations explicitly. Clifford algebra is used to perform group theoretic calculations based on the versor theorem and the Cartan-Dieudonné theorem ... shed[ding] light on geometric aspects of the H4 root system (the 600-cell) as well as other related polytopes and their symmetries ... including the construction of the Coxeter plane, which is used for visualising the complementary pairs of invariant polytopes.... This approach therefore constitutes a more systematic and general way of performing calculations concerning groups, in particular reflection groups and root systems, in a Clifford algebraic framework."
92. Grossman, Wendy A.; Sebline, Edouard, eds. (2015), Man Ray Human Equations: A journey from mathematics to Shakespeare, Hatje Cantz. See in particular mathematical object mo-6.2, p. 58; Antony and Cleopatra, SE-6, p. 59; mathematical object mo-9, p. 64; Merchant of Venice, SE-9, p. 65, and "The Hexacosichoron", Philip Ordning, p. 96.
93. Dechant 2021, pp. 22–24, §8. Snub 24-cell.
94. Sikiric, Mathieu; Myrvold, Wendy (2007). "The special cuts of 600-cell". Beiträge zur Algebra und Geometrie. 49 (1). arXiv:0708.3443.
95. Denney et al. 2020.
96. Coxeter 1991, pp. 48–49.
References
• Schläfli, Ludwig (1858), Cayley, Arthur (ed.), "An attempt to determine the twenty-seven lines upon a surface of the third order, and to derive such surfaces in species, in reference to the reality of the lines upon the surface", Quarterly Journal of Pure and Applied Mathematics, 2: 55–65, 110–120
• Coxeter, H.S.M. (1973) [1948]. Regular Polytopes (3rd ed.). New York: Dover.
• Coxeter, H.S.M. (1991). Regular Complex Polytopes (2nd ed.). Cambridge: Cambridge University Press.
• Coxeter, H.S.M. (1995). Sherk, F. Arthur; McMullen, Peter; Thompson, Anthony C.; Weiss, Asia Ivic (eds.). Kaleidoscopes: Selected Writings of H.S.M. Coxeter (2nd ed.). Wiley-Interscience Publication. ISBN 978-0-471-01003-6.
• (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
• (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
• (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
• Coxeter, H.S.M. (1970), "Twisted Honeycombs", Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, Providence, Rhode Island: American Mathematical Society, 4
• Borovik, Alexandre (2006). "Coxeter Theory: The Cognitive Aspects". In Davis, Chandler; Ellers, Erich (eds.). The Coxeter Legacy. Providence, Rhode Island: American Mathematical Society. pp. 17–43. ISBN 978-0821837221.
• J.H. Conway and M.J.T. Guy: Four-Dimensional Archimedean Polytopes, Proceedings of the Colloquium on Convexity at Copenhagen, page 38 und 39, 1965
• N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
• Four-dimensional Archimedean Polytopes (German), Marco Möller, 2004 PhD dissertation Archived 2005-03-22 at the Wayback Machine
• Oss, Salomon Levi van (1899). "Das regelmässige 600-Zell und seine selbstdeckenden Bewegungen". Verhandelingen der Koninklijke (Nederlandse) Akademie van Wetenschappen, Sectie 1 (Afdeeling Natuurkunde). Amsterdam. 7 (1): 1–18.
• Buekenhout, F.; Parker, M. (15 May 1998). "The number of nets of the regular convex polytopes in dimension <= 4". Discrete Mathematics. 186 (1–3): 69–94. doi:10.1016/S0012-365X(97)00225-2.
• Denney, Tomme; Hooker, Da'Shay; Johnson, De'Janeke; Robinson, Tianna; Butler, Majid; Claiborne, Sandernishe (2020). "The geometry of H4 polytopes". Advances in Geometry. 20 (3): 433–444. arXiv:1912.06156. doi:10.1515/advgeom-2020-0005. S2CID 220367622.
• Steinbach, Peter (1997). "Golden fields: A case for the Heptagon". Mathematics Magazine. 70 (Feb 1997): 22–31. doi:10.1080/0025570X.1997.11996494. JSTOR 2691048.
• Copher, Jessica (2019). "Sums and Products of Regular Polytopes' Squared Chord Lengths". arXiv:1903.06971 [math.MG].
• Miyazaki, Koji (1990). "Primary Hypergeodesic Polytopes". International Journal of Space Structures. 5 (3–4): 309–323. doi:10.1177/026635119000500312. S2CID 113846838.
• van Ittersum, Clara (2020). Symmetry groups of regular polytopes in three and four dimensions (Thesis). Delft University of Technology.
• Waegell, Mordecai; Aravind, P. K. (2009-11-12). "Critical noncolorings of the 600-cell proving the Bell-Kochen-Specker theorem". Journal of Physics A. 43 (10): 105304. arXiv:0911.2289. doi:10.1088/1751-8113/43/10/105304. S2CID 118501180.
• Zamboj, Michal (8 Jan 2021). "Synthetic construction of the Hopf fibration in a double orthogonal projection of 4-space". Journal of Computational Design and Engineering. 8 (3): 836–854. arXiv:2003.09236. doi:10.1093/jcde/qwab018.
• Kim, Heuna; Rote, G. (2016). "Congruence Testing of Point Sets in 4 Dimensions". arXiv:1603.07269 [cs.CG].
• Sadoc, Jean-Francois (2001). "Helices and helix packings derived from the {3,3,5} polytope". European Physical Journal E. 5: 575–582. doi:10.1007/s101890170040. S2CID 121229939.
• Sadoc, J F; Charvolin, J (2009). "3-sphere fibrations: A tool for analyzing twisted materials in condensed matter". Journal of Physics A. 42. doi:10.1088/1751-8113/42/46/465209.
• Lemmens, P.W.H.; Seidel, J.J. (1973). "Equi-isoclinic subspaces of Euclidean spaces". Indagationes Mathematicae (Proceedings). 76 (2): 98–107. doi:10.1016/1385-7258(73)90042-5. ISSN 1385-7258.
• Tyrrell, J. A.; Semple, J.G. (1971). Generalized Clifford parallelism. Cambridge University Press. ISBN 0-521-08042-8.
• Perez-Gracia, Alba; Thomas, Federico (2017). "On Cayley's Factorization of 4D Rotations and Applications" (PDF). Adv. Appl. Clifford Algebras. 27: 523–538. doi:10.1007/s00006-016-0683-9. hdl:2117/113067. S2CID 12350382.
• Mamone, Salvatore; Pileio, Giuseppe; Levitt, Malcolm H. (2010). "Orientational Sampling Schemes Based on Four Dimensional Polytopes". Symmetry. 2 (3): 1423–1449. Bibcode:2010Symm....2.1423M. doi:10.3390/sym2031423.
• Stillwell, John (January 2001). "The Story of the 120-Cell" (PDF). Notices of the AMS. 48 (1): 17–25.
• Dorst, Leo (2019). "Conformal Villarceau Rotors". Advances in Applied Clifford Algebras. 29 (44). doi:10.1007/s00006-019-0960-5. S2CID 253592159.
• Banchoff, Thomas F. (2013). "Torus Decompostions of Regular Polytopes in 4-space". In Senechal, Marjorie (ed.). Shaping Space. Springer New York. pp. 257–266. doi:10.1007/978-0-387-92714-5_20. ISBN 978-0-387-92713-8.
• Banchoff, Thomas (1988). "Geometry of the Hopf Mapping and Pinkall's Tori of Given Conformal Type". In Tangora, Martin (ed.). Computers in Algebra. New York and Basel: Marcel Dekker. pp. 57–62.
• Koca, Mehmet; Ozdes Koca, Nazife; Al-Barwani, Muataz (2012). "Snub 24-Cell Derived from the Coxeter-Weyl Group W(D4)". Int. J. Geom. Methods Mod. Phys. 09 (8). arXiv:1106.3433. doi:10.1142/S0219887812500685. S2CID 119288632.
• Koca, Mehmet; Al-Ajmi, Mudhahir; Ozdes Koca, Nazife (2011). "Quaternionic representation of snub 24-cell and its dual polytope derived from E8 root system". Linear Algebra and Its Applications. 434 (4): 977–989. doi:10.1016/j.laa.2010.10.005. ISSN 0024-3795. S2CID 18278359.
• Dechant, Pierre-Philippe (2021). "Clifford Spinors and Root System Induction: H4 and the Grand Antiprism". Advances in Applied Clifford Algebras. Springer Science and Business Media. 31 (3). doi:10.1007/s00006-021-01139-2. S2CID 232232920.
• Dechant, Pierre-Philippe (2017). "The E8 Geometry from a Clifford Perspective". Advances in Applied Clifford Algebras. 27: 397–421. doi:10.1007/s00006-016-0675-9. S2CID 253595386.
• Itoh, Jin-ichi; Nara, Chie (2021). "Continuous flattening of the 2-dimensional skeleton of a regular 24-cell". Journal of Geometry. 112 (13). doi:10.1007/s00022-021-00575-6.
External links
• Weisstein, Eric W. "600-Cell". MathWorld.
• Klitzing, Richard. "4D uniform polytopes (polychora) x3o3o5o - ex".
• Der 600-Zeller (600-cell) Marco Möller's Regular polytopes in R4 (German)
• The 600-Cell Vertex centered expansion of the 600-cell
Regular 4-polytopes
Convex
5-cell8-cell16-cell24-cell120-cell600-cell
• {3,3,3}
• pentachoron
• 4-simplex
• {4,3,3}
• tesseract
• 4-cube
• {3,3,4}
• hexadecachoron
• 4-orthoplex
• {3,4,3}
• icositetrachoron
• octaplex
• {5,3,3}
• hecatonicosachoron
• dodecaplex
• {3,3,5}
• hexacosichoron
• tetraplex
Star
icosahedral
120-cell
small
stellated
120-cell
great
120-cell
grand
120-cell
great
stellated
120-cell
grand
stellated
120-cell
great grand
120-cell
great
icosahedral
120-cell
grand
600-cell
great grand
stellated 120-cell
• {3,5,5/2}
• icosaplex
• {5/2,5,3}
• stellated dodecaplex
• {5,5/2,5}
• great dodecaplex
• {5,3,5/2}
• grand dodecaplex
• {5/2,3,5}
• great stellated dodecaplex
• {5/2,5,5/2}
• grand stellated dodecaplex
• {5,5/2,3}
• great grand dodecaplex
• {3,5/2,5}
• great icosaplex
• {3,3,5/2}
• grand tetraplex
• {5/2,3,3}
• great grand stellated dodecaplex
Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron
Uniform polychoron Pentachoron 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell
Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221
Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321
Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421
Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds
| Wikipedia |
Wallpaper group
A wallpaper is a mathematical object covering a whole Euclidean plane by repeating a motif indefinitely, in manner that certain isometries keep the drawing unchanged. For each wallpaper there corresponds a group of congruent transformations, with function composition as the group operation. Thus, a wallpaper group (or plane symmetry group or plane crystallographic group) is a mathematical classification of a two‑dimensional repetitive pattern, based on the symmetries in the pattern. Such patterns occur frequently in architecture and decorative art, especially in textiles, tessellations, tiles and physical wallpaper.
What this page calls pattern
Image 1.
Examples of repetitive surfaces
on a Pythagorean tiling.
Image 2.
The minimal area of any of possible repetitive surfaces
${\text{is either }}\;a\;$ by disregarding the colors ${\text{ or otherwise }}\;4\,a.$
Such Pythagorean tilings can be seen as wallpapers because they are periodic.
Any periodic tiling can be seen as a wallpaper. More particularly, we can consider as a wallpaper a tiling by identical tiles edge‑to‑edge, necessarily periodic, and conceive from it a wallpaper by decorating in the same manner every tiling element, and eventually erase partly or entirely the boundaries between these tiles. Conversely, from every wallpaper we can construct such a tiling by identical tiles edge‑to‑edge, which bear each identical ornaments, the identical outlines of these tiles being not necessarily visible on the original wallpaper. Such repeated boundaries delineate a repetitive surface added here in dashed lines.
Such pseudo‑tilings connected to a given wallpaper are in infinite number. For example image 1 shows two models of repetitive squares in two different positions, which have ${\text{equal areas of }}\,a.$ Another repetitive square has an ${\text{area of 5 times}}\,~a.~$ We could indefinitely conceive such repetitive squares larger and larger. An infinity of shapes of repetitive zones are possible for this Pythagorean tiling, in an infinity of positions on this wallpaper. For example in red on the bottom right‑hand corner of image 1, we could glide its repetitive parallelogram in one or another position. In common on the first two images: a repetitive square concentric with each small square tile, their common center being a point symmetry of the wallpaper.
Between identical tiles edge‑to‑edge, an edge is not necessarily a segment of a straight line. On the top left‑hand corner of image 3, point C is a vertex of a repetitive pseudo‑rhombus with thick stripes on its whole surface, called pseudo‑rhombus because of a concentric repetitive rhombus ${\text{of same area}}\,~a,~$ constructed from it by taking out a bit of surface somewhere to append it elsewhere, and keep the area unchanged. By the same process on image 4, a repetitive regular hexagon filled with vertical stripes is constructed from a rhombic repetitive zone ${\text{of area }}\,~a.~$ Conversely, from elementary geometric tiles edge‑to‑edge, an artist like M. C. Escher created attractive surfaces many times repeated. On image 2, $a~{\text{ represents}}$ the minimum area of a repetitive surface by disregarding colors, each repetitive zone in dashed lines consisting of five pieces in a certain arrangement, to be either a square or a hexagon, like in a proof of the Pythagorean theorem.
In the present article, a pattern is a repetitive parallelogram of minimal area in a determined position on the wallpaper. Image 1 shows two parallelogram‑shaped patterns — a square is a particular parallelogram —. Image 3 shows rhombic patterns — a rhombus is a particular parallelogram —.
On this page, all repetitive patterns (of minimal area) are constructed from two translations that generate the group of all translations under which the wallpaper is invariant. With the circle shaped symbol ⵔ of function composition, a pair like $\{T,U\}$ or $\{\,U,\;T\circ U^{\,-1}\}$ generates the group of all translations that transform the Pythagorean tiling into itself.
Image 3.
In one or the other orientation, every rhombus
in dark dashed lines instances a same pattern,
because the rotation of center S and ‑120° angle
leaves the wallpaper unchanged.
Image 4.
The same wallpaper as previously by disregarding its colors.
Otherwise if the colors are considered, there is no longer
a center of rotation that leaves the wallpaper unchanged,
either at point S or C or H.
Is considered as the same pattern its image
under an isometry keeping the wallpaper unchanged.
Possible groups linked to a pattern
A wallpaper remains on the whole unchanged under certain isometries, starting with certain translations that confer on the wallpaper a repetitive nature. One of the reasons to be unchanged under certain translations is that it covers the whole plane. No mathematical object in our minds is stuck onto a motionless wall! On the contrary an observer or his eye is motionless in front of a transformation, which glides or rotates or flips a wallpaper, eventually could distort it, but that would be out of our subject.
If an isometry leaves unchanged a given wallpaper, then the inverse isometry keeps it also unchanged, like translation ${\mathit {T}}{\text{ or }}{\mathit {T}}^{\,-1}~$ on image 1, 3 or 4, or a ± 120° rotation around a point like S on image 3 or 4. If they have both this property to leave unchanged a wallpaper, two isometries composed in one or the other order have then this same property to leave unchanged the wallpaper. To be exhaustive about the concepts of group and subgroups under the function composition, represented by the circle shaped symbol ⵔ, here is a traditional truism in mathematics: everything remains itself under the identity transformation. This identity function can be called translation of zero vector or rotation of 360°.
A glide can be represented by one or several arrows if parallel and of same length and same sense, in same way a wallpaper can be represented either by a few patterns or by only one pattern, considered as a pseudo‑tile imagined repeated edge‑to‑edge with an infinite number of replicas. Image 3 shows two patterns with two different contents, and the one in dark dashed lines or one of its images under $\,{\text{ rotation }}\,{\mathit {R}}\,{\text{ or }}\,{\mathit {R}}\,^{-1}\;$ represents the same wallpaper on the following image 4, by disregarding the colors. Certainly a color is perceived subjectively whereas a wallpaper is an ideal object, however any color can be seen as a label that characterizes certain surfaces, we might think of a hexadecimal code of color as a label specific to certain zones. It may be added that a well‑known theorem deals with colors.
Groups are registered in the catalog by examining properties of a parallelogram, edge‑to‑edge with its replicas. For example its diagonals intersect at their common midpoints, center and symmetry point of any parallelogram, not necessarily symmetry point of its content. Other example, the midpoint of a full side shared by two patterns is the center of a new repetitive parallelogram formed by the two together, center which is not necessarily symmetry point of the content of this double parallelogram. Other possible symmetry point, two patterns symmetric one to the other with respect to their common vertex form together a new repetitive surface, the center of which is not necessarily symmetry point of its content.
Certain rotational symmetries are possible only for certain shapes of pattern. For example on image 2, a Pythagorean tiling is sometimes called pinwheel tilings because of its rotational symmetry of 90 degrees about the center of a tile, either small or large, or about the center of any replica of tile, of course. Also when two equilateral triangles form edge‑to‑edge a rhombic pattern, like on image 4 or 5 (future image 5), a rotational symmetry of 120 degrees about a vertex of a 120° angle, formed by two sides of pattern, is not always a symmetry point of the content of the regular hexagon formed by three patterns together sharing a vertex, because it does not always contain the same motif.
First examples of groups
The simplest wallpaper group, Group p1, applies when there is no symmetry other than the fact that a pattern repeats over regular intervals in two dimensions, as shown in the section on p1 below.
The following examples are patterns with more forms of symmetry:
• Example A: Cloth, Tahiti
• Example B: Ornamental painting, Nineveh, Assyria
• Example C: Painted porcelain, China
Examples A and B have the same wallpaper group; it is called p4m in the IUCr notation and *442 in the orbifold notation. Example C has a different wallpaper group, called p4g or 4*2 . The fact that A and B have the same wallpaper group means that they have the same symmetries, regardless of details of the designs, whereas C has a different set of symmetries despite any superficial similarities.
The number of symmetry groups depends on the number of dimensions in the patterns. Wallpaper groups apply to the two-dimensional case, intermediate in complexity between the simpler frieze groups and the three-dimensional space groups. Subtle differences may place similar patterns in different groups, while patterns that are very different in style, color, scale or orientation may belong to the same group.
A proof that there are only 17 distinct groups of such planar symmetries was first carried out by Evgraf Fedorov in 1891[1] and then derived independently by George Pólya in 1924.[2] The proof that the list of wallpaper groups is complete only came after the much harder case of space groups had been done. The seventeen possible wallpaper groups are listed below in § The seventeen groups.
Symmetries of patterns
A symmetry of a pattern is, loosely speaking, a way of transforming the pattern so that it looks exactly the same after the transformation. For example, translational symmetry is present when the pattern can be translated (in other words, shifted) some finite distance and appear unchanged. Think of shifting a set of vertical stripes horizontally by one stripe. The pattern is unchanged. Strictly speaking, a true symmetry only exists in patterns that repeat exactly and continue indefinitely. A set of only, say, five stripes does not have translational symmetry—when shifted, the stripe on one end "disappears" and a new stripe is "added" at the other end. In practice, however, classification is applied to finite patterns, and small imperfections may be ignored.
The types of transformations that are relevant here are called Euclidean plane isometries. For example:
• If one shifts example B one unit to the right, so that each square covers the square that was originally adjacent to it, then the resulting pattern is exactly the same as the starting pattern. This type of symmetry is called a translation. Examples A and C are similar, except that the smallest possible shifts are in diagonal directions.
• If one turns example B clockwise by 90°, around the centre of one of the squares, again one obtains exactly the same pattern. This is called a rotation. Examples A and C also have 90° rotations, although it requires a little more ingenuity to find the correct centre of rotation for C.
• One can also flip example B across a horizontal axis that runs across the middle of the image. This is called a reflection. Example C also has reflections across a vertical axis, and across two diagonal axes. The same can be said for A.
However, example C is different. It only has reflections in horizontal and vertical directions, not across diagonal axes. If one flips across a diagonal line, one does not get the same pattern back, but the original pattern shifted across by a certain distance. This is part of the reason that the wallpaper group of A and B is different from the wallpaper group of C.
Another transformation is "Glide", a combination of reflection and translation parallel to the line of reflection.
Formal definition and discussion
Mathematically, a wallpaper group or plane crystallographic group is a type of topologically discrete group of isometries of the Euclidean plane that contains two linearly independent translations.
Two such isometry groups are of the same type (of the same wallpaper group) if they are the same up to an affine transformation of the plane. Thus e.g. a translation of the plane (hence a translation of the mirrors and centres of rotation) does not affect the wallpaper group. The same applies for a change of angle between translation vectors, provided that it does not add or remove any symmetry (this is only the case if there are no mirrors and no glide reflections, and rotational symmetry is at most of order 2).
Unlike in the three-dimensional case, one can equivalently restrict the affine transformations to those that preserve orientation.
It follows from the Bieberbach theorem that all wallpaper groups are different even as abstract groups (as opposed to e.g. frieze groups, of which two are isomorphic with Z).
2D patterns with double translational symmetry can be categorized according to their symmetry group type.
Isometries of the Euclidean plane
Isometries of the Euclidean plane fall into four categories (see the article Euclidean plane isometry for more information).
• Translations, denoted by Tv, where v is a vector in R2. This has the effect of shifting the plane applying displacement vector v.
• Rotations, denoted by Rc,θ, where c is a point in the plane (the centre of rotation), and θ is the angle of rotation.
• Reflections, or mirror isometries, denoted by FL, where L is a line in R2. (F is for "flip"). This has the effect of reflecting the plane in the line L, called the reflection axis or the associated mirror.
• Glide reflections, denoted by GL,d, where L is a line in R2 and d is a distance. This is a combination of a reflection in the line L and a translation along L by a distance d.
The independent translations condition
The condition on linearly independent translations means that there exist linearly independent vectors v and w (in R2) such that the group contains both Tv and Tw.
The purpose of this condition is to distinguish wallpaper groups from frieze groups, which possess a translation but not two linearly independent ones, and from two-dimensional discrete point groups, which have no translations at all. In other words, wallpaper groups represent patterns that repeat themselves in two distinct directions, in contrast to frieze groups, which only repeat along a single axis.
(It is possible to generalise this situation. One could for example study discrete groups of isometries of Rn with m linearly independent translations, where m is any integer in the range 0 ≤ m ≤ n.)
The discreteness condition
The discreteness condition means that there is some positive real number ε, such that for every translation Tv in the group, the vector v has length at least ε (except of course in the case that v is the zero vector, but the independent translations condition prevents this, since any set that contains the zero vector is linearly dependent by definition and thus disallowed).
The purpose of this condition is to ensure that the group has a compact fundamental domain, or in other words, a "cell" of nonzero, finite area, which is repeated through the plane. Without this condition, one might have for example a group containing the translation Tx for every rational number x, which would not correspond to any reasonable wallpaper pattern.
One important and nontrivial consequence of the discreteness condition in combination with the independent translations condition is that the group can only contain rotations of order 2, 3, 4, or 6; that is, every rotation in the group must be a rotation by 180°, 120°, 90°, or 60°. This fact is known as the crystallographic restriction theorem,[3] and can be generalised to higher-dimensional cases.
Crystallographic notation
Crystallography has 230 space groups to distinguish, far more than the 17 wallpaper groups, but many of the symmetries in the groups are the same. Thus one can use a similar notation for both kinds of groups, that of Carl Hermann and Charles-Victor Mauguin. An example of a full wallpaper name in Hermann-Mauguin style (also called IUCr notation) is p31m, with four letters or digits; more usual is a shortened name like cmm or pg.
For wallpaper groups the full notation begins with either p or c, for a primitive cell or a face-centred cell; these are explained below. This is followed by a digit, n, indicating the highest order of rotational symmetry: 1-fold (none), 2-fold, 3-fold, 4-fold, or 6-fold. The next two symbols indicate symmetries relative to one translation axis of the pattern, referred to as the "main" one; if there is a mirror perpendicular to a translation axis that is the main one (or if there are two, one of them). The symbols are either m, g, or 1, for mirror, glide reflection, or none. The axis of the mirror or glide reflection is perpendicular to the main axis for the first letter, and either parallel or tilted 180°/n (when n > 2) for the second letter. Many groups include other symmetries implied by the given ones. The short notation drops digits or an m that can be deduced, so long as that leaves no confusion with another group.
A primitive cell is a minimal region repeated by lattice translations. All but two wallpaper symmetry groups are described with respect to primitive cell axes, a coordinate basis using the translation vectors of the lattice. In the remaining two cases symmetry description is with respect to centred cells that are larger than the primitive cell, and hence have internal repetition; the directions of their sides is different from those of the translation vectors spanning a primitive cell. Hermann-Mauguin notation for crystal space groups uses additional cell types.
Examples
• p2 (p2): Primitive cell, 2-fold rotation symmetry, no mirrors or glide reflections.
• p4gm (p4mm): Primitive cell, 4-fold rotation, glide reflection perpendicular to main axis, mirror axis at 45°.
• c2mm (c2mm): Centred cell, 2-fold rotation, mirror axes both perpendicular and parallel to main axis.
• p31m (p31m): Primitive cell, 3-fold rotation, mirror axis at 60°.
Here are all the names that differ in short and full notation.
Crystallographic short and full names
Short pm pg cm pmm pmg pgg cmm p4m p4g p6m
Fullp1m1p1g1c1m1p2mmp2mgp2ggc2mmp4mmp4gmp6mm
The remaining names are p1, p2, p3, p3m1, p31m, p4, and p6.
Orbifold notation
Orbifold notation for wallpaper groups, advocated by John Horton Conway (Conway, 1992) (Conway 2008), is based not on crystallography, but on topology. One can fold the infinite periodic tiling of the plane into its essence, an orbifold, then describe that with a few symbols.
• A digit, n, indicates a centre of n-fold rotation corresponding to a cone point on the orbifold. By the crystallographic restriction theorem, n must be 2, 3, 4, or 6.
• An asterisk, *, indicates a mirror symmetry corresponding to a boundary of the orbifold. It interacts with the digits as follows:
1. Digits before * denote centres of pure rotation (cyclic).
2. Digits after * denote centres of rotation with mirrors through them, corresponding to "corners" on the boundary of the orbifold (dihedral).
• A cross, ×, occurs when a glide reflection is present and indicates a crosscap on the orbifold. Pure mirrors combine with lattice translation to produce glides, but those are already accounted for so need no notation.
• The "no symmetry" symbol, o, stands alone, and indicates there are only lattice translations with no other symmetry. The orbifold with this symbol is a torus; in general the symbol o denotes a handle on the orbifold.
The group denoted in crystallographic notation by cmm will, in Conway's notation, be 2*22. The 2 before the * says there is a 2-fold rotation centre with no mirror through it. The * itself says there is a mirror. The first 2 after the * says there is a 2-fold rotation centre on a mirror. The final 2 says there is an independent second 2-fold rotation centre on a mirror, one that is not a duplicate of the first one under symmetries.
The group denoted by pgg will be 22×. There are two pure 2-fold rotation centres, and a glide reflection axis. Contrast this with pmg, Conway 22*, where crystallographic notation mentions a glide, but one that is implicit in the other symmetries of the orbifold.
Coxeter's bracket notation is also included, based on reflectional Coxeter groups, and modified with plus superscripts accounting for rotations, improper rotations and translations.
Conway, Coxeter and crystallographic correspondence
Conway o××*×**632*632
Coxeter [∞+,2,∞+][(∞,2)+,∞+][∞,2+,∞+][∞,2,∞+][6,3]+[6,3]
Crystallographic p1pgcmpmp6p6m
Conway 333*3333*3442*4424*2
Coxeter [3[3]]+[3[3]][3+,6][4,4]+ [4,4][4+,4]
Crystallographic p3p3m1p31mp4 p4mp4g
Conway 222222×22**22222*22
Coxeter [∞,2,∞]+[((∞,2)+,(∞,2)+)][(∞,2)+,∞][∞,2,∞][∞,2+,∞]
Crystallographic p2pggpmgpmmcmm
Why there are exactly seventeen groups
An orbifold can be viewed as a polygon with face, edges, and vertices which can be unfolded to form a possibly infinite set of polygons which tile either the sphere, the plane or the hyperbolic plane. When it tiles the plane it will give a wallpaper group and when it tiles the sphere or hyperbolic plane it gives either a spherical symmetry group or Hyperbolic symmetry group. The type of space the polygons tile can be found by calculating the Euler characteristic, χ = V − E + F, where V is the number of corners (vertices), E is the number of edges and F is the number of faces. If the Euler characteristic is positive then the orbifold has an elliptic (spherical) structure; if it is zero then it has a parabolic structure, i.e. a wallpaper group; and if it is negative it will have a hyperbolic structure. When the full set of possible orbifolds is enumerated it is found that only 17 have Euler characteristic 0.
When an orbifold replicates by symmetry to fill the plane, its features create a structure of vertices, edges, and polygon faces, which must be consistent with the Euler characteristic. Reversing the process, one can assign numbers to the features of the orbifold, but fractions, rather than whole numbers. Because the orbifold itself is a quotient of the full surface by the symmetry group, the orbifold Euler characteristic is a quotient of the surface Euler characteristic by the order of the symmetry group.
The orbifold Euler characteristic is 2 minus the sum of the feature values, assigned as follows:
• A digit n without or before a * counts as n − 1/n.
• A digit n after a * counts as n − 1/2n.
• Both * and × count as 1.
• The "no symmetry" o counts as 2.
For a wallpaper group, the sum for the characteristic must be zero; thus the feature sum must be 2.
Examples
• 632: 5/6 + 2/3 + 1/2 = 2
• 3*3: 2/3 + 1 + 2/6 = 2
• 4*2: 3/4 + 1 + 1/4 = 2
• 22×: 1/2 + 1/2 + 1 = 2
Now enumeration of all wallpaper groups becomes a matter of arithmetic, of listing all feature strings with values summing to 2.
Feature strings with other sums are not nonsense; they imply non-planar tilings, not discussed here. (When the orbifold Euler characteristic is negative, the tiling is hyperbolic; when positive, spherical or bad).
Guide to recognizing wallpaper groups
To work out which wallpaper group corresponds to a given design, one may use the following table.[4]
Size of smallest
rotation
Has reflection?
YesNo
360° / 6p6m (*632)p6 (632)
360° / 4Has mirrors at 45°?p4 (442)
Yes: p4m (*442)No: p4g (4*2)
360° / 3Has rot. centre off mirrors?p3 (333)
Yes: p31m (3*3)No: p3m1 (*333)
360° / 2Has perpendicular reflections?Has glide reflection?
YesNo
Has rot. centre off mirrors?pmg (22*)Yes: pgg (22×)No: p2 (2222)
Yes: cmm (2*22)No: pmm (*2222)
noneHas glide axis off mirrors?Has glide reflection?
Yes: cm (*×)No: pm (**)Yes: pg (××)No: p1 (o)
See also this overview with diagrams.
The seventeen groups
Each of the groups in this section has two cell structure diagrams, which are to be interpreted as follows (it is the shape that is significant, not the colour):
a centre of rotation of order two (180°).
a centre of rotation of order three (120°).
a centre of rotation of order four (90°).
a centre of rotation of order six (60°).
an axis of reflection.
an axis of glide reflection.
On the right-hand side diagrams, different equivalence classes of symmetry elements are colored (and rotated) differently.
The brown or yellow area indicates a fundamental domain, i.e. the smallest part of the pattern that is repeated.
The diagrams on the right show the cell of the lattice corresponding to the smallest translations; those on the left sometimes show a larger area.
Group p1 (o)
Cell structures for p1 by lattice type
Oblique
Hexagonal
Rectangular
Rhombic
Square
• Orbifold signature: o
• Coxeter notation (rectangular): [∞+,2,∞+] or [∞]+×[∞]+
• Lattice: oblique
• Point group: C1
• The group p1 contains only translations; there are no rotations, reflections, or glide reflections.
Examples of group p1
• Computer generated
• Medieval wall diapering
The two translations (cell sides) can each have different lengths, and can form any angle.
Group p2 (2222)
Cell structures for p2 by lattice type
Oblique
Hexagonal
Rectangular
Rhombic
Square
• Orbifold signature: 2222
• Coxeter notation (rectangular): [∞,2,∞]+
• Lattice: oblique
• Point group: C2
• The group p2 contains four rotation centres of order two (180°), but no reflections or glide reflections.
Examples of group p2
• Computer generated
• Cloth, Sandwich Islands (Hawaii)
• Mat on which an Egyptian king stood
• Egyptian mat (detail)
• Ceiling of an Egyptian tomb
• Wire fence, U.S.
Group pm (**)
Cell structure for pm
Horizontal mirrors
Vertical mirrors
• Orbifold signature: **
• Coxeter notation: [∞,2,∞+] or [∞+,2,∞]
• Lattice: rectangular
• Point group: D1
• The group pm has no rotations. It has reflection axes, they are all parallel.
Examples of group pm
(The first three have a vertical symmetry axis, and the last two each have a different diagonal one.)
• Computer generated
• Dress of a figure in a tomb at Biban el Moluk, Egypt
• Egyptian tomb, Thebes
• Ceiling of a tomb at Gourna, Egypt. Reflection axis is diagonal
• Indian metalwork at the Great Exhibition in 1851. This is almost pm (ignoring short diagonal lines between ovals motifs, which make it p1)
Group pg (××)
Cell structures for pg
Horizontal glides
Vertical glides
Rectangular
• Orbifold signature: ××
• Coxeter notation: [(∞,2)+,∞+] or [∞+,(2,∞)+]
• Lattice: rectangular
• Point group: D1
• The group pg contains glide reflections only, and their axes are all parallel. There are no rotations or reflections.
Examples of group pg
• Computer generated
• Mat with herringbone pattern on which Egyptian king stood
• Egyptian mat (detail)
• Pavement with herringbone pattern in Salzburg. Glide reflection axis runs northeast–southwest
• One of the colorings of the snub square tiling; the glide reflection lines are in the direction upper left / lower right; ignoring colors there is much more symmetry than just pg, then it is p4g (see there for this image with equally colored triangles)[5]
Without the details inside the zigzag bands the mat is pmg; with the details but without the distinction between brown and black it is pgg.
Ignoring the wavy borders of the tiles, the pavement is pgg.
Group cm (*×)
Cell structure for cm
Horizontal mirrors
Vertical mirrors
Rhombic
• Orbifold signature: *×
• Coxeter notation: [∞+,2+,∞] or [∞,2+,∞+]
• Lattice: rhombic
• Point group: D1
• The group cm contains no rotations. It has reflection axes, all parallel. There is at least one glide reflection whose axis is not a reflection axis; it is halfway between two adjacent parallel reflection axes.
• This group applies for symmetrically staggered rows (i.e. there is a shift per row of half the translation distance inside the rows) of identical objects, which have a symmetry axis perpendicular to the rows.
Examples of group cm
• Computer generated
• Dress of Amun, from Abu Simbel, Egypt
• Dado from Biban el Moluk, Egypt
• Bronze vessel in Nimroud, Assyria
• Spandrels of arches, the Alhambra, Spain
• Soffitt of arch, the Alhambra, Spain
• Persian tapestry
• Indian metalwork at the Great Exhibition in 1851
• Dress of a figure in a tomb at Biban el Moluk, Egypt
Group pmm (*2222)
Cell structure for pmm
rectangular
square
• Orbifold signature: *2222
• Coxeter notation (rectangular): [∞,2,∞] or [∞]×[∞]
• Coxeter notation (square): [4,1+,4] or [1+,4,4,1+]
• Lattice: rectangular
• Point group: D2
• The group pmm has reflections in two perpendicular directions, and four rotation centres of order two (180°) located at the intersections of the reflection axes.
Examples of group pmm
• 2D image of lattice fence, U.S. (in 3D there is additional symmetry)
• Mummy case stored in The Louvre
• Mummy case stored in The Louvre. Would be type p4m except for the mismatched coloring
Group pmg (22*)
Cell structures for pmg
Horizontal mirrors
Vertical mirrors
• Orbifold signature: 22*
• Coxeter notation: [(∞,2)+,∞] or [∞,(2,∞)+]
• Lattice: rectangular
• Point group: D2
• The group pmg has two rotation centres of order two (180°), and reflections in only one direction. It has glide reflections whose axes are perpendicular to the reflection axes. The centres of rotation all lie on glide reflection axes.
Examples of group pmg
• Computer generated
• Cloth, Sandwich Islands (Hawaii)
• Ceiling of Egyptian tomb
• Floor tiling in Prague, the Czech Republic
• Bowl from Kerma
• Pentagon packing
Group pgg (22×)
Cell structures for pgg by lattice type
Rectangular
Square
• Orbifold signature: 22×
• Coxeter notation (rectangular): [((∞,2)+,(∞,2)+)]
• Coxeter notation (square): [4+,4+]
• Lattice: rectangular
• Point group: D2
• The group pgg contains two rotation centres of order two (180°), and glide reflections in two perpendicular directions. The centres of rotation are not located on the glide reflection axes. There are no reflections.
Examples of group pgg
• Computer generated
• Bronze vessel in Nimroud, Assyria
• Pavement in Budapest, Hungary
Group cmm (2*22)
Cell structures for cmm by lattice type
Rhombic
Square
• Orbifold signature: 2*22
• Coxeter notation (rhombic): [∞,2+,∞]
• Coxeter notation (square): [(4,4,2+)]
• Lattice: rhombic
• Point group: D2
• The group cmm has reflections in two perpendicular directions, and a rotation of order two (180°) whose centre is not on a reflection axis. It also has two rotations whose centres are on a reflection axis.
• This group is frequently seen in everyday life, since the most common arrangement of bricks in a brick building (running bond) utilises this group (see example below).
The rotational symmetry of order 2 with centres of rotation at the centres of the sides of the rhombus is a consequence of the other properties.
The pattern corresponds to each of the following:
• symmetrically staggered rows of identical doubly symmetric objects
• a checkerboard pattern of two alternating rectangular tiles, of which each, by itself, is doubly symmetric
• a checkerboard pattern of alternatingly a 2-fold rotationally symmetric rectangular tile and its mirror image
Examples of group cmm
• Computer generated
• Elongated triangular tiling
• Suburban brick wall using running bond arrangement, U.S.
• Ceiling of Egyptian tomb. Ignoring colors, this would be p4g
• Egyptian
• Persian tapestry
• Egyptian tomb
• Turkish dish
• A compact packing of two sizes of circle
• Another compact packing of two sizes of circle
• Another compact packing of two sizes of circle
Group p4 (442)
• Orbifold signature: 442
• Coxeter notation: [4,4]+
• Lattice: square
• Point group: C4
• The group p4 has two rotation centres of order four (90°), and one rotation centre of order two (180°). It has no reflections or glide reflections.
Examples of group p4
A p4 pattern can be looked upon as a repetition in rows and columns of equal square tiles with 4-fold rotational symmetry. Also it can be looked upon as a checkerboard pattern of two such tiles, a factor √2 smaller and rotated 45°.
• Computer generated
• Ceiling of Egyptian tomb; ignoring colors this is p4, otherwise p2
• Ceiling of Egyptian tomb
• Overlaid patterns
• Frieze, the Alhambra, Spain. Requires close inspection to see why there are no reflections
• Viennese cane
• Renaissance earthenware
• Pythagorean tiling
• Generated from a photograph
Group p4m (*442)
• Orbifold signature: *442
• Coxeter notation: [4,4]
• Lattice: square
• Point group: D4
• The group p4m has two rotation centres of order four (90°), and reflections in four distinct directions (horizontal, vertical, and diagonals). It has additional glide reflections whose axes are not reflection axes; rotations of order two (180°) are centred at the intersection of the glide reflection axes. All rotation centres lie on reflection axes.
This corresponds to a straightforward grid of rows and columns of equal squares with the four reflection axes. Also it corresponds to a checkerboard pattern of two of such squares.
Examples of group p4m
Examples displayed with the smallest translations horizontal and vertical (like in the diagram):
• Computer generated
• Square tiling
• Tetrakis square tiling; ignoring colors, this is p4m, otherwise c2m
• Truncated square tiling (ignoring color also, with smaller translations)
• Ornamental painting, Nineveh, Assyria
• Storm drain, U.S.
• Egyptian mummy case
• Persian glazed tile
• Compact packing of two sizes of circle
Examples displayed with the smallest translations diagonal:
• checkerboard
• Cloth, Otaheite (Tahiti)
• Egyptian tomb
• Cathedral of Bourges
• Dish from Turkey, Ottoman period
Group p4g (4*2)
• Orbifold signature: 4*2
• Coxeter notation: [4+,4]
• Lattice: square
• Point group: D4
• The group p4g has two centres of rotation of order four (90°), which are each other's mirror image, but it has reflections in only two directions, which are perpendicular. There are rotations of order two (180°) whose centres are located at the intersections of reflection axes. It has glide reflections axes parallel to the reflection axes, in between them, and also at an angle of 45° with these.
A p4g pattern can be looked upon as a checkerboard pattern of copies of a square tile with 4-fold rotational symmetry, and its mirror image. Alternatively it can be looked upon (by shifting half a tile) as a checkerboard pattern of copies of a horizontally and vertically symmetric tile and its 90° rotated version. Note that neither applies for a plain checkerboard pattern of black and white tiles, this is group p4m (with diagonal translation cells).
Examples of group p4g
• Bathroom linoleum, U.S.
• Painted porcelain, China
• Fly screen, U.S.
• Painting, China
• one of the colorings of the snub square tiling (see also at pg)
Group p3 (333)
• Orbifold signature: 333
• Coxeter notation: [(3,3,3)]+ or [3[3]]+
• Lattice: hexagonal
• Point group: C3
• The group p3 has three different rotation centres of order three (120°), but no reflections or glide reflections.
Imagine a tessellation of the plane with equilateral triangles of equal size, with the sides corresponding to the smallest translations. Then half of the triangles are in one orientation, and the other half upside down. This wallpaper group corresponds to the case that all triangles of the same orientation are equal, while both types have rotational symmetry of order three, but the two are not equal, not each other's mirror image, and not both symmetric (if the two are equal it is p6, if they are each other's mirror image it is p31m, if they are both symmetric it is p3m1; if two of the three apply then the third also, and it is p6m). For a given image, three of these tessellations are possible, each with rotation centres as vertices, i.e. for any tessellation two shifts are possible. In terms of the image: the vertices can be the red, the blue or the green triangles.
Equivalently, imagine a tessellation of the plane with regular hexagons, with sides equal to the smallest translation distance divided by √3. Then this wallpaper group corresponds to the case that all hexagons are equal (and in the same orientation) and have rotational symmetry of order three, while they have no mirror image symmetry (if they have rotational symmetry of order six it is p6, if they are symmetric with respect to the main diagonals it is p31m, if they are symmetric with respect to lines perpendicular to the sides it is p3m1; if two of the three apply then the third also, it is p6m). For a given image, three of these tessellations are possible, each with one third of the rotation centres as centres of the hexagons. In terms of the image: the centres of the hexagons can be the red, the blue or the green triangles.
Examples of group p3
• Computer generated
• Snub trihexagonal tiling (ignoring the colors: p6); the translation vectors are rotated a little to the right compared with the directions in the underlying hexagonal lattice of the image
• Street pavement in Zakopane, Poland
• Wall tiling in the Alhambra, Spain (and the whole wall); ignoring all colors this is p3 (ignoring only star colors it is p1)
Group p3m1 (*333)
• Orbifold signature: *333
• Coxeter notation: [(3,3,3)] or [3[3]]
• Lattice: hexagonal
• Point group: D3
• The group p3m1 has three different rotation centres of order three (120°). It has reflections in the three sides of an equilateral triangle. The centre of every rotation lies on a reflection axis. There are additional glide reflections in three distinct directions, whose axes are located halfway between adjacent parallel reflection axes.
Like for p3, imagine a tessellation of the plane with equilateral triangles of equal size, with the sides corresponding to the smallest translations. Then half of the triangles are in one orientation, and the other half upside down. This wallpaper group corresponds to the case that all triangles of the same orientation are equal, while both types have rotational symmetry of order three, and both are symmetric, but the two are not equal, and not each other's mirror image. For a given image, three of these tessellations are possible, each with rotation centres as vertices. In terms of the image: the vertices can be the red, the blue or the green triangles.
Examples of group p3m1
• Triangular tiling (ignoring colors: p6m)
• Hexagonal tiling (ignoring colors: p6m)
• Truncated hexagonal tiling (ignoring colors: p6m)
• Persian glazed tile (ignoring colors: p6m)
• Persian ornament
• Painting, China (see detailed image)
Group p31m (3*3)
• Orbifold signature: 3*3
• Coxeter notation: [6,3+]
• Lattice: hexagonal
• Point group: D3
• The group p31m has three different rotation centres of order three (120°), of which two are each other's mirror image. It has reflections in three distinct directions. It has at least one rotation whose centre does not lie on a reflection axis. There are additional glide reflections in three distinct directions, whose axes are located halfway between adjacent parallel reflection axes.
Like for p3 and p3m1, imagine a tessellation of the plane with equilateral triangles of equal size, with the sides corresponding to the smallest translations. Then half of the triangles are in one orientation, and the other half upside down. This wallpaper group corresponds to the case that all triangles of the same orientation are equal, while both types have rotational symmetry of order three and are each other's mirror image, but not symmetric themselves, and not equal. For a given image, only one such tessellation is possible. In terms of the image: the vertices must be the red triangles, not the blue triangles.
Examples of group p31m
• Persian glazed tile
• Painted porcelain, China
• Painting, China
• Compact packing of two sizes of circle
Group p6 (632)
• Orbifold signature: 632
• Coxeter notation: [6,3]+
• Lattice: hexagonal
• Point group: C6
• The group p6 has one rotation centre of order six (60°); two rotation centres of order three (120°), which are each other's images under a rotation of 60°; and three rotation centres of order two (180°) which are also each other's images under a rotation of 60°. It has no reflections or glide reflections.
A pattern with this symmetry can be looked upon as a tessellation of the plane with equal triangular tiles with C3 symmetry, or equivalently, a tessellation of the plane with equal hexagonal tiles with C6 symmetry (with the edges of the tiles not necessarily part of the pattern).
Examples of group p6
• Computer generated
• Regular polygons
• Wall panelling, the Alhambra, Spain
• Persian ornament
Group p6m (*632)
• Orbifold signature: *632
• Coxeter notation: [6,3]
• Lattice: hexagonal
• Point group: D6
• The group p6m has one rotation centre of order six (60°); it has two rotation centres of order three, which only differ by a rotation of 60° (or, equivalently, 180°), and three of order two, which only differ by a rotation of 60°. It has also reflections in six distinct directions. There are additional glide reflections in six distinct directions, whose axes are located halfway between adjacent parallel reflection axes.
A pattern with this symmetry can be looked upon as a tessellation of the plane with equal triangular tiles with D3 symmetry, or equivalently, a tessellation of the plane with equal hexagonal tiles with D6 symmetry (with the edges of the tiles not necessarily part of the pattern). Thus the simplest examples are a triangular lattice with or without connecting lines, and a hexagonal tiling with one color for outlining the hexagons and one for the background.
Examples of group p6m
• Computer generated
• Trihexagonal tiling
• Small rhombitrihexagonal tiling
• Great rhombitrihexagonal tiling
• Persian glazed tile
• King's dress, Khorsabad, Assyria; this is almost p6m (ignoring inner parts of flowers, which make it cmm)
• Bronze vessel in Nimroud, Assyria
• Byzantine marble pavement, Rome
• Painted porcelain, China
• Painted porcelain, China
• Compact packing of two sizes of circle
• Another compact packing of two sizes of circle
Lattice types
There are five lattice types or Bravais lattices, corresponding to the five possible wallpaper groups of the lattice itself. The wallpaper group of a pattern with this lattice of translational symmetry cannot have more, but may have less symmetry than the lattice itself.
• In the 5 cases of rotational symmetry of order 3 or 6, the unit cell consists of two equilateral triangles (hexagonal lattice, itself p6m). They form a rhombus with angles 60° and 120°.
• In the 3 cases of rotational symmetry of order 4, the cell is a square (square lattice, itself p4m).
• In the 5 cases of reflection or glide reflection, but not both, the cell is a rectangle (rectangular lattice, itself pmm). It may also be interpreted as a centered rhombic lattice. Special cases: square.
• In the 2 cases of reflection combined with glide reflection, the cell is a rhombus (rhombic lattice, itself cmm). It may also be interpreted as a centered rectangular lattice. Special cases: square, hexagonal unit cell.
• In the case of only rotational symmetry of order 2, and the case of no other symmetry than translational, the cell is in general a parallelogram (parallelogrammatic or oblique lattice, itself p2). Special cases: rectangle, square, rhombus, hexagonal unit cell.
Symmetry groups
The actual symmetry group should be distinguished from the wallpaper group. Wallpaper groups are collections of symmetry groups. There are 17 of these collections, but for each collection there are infinitely many symmetry groups, in the sense of actual groups of isometries. These depend, apart from the wallpaper group, on a number of parameters for the translation vectors, the orientation and position of the reflection axes and rotation centers.
The numbers of degrees of freedom are:
• 6 for p2
• 5 for pmm, pmg, pgg, and cmm
• 4 for the rest.
However, within each wallpaper group, all symmetry groups are algebraically isomorphic.
Some symmetry group isomorphisms:
• p1: Z2
• pm: Z × D∞
• pmm: D∞ × D∞.
Dependence of wallpaper groups on transformations
• The wallpaper group of a pattern is invariant under isometries and uniform scaling (similarity transformations).
• Translational symmetry is preserved under arbitrary bijective affine transformations.
• Rotational symmetry of order two ditto; this means also that 4- and 6-fold rotation centres at least keep 2-fold rotational symmetry.
• Reflection in a line and glide reflection are preserved on expansion/contraction along, or perpendicular to, the axis of reflection and glide reflection. It changes p6m, p4g, and p3m1 into cmm, p3m1 into cm, and p4m, depending on direction of expansion/contraction, into pmm or cmm. A pattern of symmetrically staggered rows of points is special in that it can convert by expansion/contraction from p6m to p4m.
Note that when a transformation decreases symmetry, a transformation of the same kind (the inverse) obviously for some patterns increases the symmetry. Such a special property of a pattern (e.g. expansion in one direction produces a pattern with 4-fold symmetry) is not counted as a form of extra symmetry.
Change of colors does not affect the wallpaper group if any two points that have the same color before the change, also have the same color after the change, and any two points that have different colors before the change, also have different colors after the change.
If the former applies, but not the latter, such as when converting a color image to one in black and white, then symmetries are preserved, but they may increase, so that the wallpaper group can change.
Web demo and software
Several software graphic tools will let you create 2D patterns using wallpaper symmetry groups. Usually you can edit the original tile and its copies in the entire pattern are updated automatically.
• MadPattern, a free set of Adobe Illustrator templates that support the 17 wallpaper groups
• Tess, a shareware tessellation program for multiple platforms, supports all wallpaper, frieze, and rosette groups, as well as Heesch tilings.
• Wallpaper Symmetry is a free online JavaScript drawing tool supporting the 17 groups. The main page has an explanation of the wallpaper groups, as well as drawing tools and explanations for the other planar symmetry groups as well.
• TALES GAME, a free software designed for educational purposes which includes the tessellation function.
• Kali Archived 2018-12-16 at the Wayback Machine, online graphical symmetry editor Java applet (not supported by default in browsers).
• Kali Archived 2020-11-21 at the Wayback Machine, free downloadable Kali for Windows and Mac Classic.
• Inkscape, a free vector graphics editor, supports all 17 groups plus arbitrary scales, shifts, rotates, and color changes per row or per column, optionally randomized to a given degree. (See )
• SymmetryWorks is a commercial plugin for Adobe Illustrator, supports all 17 groups.
• EscherSketch is a free online JavaScript drawing tool supporting the 17 groups.
• Repper is a commercial online drawing tool supporting the 17 groups plus a number of non-periodic tilings
See also
Wikimedia Commons has media related to Wallpaper group diagrams.
• List of planar symmetry groups (summary of this page)
• Aperiodic tiling
• Crystallography
• Layer group
• Mathematics and art
• M. C. Escher
• Point group
• Symmetry groups in one dimension
• Tessellation
Notes
1. E. Fedorov (1891) "Симметрія на плоскости" (Simmetrija na ploskosti, Symmetry in the plane), Записки Императорского С.-Петербургского минералогического общества (Zapiski Imperatorskogo Sant-Petersburgskogo Mineralogicheskogo Obshchestva, Proceedings of the Imperial St. Petersburg Mineralogical Society), series 2, 28 : 345–390 (in Russian).
2. Pólya, George (November 1924). "Über die Analogie der Kristallsymmetrie in der Ebene" [On the analog of crystal symmetry in the plane]. Zeitschrift für Kristallographie (in German). 60 (1–6): 278–282. doi:10.1524/zkri.1924.60.1.278. S2CID 102174323.
3. Klarreich, Erica (5 March 2013). "How to Make Impossible Wallpaper". Quanta Magazine. Retrieved 2021-04-07.
4. Radaelli, Paulo G. Symmetry in Crystallography. Oxford University Press.
5. If one thinks of the squares as the background, then one can see a simple patterns of rows of rhombuses.
References
• The Grammar of Ornament (1856), by Owen Jones. Many of the images in this article are from this book; it contains many more.
• John H. Conway (1992). "The Orbifold Notation for Surface Groups". In: M. W. Liebeck and J. Saxl (eds.), Groups, Combinatorics and Geometry, Proceedings of the L.M.S. Durham Symposium, July 5–15, Durham, UK, 1990; London Math. Soc. Lecture Notes Series 165. Cambridge University Press, Cambridge. pp. 438–447
• John H. Conway, Heidi Burgiel and Chaim Goodman-Strauss (2008): The Symmetries of Things. Worcester MA: A.K. Peters. ISBN 1-56881-220-5.
• Branko Grünbaum and G. C. Shephard (1987): Tilings and Patterns. New York: Freeman. ISBN 0-7167-1193-1.
• Pattern Design, Lewis F. Day
External links
• International Tables for Crystallography Volume A: Space-group symmetry by the International Union of Crystallography
• The 17 plane symmetry groups by David E. Joyce
• Introduction to wallpaper patterns by Chaim Goodman-Strauss and Heidi Burgiel
• Description by Silvio Levy
• Example tiling for each group, with dynamic demos of properties
• Overview with example tiling for each group, by Brian Sanderson
• Escher Web Sketch, a java applet with interactive tools for drawing in all 17 plane symmetry groups
• Burak, a Java applet for drawing symmetry groups. Archived 2009-02-18 at the Wayback Machine
• A JavaScript app for drawing wallpaper patterns
• Circle-Pattern on Roman Mosaics in Greece
• Seventeen Kinds of Wallpaper Patterns Archived 2017-10-12 at the Wayback Machine the 17 symmetries found in traditional Japanese patterns.
• Baloglou, George (2002). "An elementary, purely geometrical classification of the 17 planar crystallographic groups (wallpaper patterns)". Archived from the original on 2018-08-07. Retrieved 2018-07-22.
Tessellation
Periodic
• Pythagorean
• Rhombille
• Schwarz triangle
• Rectangle
• Domino
• Uniform tiling and honeycomb
• Coloring
• Convex
• Kisrhombille
• Wallpaper group
• Wythoff
Aperiodic
• Ammann–Beenker
• Aperiodic set of prototiles
• List
• Einstein problem
• Socolar–Taylor
• Gilbert
• Penrose
• Pentagonal
• Pinwheel
• Quaquaversal
• Rep-tile and Self-tiling
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• Truchet
Other
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• Circle Limit III
• Computer graphics
• Honeycomb
• Isotoxal
• List
• Packing
• Problems
• Domino
• Wang
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• Squaring
• Dividing a square into similar rectangles
• Prototile
• Conway criterion
• Girih
• Regular Division of the Plane
• Regular grid
• Substitution
• Voronoi
• Voderberg
By vertex type
Spherical
• 2n
• 33.n
• V33.n
• 42.n
• V42.n
Regular
• 2∞
• 36
• 44
• 63
Semi-
regular
• 32.4.3.4
• V32.4.3.4
• 33.42
• 33.∞
• 34.6
• V34.6
• 3.4.6.4
• (3.6)2
• 3.122
• 42.∞
• 4.6.12
• 4.82
Hyper-
bolic
• 32.4.3.5
• 32.4.3.6
• 32.4.3.7
• 32.4.3.8
• 32.4.3.∞
• 32.5.3.5
• 32.5.3.6
• 32.6.3.6
• 32.6.3.8
• 32.7.3.7
• 32.8.3.8
• 33.4.3.4
• 32.∞.3.∞
• 34.7
• 34.8
• 34.∞
• 35.4
• 37
• 38
• 3∞
• (3.4)3
• (3.4)4
• 3.4.62.4
• 3.4.7.4
• 3.4.8.4
• 3.4.∞.4
• 3.6.4.6
• (3.7)2
• (3.8)2
• 3.142
• 3.162
• (3.∞)2
• 3.∞2
• 42.5.4
• 42.6.4
• 42.7.4
• 42.8.4
• 42.∞.4
• 45
• 46
• 47
• 48
• 4∞
• (4.5)2
• (4.6)2
• 4.6.12
• 4.6.14
• V4.6.14
• 4.6.16
• V4.6.16
• 4.6.∞
• (4.7)2
• (4.8)2
• 4.8.10
• V4.8.10
• 4.8.12
• 4.8.14
• 4.8.16
• 4.8.∞
• 4.102
• 4.10.12
• 4.122
• 4.12.16
• 4.142
• 4.162
• 4.∞2
• (4.∞)2
• 54
• 55
• 56
• 5∞
• 5.4.6.4
• (5.6)2
• 5.82
• 5.102
• 5.122
• (5.∞)2
• 64
• 65
• 66
• 68
• 6.4.8.4
• (6.8)2
• 6.82
• 6.102
• 6.122
• 6.162
• 73
• 74
• 77
• 7.62
• 7.82
• 7.142
• 83
• 84
• 86
• 88
• 8.62
• 8.122
• 8.162
• ∞3
• ∞4
• ∞5
• ∞∞
• ∞.62
• ∞.82
Mathematics and art
Concepts
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Artists
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19th–20th
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• William Blake
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• Danseuse au café
• L'Oiseau bleu
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Theorists
Ancient
• Polykleitos
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Renaissance
• Filippo Brunelleschi
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• I quattro libri dell'architettura
Romantic
• Samuel Colman
• Nature's Harmonic Unity
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• Ad Quadratum
• Jay Hambidge
• The Greek Vase
Modern
• Owen Jones
• The Grammar of Ornament
• Ernest Hanbury Hankin
• The Drawing of Geometric Patterns in Saracenic Art
• G. H. Hardy
• A Mathematician's Apology
• George David Birkhoff
• Aesthetic Measure
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• The 'Life' of a Carpet
Publications
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Organizations
• Ars Mathematica
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• Institute For Figuring
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Related
• Droste effect
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• Sacred geometry
• Category
| Wikipedia |
693 (number)
693 (six hundred [and] ninety-three) is the natural number following 692 and preceding 694.
← 692 693 694 →
• List of numbers
• Integers
← 0 100 200 300 400 500 600 700 800 900 →
Cardinalsix hundred ninety-three
Ordinal693rd
(six hundred ninety-third)
Factorization32 × 7 × 11
Greek numeralΧϞΓ´
Roman numeralDCXCIII
Binary10101101012
Ternary2212003
Senary31136
Octal12658
Duodecimal49912
Hexadecimal2B516
In mathematics
693 has twelve divisors: 1, 3, 7, 9, 11, 21, 33, 63, 77, 99, 231, and 693. Thus, 693 is tied with 315 for the highest number of divisors for any odd natural number below 900. The smallest positive odd integer with more divisors is 945, which has 16 divisors. Consequently, 945 is also the smallest odd abundant number, having an abundancy index of 1920/945 ≈ 2.03175.
693 appears as the first three digits after the decimal point in the decimal form for the natural logarithm of 2. To 10 digits, this number is 0.6931471805. As a result, if an event has a constant probability of 0.1% of occurring, 693 is the smallest number of trials that must be performed for there to be at least a 50% chance that the event occurs at least once. More generally, for any probability p, the probability that the event occurs at least once in a sample of n items, assuming the items are independent, is given by the following formula:
1 − (1 − p)n
For p = 10−3 = 0.001, plugging in n = 692 gives, to four decimal places, 0.4996, while n = 693 yields 0.5001.
693 is the lowest common multiple of 7, 9, and 11. Multiplying 693 by 5 gives 3465, the smallest positive integer divisible by 3, 5, 7, 9, and 11.[1]
693 is a palindrome in bases 32, 62, 76, 98, 230, and 692. It is also a palindrome in binary: 1010110101.
The reciprocal of 693 has a period of six: 1/693 = 0.001443.
693 is a triangular matchstick number.[2]
References
1. "Least common multiple of 1,3,5,...,2n-1". OEIS Foundation.
2. Sloane, N. J. A. (ed.). "Sequence A045943". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-05-31.
| Wikipedia |
63 knot
In knot theory, the 63 knot is one of three prime knots with crossing number six, the others being the stevedore knot and the 62 knot. It is alternating, hyperbolic, and fully amphichiral. It can be written as the braid word
$\sigma _{1}^{-1}\sigma _{2}^{2}\sigma _{1}^{-2}\sigma _{2}.\,$[1]
63 knot
Arf invariant1
Braid length6
Braid no.3
Bridge no.2
Crosscap no.3
Crossing no.6
Genus2
Hyperbolic volume5.69302
Stick no.8
Unknotting no.1
Conway notation[2112]
A–B notation63
Dowker notation4, 8, 10, 2, 12, 6
Last /Next62 / 71
Other
alternating, hyperbolic, fibered, prime, fully amphichiral
Symmetry
Like the figure-eight knot, the 63 knot is fully amphichiral. This means that the 63 knot is amphichiral,[2] meaning that it is indistinguishable from its own mirror image. In addition, it is also invertible, meaning that orienting the curve in either direction yields the same oriented knot.
Invariants
The Alexander polynomial of the 63 knot is
$\Delta (t)=t^{2}-3t+5-3t^{-1}+t^{-2},\,$
Conway polynomial is
$\nabla (z)=z^{4}+z^{2}+1,\,$
Jones polynomial is
$V(q)=-q^{3}+2q^{2}-2q+3-2q^{-1}+2q^{-2}-q^{-3},\,$
and the Kauffman polynomial is
$L(a,z)=az^{5}+z^{5}a^{-1}+2a^{2}z^{4}+2z^{4}a^{-2}+4z^{4}+a^{3}z^{3}+az^{3}+z^{3}a^{-1}+z^{3}a^{-3}-3a^{2}z^{2}-3z^{2}a^{-2}-6z^{2}-a^{3}z-2az-2za^{-1}-za^{}-3+a^{2}+a^{-2}+3.\,$ [3]
The 63 knot is a hyperbolic knot, with its complement having a volume of approximately 5.69302.
References
1. "6_3 knot - Wolfram|Alpha".
2. Weisstein, Eric W. "Amphichiral Knot". MathWorld. Accessed: May 12, 2014.
3. "6_3", The Knot Atlas.
Knot theory (knots and links)
Hyperbolic
• Figure-eight (41)
• Three-twist (52)
• Stevedore (61)
• 62
• 63
• Endless (74)
• Carrick mat (818)
• Perko pair (10161)
• (−2,3,7) pretzel (12n242)
• Whitehead (52
1
)
• Borromean rings (63
2
)
• L10a140
• Conway knot (11n34)
Satellite
• Composite knots
• Granny
• Square
• Knot sum
Torus
• Unknot (01)
• Trefoil (31)
• Cinquefoil (51)
• Septafoil (71)
• Unlink (02
1
)
• Hopf (22
1
)
• Solomon's (42
1
)
Invariants
• Alternating
• Arf invariant
• Bridge no.
• 2-bridge
• Brunnian
• Chirality
• Invertible
• Crosscap no.
• Crossing no.
• Finite type invariant
• Hyperbolic volume
• Khovanov homology
• Genus
• Knot group
• Link group
• Linking no.
• Polynomial
• Alexander
• Bracket
• HOMFLY
• Jones
• Kauffman
• Pretzel
• Prime
• list
• Stick no.
• Tricolorability
• Unknotting no. and problem
Notation
and operations
• Alexander–Briggs notation
• Conway notation
• Dowker–Thistlethwaite notation
• Flype
• Mutation
• Reidemeister move
• Skein relation
• Tabulation
Other
• Alexander's theorem
• Berge
• Braid theory
• Conway sphere
• Complement
• Double torus
• Fibered
• Knot
• List of knots and links
• Ribbon
• Slice
• Sum
• Tait conjectures
• Twist
• Wild
• Writhe
• Surgery theory
• Category
• Commons
| Wikipedia |
6-j symbol
Wigner's 6-j symbols were introduced by Eugene Paul Wigner in 1940 and published in 1965. They are defined as a sum over products of four Wigner 3-j symbols,
${\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\j_{4}&j_{5}&j_{6}\end{Bmatrix}}=\sum _{m_{1},\dots ,m_{6}}(-1)^{\sum _{k=1}^{6}(j_{k}-m_{k})}{\begin{pmatrix}j_{1}&j_{2}&j_{3}\\-m_{1}&-m_{2}&-m_{3}\end{pmatrix}}{\begin{pmatrix}j_{1}&j_{5}&j_{6}\\m_{1}&-m_{5}&m_{6}\end{pmatrix}}{\begin{pmatrix}j_{4}&j_{2}&j_{6}\\m_{4}&m_{2}&-m_{6}\end{pmatrix}}{\begin{pmatrix}j_{4}&j_{5}&j_{3}\\-m_{4}&m_{5}&m_{3}\end{pmatrix}}.$
The summation is over all six mi allowed by the selection rules of the 3-j symbols.
They are closely related to the Racah W-coefficients, which are used for recoupling 3 angular momenta, although Wigner 6-j symbols have higher symmetry and therefore provide a more efficient means of storing the recoupling coefficients.[1] Their relationship is given by:
${\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\j_{4}&j_{5}&j_{6}\end{Bmatrix}}=(-1)^{j_{1}+j_{2}+j_{4}+j_{5}}W(j_{1}j_{2}j_{5}j_{4};j_{3}j_{6}).$
Symmetry relations
The 6-j symbol is invariant under any permutation of the columns:
${\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\j_{4}&j_{5}&j_{6}\end{Bmatrix}}={\begin{Bmatrix}j_{2}&j_{1}&j_{3}\\j_{5}&j_{4}&j_{6}\end{Bmatrix}}={\begin{Bmatrix}j_{1}&j_{3}&j_{2}\\j_{4}&j_{6}&j_{5}\end{Bmatrix}}={\begin{Bmatrix}j_{3}&j_{2}&j_{1}\\j_{6}&j_{5}&j_{4}\end{Bmatrix}}=\cdots $
The 6-j symbol is also invariant if upper and lower arguments are interchanged in any two columns:
${\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\j_{4}&j_{5}&j_{6}\end{Bmatrix}}={\begin{Bmatrix}j_{4}&j_{5}&j_{3}\\j_{1}&j_{2}&j_{6}\end{Bmatrix}}={\begin{Bmatrix}j_{1}&j_{5}&j_{6}\\j_{4}&j_{2}&j_{3}\end{Bmatrix}}={\begin{Bmatrix}j_{4}&j_{2}&j_{6}\\j_{1}&j_{5}&j_{3}\end{Bmatrix}}.$
These equations reflect the 24 symmetry operations of the automorphism group that leave the associated tetrahedral Yutsis graph with 6 edges invariant: mirror operations that exchange two vertices and a swap an adjacent pair of edges.
The 6-j symbol
${\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\j_{4}&j_{5}&j_{6}\end{Bmatrix}}$
is zero unless j1, j2, and j3 satisfy triangle conditions, i.e.,
$j_{1}=|j_{2}-j_{3}|,\ldots ,j_{2}+j_{3}$
In combination with the symmetry relation for interchanging upper and lower arguments this shows that triangle conditions must also be satisfied for the triads (j1, j5, j6), (j4, j2, j6), and (j4, j5, j3). Furthermore, the sum of each of the elements of a triad must be an integer. Therefore, the members of each triad are either all integers or contain one integer and two half-integers.
Special case
When j6 = 0 the expression for the 6-j symbol is:
${\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\j_{4}&j_{5}&0\end{Bmatrix}}={\frac {\delta _{j_{2},j_{4}}\delta _{j_{1},j_{5}}}{\sqrt {(2j_{1}+1)(2j_{2}+1)}}}(-1)^{j_{1}+j_{2}+j_{3}}{\begin{Bmatrix}j_{1}&j_{2}&j_{3}\end{Bmatrix}}.$
The triangular delta {j1 j2 j3} is equal to 1 when the triad (j1, j2, j3) satisfies the triangle conditions, and zero otherwise. The symmetry relations can be used to find the expression when another j is equal to zero.
Orthogonality relation
The 6-j symbols satisfy this orthogonality relation:
$\sum _{j_{3}}(2j_{3}+1){\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\j_{4}&j_{5}&j_{6}\end{Bmatrix}}{\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\j_{4}&j_{5}&j_{6}'\end{Bmatrix}}={\frac {\delta _{j_{6}^{}j_{6}'}}{2j_{6}+1}}{\begin{Bmatrix}j_{1}&j_{5}&j_{6}\end{Bmatrix}}{\begin{Bmatrix}j_{4}&j_{2}&j_{6}\end{Bmatrix}}.$
Asymptotics
A remarkable formula for the asymptotic behavior of the 6-j symbol was first conjectured by Ponzano and Regge[2] and later proven by Roberts.[3] The asymptotic formula applies when all six quantum numbers j1, ..., j6 are taken to be large and associates to the 6-j symbol the geometry of a tetrahedron. If the 6-j symbol is determined by the quantum numbers j1, ..., j6 the associated tetrahedron has edge lengths Ji = ji+1/2 (i=1,...,6) and the asymptotic formula is given by,
${\begin{Bmatrix}j_{1}&j_{2}&j_{3}\\j_{4}&j_{5}&j_{6}\end{Bmatrix}}\sim {\frac {1}{\sqrt {12\pi |V|}}}\cos {\left(\sum _{i=1}^{6}J_{i}\theta _{i}+{\frac {\pi }{4}}\right)}.$
The notation is as follows: Each θi is the external dihedral angle about the edge Ji of the associated tetrahedron and the amplitude factor is expressed in terms of the volume, V, of this tetrahedron.
Mathematical interpretation
In representation theory, 6-j symbols are matrix coefficients of the associator isomorphism in a tensor category.[4] For example, if we are given three representations Vi, Vj, Vk of a group (or quantum group), one has a natural isomorphism
$(V_{i}\otimes V_{j})\otimes V_{k}\to V_{i}\otimes (V_{j}\otimes V_{k})$
of tensor product representations, induced by coassociativity of the corresponding bialgebra. One of the axioms defining a monoidal category is that associators satisfy a pentagon identity, which is equivalent to the Biedenharn-Elliot identity for 6-j symbols.
When a monoidal category is semisimple, we can restrict our attention to irreducible objects, and define multiplicity spaces
$H_{i,j}^{\ell }=\operatorname {Hom} (V_{\ell },V_{i}\otimes V_{j})$
so that tensor products are decomposed as:
$V_{i}\otimes V_{j}=\bigoplus _{\ell }H_{i,j}^{\ell }\otimes V_{\ell }$
where the sum is over all isomorphism classes of irreducible objects. Then:
$(V_{i}\otimes V_{j})\otimes V_{k}\cong \bigoplus _{\ell ,m}H_{i,j}^{\ell }\otimes H_{\ell ,k}^{m}\otimes V_{m}\qquad {\text{while}}\qquad V_{i}\otimes (V_{j}\otimes V_{k})\cong \bigoplus _{m,n}H_{i,n}^{m}\otimes H_{j,k}^{n}\otimes V_{m}$
The associativity isomorphism induces a vector space isomorphism
$\Phi _{i,j}^{k,m}:\bigoplus _{\ell }H_{i,j}^{\ell }\otimes H_{\ell ,k}^{m}\to \bigoplus _{n}H_{i,n}^{m}\otimes H_{j,k}^{n}$
and the 6j symbols are defined as the component maps:
${\begin{Bmatrix}i&j&\ell \\k&m&n\end{Bmatrix}}=(\Phi _{i,j}^{k,m})_{\ell ,n}$
When the multiplicity spaces have canonical basis elements and dimension at most one (as in the case of SU(2) in the traditional setting), these component maps can be interpreted as numbers, and the 6-j symbols become ordinary matrix coefficients.
In abstract terms, the 6-j symbols are precisely the information that is lost when passing from a semisimple monoidal category to its Grothendieck ring, since one can reconstruct a monoidal structure using the associator. For the case of representations of a finite group, it is well known that the character table alone (which determines the underlying abelian category and the Grothendieck ring structure) does not determine a group up to isomorphism, while the symmetric monoidal category structure does, by Tannaka-Krein duality. In particular, the two nonabelian groups of order 8 have equivalent abelian categories of representations and isomorphic Grothdendieck rings, but the 6-j symbols of their representation categories are distinct, meaning their representation categories are inequivalent as monoidal categories. Thus, the 6-j symbols give an intermediate level of information, that in fact uniquely determines the groups in many cases, such as when the group is odd order or simple.[5]
See also
• Clebsch–Gordan coefficients
• 3-j symbol
• Racah W-coefficient
• 9-j symbol
Notes
1. Rasch, J.; Yu, A. C. H. (2003). "Efficient Storage Scheme for Pre-calculated Wigner 3j, 6j and Gaunt Coefficients". SIAM J. Sci. Comput. 25 (4): 1416–1428. doi:10.1137/s1064827503422932.
2. Ponzano, G.; Regge, T. (1968). "Semiclassical Limit of Racah Coefficients". Spectroscopy and Group Theoretical Methods in Physics. Elsevier. pp. 1–58. ISBN 978-0-444-10147-1.
3. Roberts J (1999). "Classical 6j-symbols and the tetrahedron". Geometry and Topology. 3: 21–66. arXiv:math-ph/9812013. doi:10.2140/gt.1999.3.21. S2CID 9678271.
4. Etingof, P.; Gelaki, S.; Nikshych, D.; Ostrik, V. (2009). Tensor Categories. Lecture notes for MIT 18.769 (PDF).
5. Etingof, P.; Gelaki, S. (2001). "Isocategorical Groups". International Mathematics Research Notices. 2001 (2): 59–76. arXiv:math/0007196. CiteSeerX 10.1.1.239.6293. doi:10.1155/S1073792801000046.
References
• Biedenharn, L. C.; van Dam, H. (1965). Quantum Theory of Angular Momentum: A collection of Reprints and Original Papers. Academic Press. ISBN 0-12-096056-7.
• Edmonds, A. R. (1957). Angular Momentum in Quantum Mechanics. Princeton University Press. ISBN 0-691-07912-9.
• Condon, Edward U.; Shortley, G. H. (1970). "3. Angular Momentum". The Theory of Atomic Spectra. Cambridge University Press. ISBN 0-521-09209-4.
• Maximon, Leonard C. (2010), "3j,6j,9j Symbols", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248.
• Messiah, Albert (1981). Quantum Mechanics. Vol. II (12th ed.). North Holland Publishing. ISBN 0-7204-0045-7.
• Brink, D. M.; Satchler, G. R. (1993). "2. Representations of the Rotation Group". Angular Momentum (3rd ed.). Clarendon Press. ISBN 0-19-851759-9.
• Zare, Richard N. (1988). "2. Coupling of two Angular Momentum Vectors". Angular Momentum. Wiley. ISBN 0-471-85892-7.
• Biedenharn, L. C.; Louck, J. D. (1981). Angular Momentum in Quantum Physics. Addison-Wesley. ISBN 0-201-13507-8.
External links
• Regge, T. (1959). "Simmetry Properties of Racah's Coefficients". Nuovo Cimento. 11 (1): 116–7. Bibcode:1959NCim...11..116R. doi:10.1007/BF02724914. S2CID 121333785.
• Stone, Anthony. "Wigner coefficient calculator". (Gives exact answer)
• Simons, Frederik J. "Matlab software archive, the code SIXJ.M".
• Volya, A. "Clebsch-Gordan, 3-j and 6-j Coefficient Web Calculator". Archived from the original on 2012-12-20.
• Plasma Laboratory of Weizmann Institute of Science. "369j-symbol calculator".
• GNU scientific library. "Coupling coefficients".
• Johansson, H.T.; Forssén, C. "(WIGXJPF)". (accurate; C, fortran, python)
• Johansson, H.T. "(FASTWIGXJ)". (fast lookup, accurate; C, fortran)
| Wikipedia |
Stevedore knot (mathematics)
In knot theory, the stevedore knot is one of three prime knots with crossing number six, the others being the 62 knot and the 63 knot. The stevedore knot is listed as the 61 knot in the Alexander–Briggs notation, and it can also be described as a twist knot with four half twists, or as the (5,−1,−1) pretzel knot.
Stevedore knot
Common nameStevedore knot
Arf invariant0
Braid length7
Braid no.4
Bridge no.2
Crosscap no.2
Crossing no.6
Genus1
Hyperbolic volume3.16396
Stick no.8
Unknotting no.1
Conway notation[42]
A–B notation61
Dowker notation4, 8, 12, 10, 2, 6
Last /Next52 / 62
Other
alternating, hyperbolic, pretzel, prime, slice, reversible, twist
The mathematical stevedore knot is named after the common stevedore knot, which is often used as a stopper at the end of a rope. The mathematical version of the knot can be obtained from the common version by joining together the two loose ends of the rope, forming a knotted loop.
The stevedore knot is invertible but not amphichiral. Its Alexander polynomial is
$\Delta (t)=-2t+5-2t^{-1},\,$
its Conway polynomial is
$\nabla (z)=1-2z^{2},\,$
and its Jones polynomial is
$V(q)=q^{2}-q+2-2q^{-1}+q^{-2}-q^{-3}+q^{-4}.\,$[1]
The Alexander polynomial and Conway polynomial are the same as those for the knot 946, but the Jones polynomials for these two knots are different.[2] Because the Alexander polynomial is not monic, the stevedore knot is not fibered.
The stevedore knot is a ribbon knot, and is therefore also a slice knot.
The stevedore knot is a hyperbolic knot, with its complement having a volume of approximately 3.16396.
See also
• Figure-eight knot (mathematics)
References
1. "6_1", The Knot Atlas.
2. Weisstein, Eric W. "Stevedore's Knot". MathWorld.
Knot theory (knots and links)
Hyperbolic
• Figure-eight (41)
• Three-twist (52)
• Stevedore (61)
• 62
• 63
• Endless (74)
• Carrick mat (818)
• Perko pair (10161)
• (−2,3,7) pretzel (12n242)
• Whitehead (52
1
)
• Borromean rings (63
2
)
• L10a140
• Conway knot (11n34)
Satellite
• Composite knots
• Granny
• Square
• Knot sum
Torus
• Unknot (01)
• Trefoil (31)
• Cinquefoil (51)
• Septafoil (71)
• Unlink (02
1
)
• Hopf (22
1
)
• Solomon's (42
1
)
Invariants
• Alternating
• Arf invariant
• Bridge no.
• 2-bridge
• Brunnian
• Chirality
• Invertible
• Crosscap no.
• Crossing no.
• Finite type invariant
• Hyperbolic volume
• Khovanov homology
• Genus
• Knot group
• Link group
• Linking no.
• Polynomial
• Alexander
• Bracket
• HOMFLY
• Jones
• Kauffman
• Pretzel
• Prime
• list
• Stick no.
• Tricolorability
• Unknotting no. and problem
Notation
and operations
• Alexander–Briggs notation
• Conway notation
• Dowker–Thistlethwaite notation
• Flype
• Mutation
• Reidemeister move
• Skein relation
• Tabulation
Other
• Alexander's theorem
• Berge
• Braid theory
• Conway sphere
• Complement
• Double torus
• Fibered
• Knot
• List of knots and links
• Ribbon
• Slice
• Sum
• Tait conjectures
• Twist
• Wild
• Writhe
• Surgery theory
• Category
• Commons
| Wikipedia |
62 knot
In knot theory, the 62 knot is one of three prime knots with crossing number six, the others being the stevedore knot and the 63 knot. This knot is sometimes referred to as the Miller Institute knot,[1] because it appears in the logo[2] of the Miller Institute for Basic Research in Science at the University of California, Berkeley.
62 knot
Arf invariant1
Braid length6
Braid no.3
Bridge no.2
Crosscap no.2
Crossing no.6
Genus2
Hyperbolic volume4.40083
Stick no.8
Unknotting no.1
Conway notation[312]
A–B notation62
Dowker notation4, 8, 10, 12, 2, 6
Last /Next61 / 63
Other
alternating, hyperbolic, fibered, prime, reversible
The 62 knot is invertible but not amphichiral. Its Alexander polynomial is
$\Delta (t)=-t^{2}+3t-3+3t^{-1}-t^{-2},\,$
its Conway polynomial is
$\nabla (z)=-z^{4}-z^{2}+1,\,$
and its Jones polynomial is
$V(q)=q-1+2q^{-1}-2q^{-2}+2q^{-3}-2q^{-4}+q^{-5}.\,$[3]
The 62 knot is a hyperbolic knot, with its complement having a volume of approximately 4.40083.
Surface
• Surface of knot 6.2
Example
Ways to assemble of knot 6.2
• Example 1
• Example 2
If a bowline is tied and the two free ends of the rope are brought together in the simplest way, the knot obtained is the 62 knot. The sequence of necessary moves are depicted here:
• From a bowline (ends connected) to the 6₂ knot.
References
1. Weisstein, Eric W. "Miller Institute Knot". MathWorld.
2. Miller Institute - Home Page
3. "6_2", The Knot Atlas.
Knot theory (knots and links)
Hyperbolic
• Figure-eight (41)
• Three-twist (52)
• Stevedore (61)
• 62
• 63
• Endless (74)
• Carrick mat (818)
• Perko pair (10161)
• (−2,3,7) pretzel (12n242)
• Whitehead (52
1
)
• Borromean rings (63
2
)
• L10a140
• Conway knot (11n34)
Satellite
• Composite knots
• Granny
• Square
• Knot sum
Torus
• Unknot (01)
• Trefoil (31)
• Cinquefoil (51)
• Septafoil (71)
• Unlink (02
1
)
• Hopf (22
1
)
• Solomon's (42
1
)
Invariants
• Alternating
• Arf invariant
• Bridge no.
• 2-bridge
• Brunnian
• Chirality
• Invertible
• Crosscap no.
• Crossing no.
• Finite type invariant
• Hyperbolic volume
• Khovanov homology
• Genus
• Knot group
• Link group
• Linking no.
• Polynomial
• Alexander
• Bracket
• HOMFLY
• Jones
• Kauffman
• Pretzel
• Prime
• list
• Stick no.
• Tricolorability
• Unknotting no. and problem
Notation
and operations
• Alexander–Briggs notation
• Conway notation
• Dowker–Thistlethwaite notation
• Flype
• Mutation
• Reidemeister move
• Skein relation
• Tabulation
Other
• Alexander's theorem
• Berge
• Braid theory
• Conway sphere
• Complement
• Double torus
• Fibered
• Knot
• List of knots and links
• Ribbon
• Slice
• Sum
• Tait conjectures
• Twist
• Wild
• Writhe
• Surgery theory
• Category
• Commons
| Wikipedia |
7-cubic honeycomb
The 7-cubic honeycomb or hepteractic honeycomb is the only regular space-filling tessellation (or honeycomb) in Euclidean 7-space.
7-cubic honeycomb
(no image)
TypeRegular 7-honeycomb
Uniform 7-honeycomb
FamilyHypercube honeycomb
Schläfli symbol{4,35,4}
{4,34,31,1}
{∞}(7)
Coxeter-Dynkin diagrams
7-face type{4,3,3,3,3,3}
6-face type{4,3,3,3,3}
5-face type{4,3,3,3}
4-face type{4,3,3}
Cell type{4,3}
Face type{4}
Face figure{4,3}
(octahedron)
Edge figure8 {4,3,3}
(16-cell)
Vertex figure128 {4,35}
(7-orthoplex)
Coxeter group[4,35,4]
Dualself-dual
Propertiesvertex-transitive, edge-transitive, face-transitive, cell-transitive
It is analogous to the square tiling of the plane and to the cubic honeycomb of 3-space.
There are many different Wythoff constructions of this honeycomb. The most symmetric form is regular, with Schläfli symbol {4,35,4}. Another form has two alternating 7-cube facets (like a checkerboard) with Schläfli symbol {4,34,31,1}. The lowest symmetry Wythoff construction has 128 types of facets around each vertex and a prismatic product Schläfli symbol {∞}(7).
Related honeycombs
The [4,35,4], , Coxeter group generates 255 permutations of uniform tessellations, 135 with unique symmetry and 134 with unique geometry. The expanded 7-cubic honeycomb is geometrically identical to the 7-cubic honeycomb.
The 7-cubic honeycomb can be alternated into the 7-demicubic honeycomb, replacing the 7-cubes with 7-demicubes, and the alternated gaps are filled by 7-orthoplex facets.
Quadritruncated 7-cubic honeycomb
A quadritruncated 7-cubic honeycomb, , contains all tritruncated 7-orthoplex facets and is the Voronoi tessellation of the D7* lattice. Facets can be identically colored from a doubled ${\tilde {C}}_{7}$×2, [[4,35,4]] symmetry, alternately colored from ${\tilde {C}}_{7}$, [4,35,4] symmetry, three colors from ${\tilde {B}}_{7}$, [4,34,31,1] symmetry, and 4 colors from ${\tilde {D}}_{7}$, [31,1,33,31,1] symmetry.
See also
• List of regular polytopes
References
• Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8 p. 296, Table II: Regular honeycombs
• Kaleidoscopes: Selected Writings of H. S. M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6
• (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
Fundamental convex regular and uniform honeycombs in dimensions 2–9
Space Family ${\tilde {A}}_{n-1}$ ${\tilde {C}}_{n-1}$ ${\tilde {B}}_{n-1}$ ${\tilde {D}}_{n-1}$ ${\tilde {G}}_{2}$ / ${\tilde {F}}_{4}$ / ${\tilde {E}}_{n-1}$
E2 Uniform tiling {3[3]} δ3 hδ3 qδ3 Hexagonal
E3 Uniform convex honeycomb {3[4]} δ4 hδ4 qδ4
E4 Uniform 4-honeycomb {3[5]} δ5 hδ5 qδ5 24-cell honeycomb
E5 Uniform 5-honeycomb {3[6]} δ6 hδ6 qδ6
E6 Uniform 6-honeycomb {3[7]} δ7 hδ7 qδ7 222
E7 Uniform 7-honeycomb {3[8]} δ8 hδ8 qδ8 133 • 331
E8 Uniform 8-honeycomb {3[9]} δ9 hδ9 qδ9 152 • 251 • 521
E9 Uniform 9-honeycomb {3[10]} δ10 hδ10 qδ10
E10 Uniform 10-honeycomb {3[11]} δ11 hδ11 qδ11
En-1 Uniform (n-1)-honeycomb {3[n]} δn hδn qδn 1k2 • 2k1 • k21
| Wikipedia |
7-demicube
In geometry, a demihepteract or 7-demicube is a uniform 7-polytope, constructed from the 7-hypercube (hepteract) with alternated vertices removed. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes.
Demihepteract
(7-demicube)
Petrie polygon projection
Type Uniform 7-polytope
Family demihypercube
Coxeter symbol 141
Schläfli symbol {3,34,1} = h{4,35}
s{21,1,1,1,1,1}
Coxeter diagrams =
6-faces7814 {31,3,1}
64 {35}
5-faces53284 {31,2,1}
448 {34}
4-faces1624280 {31,1,1}
1344 {33}
Cells2800560 {31,0,1}
2240 {3,3}
Faces2240{3}
Edges672
Vertices64
Vertex figure Rectified 6-simplex
Symmetry group D7, [34,1,1] = [1+,4,35]
[26]+
Dual ?
Properties convex
E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as HM7 for a 7-dimensional half measure polytope.
Coxeter named this polytope as 141 from its Coxeter diagram, with a ring on one of the 1-length branches, and Schläfli symbol $\left\{3{\begin{array}{l}3,3,3,3\\3\end{array}}\right\}$ or {3,34,1}.
Cartesian coordinates
Cartesian coordinates for the vertices of a demihepteract centered at the origin are alternate halves of the hepteract:
(±1,±1,±1,±1,±1,±1,±1)
with an odd number of plus signs.
Images
orthographic projections
Coxeter
plane
B7 D7 D6
Graph
Dihedral
symmetry
[14/2] [12] [10]
Coxeter plane D5 D4 D3
Graph
Dihedral
symmetry
[8] [6] [4]
Coxeter
plane
A5 A3
Graph
Dihedral
symmetry
[6] [4]
As a configuration
This configuration matrix represents the 7-demicube. The rows and columns correspond to vertices, edges, faces, cells, 4-faces, 5-faces and 6-faces. The diagonal numbers say how many of each element occur in the whole 7-demicube. The nondiagonal numbers say how many of the column's element occur in or at the row's element.[1][2]
The diagonal f-vector numbers are derived through the Wythoff construction, dividing the full group order of a subgroup order by removing one mirror at a time.[3]
D7k-facefkf0f1f2f3f4f5f6k-figuresnotes
A6( ) f0 64211053514035105214277041D7/A6 = 64*7!/7! = 64
A4A1A1{ } f1 2672105201020101052{ }×{3,3,3}D7/A4A1A1 = 64*7!/5!/2/2 = 672
A3A2100 f2 33224014466441{3,3}v( )D7/A3A2 = 64*7!/4!/3! = 2240
A3A3101 f3 464560*406040{3,3}D7/A3A3 = 64*7!/4!/4! = 560
A3A2110 464*2240133331{3}v( )D7/A3A2 = 64*7!/4!/3! = 2240
D4A2111 f4 8243288280*3030{3}D7/D4A2 = 64*7!/8/4!/2 = 280
A4A1120 5101005*13441221{ }v( )D7/A4A1 = 64*7!/5!/2 = 1344
D5A1121 f5 16801604080101684*20{ }D7/D5A1 = 64*7!/16/5!/2 = 84
A5130 6152001506*44811D7/A5 = 64*7!/6! = 448
D6131 f6 3224064016048060192123214*( )D7/D6 = 64*7!/32/6! = 14
A6140 7213503502107*64D7/A6 = 64*7!/7! = 64
Related polytopes
There are 95 uniform polytopes with D6 symmetry, 63 are shared by the B6 symmetry, and 32 are unique:
D7 polytopes
t0(141)
t0,1(141)
t0,2(141)
t0,3(141)
t0,4(141)
t0,5(141)
t0,1,2(141)
t0,1,3(141)
t0,1,4(141)
t0,1,5(141)
t0,2,3(141)
t0,2,4(141)
t0,2,5(141)
t0,3,4(141)
t0,3,5(141)
t0,4,5(141)
t0,1,2,3(141)
t0,1,2,4(141)
t0,1,2,5(141)
t0,1,3,4(141)
t0,1,3,5(141)
t0,1,4,5(141)
t0,2,3,4(141)
t0,2,3,5(141)
t0,2,4,5(141)
t0,3,4,5(141)
t0,1,2,3,4(141)
t0,1,2,3,5(141)
t0,1,2,4,5(141)
t0,1,3,4,5(141)
t0,2,3,4,5(141)
t0,1,2,3,4,5(141)
References
1. Coxeter, Regular Polytopes, sec 1.8 Configurations
2. Coxeter, Complex Regular Polytopes, p.117
3. Klitzing, Richard. "x3o3o *b3o3o3o - hax".
• H.S.M. Coxeter:
• Coxeter, Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8, p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
• H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973, p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
• Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6
• (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
• (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
• (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 26. pp. 409: Hemicubes: 1n1)
• Klitzing, Richard. "7D uniform polytopes (polyexa) x3o3o *b3o3o3o3o - hesa".
External links
• Olshevsky, George. "Demihepteract". Glossary for Hyperspace. Archived from the original on 4 February 2007.
• Multi-dimensional Glossary
Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron
Uniform polychoron Pentachoron 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell
Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221
Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321
Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421
Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds
| Wikipedia |
7-demicubic honeycomb
The 7-demicubic honeycomb, or demihepteractic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 7-space. It is constructed as an alternation of the regular 7-cubic honeycomb.
7-demicubic honeycomb
(No image)
TypeUniform 7-honeycomb
FamilyAlternated hypercube honeycomb
Schläfli symbolh{4,3,3,3,3,3,4}
h{4,3,3,3,3,31,1}
ht0,7{4,3,3,3,3,3,4}
Coxeter-Dynkin diagram =
=
Facets{3,3,3,3,3,4}
h{4,3,3,3,3,3}
Vertex figureRectified 7-orthoplex
Coxeter group${\tilde {B}}_{7}$ [4,3,3,3,3,31,1]
${\tilde {D}}_{7}$, [31,1,3,3,3,31,1]
It is composed of two different types of facets. The 7-cubes become alternated into 7-demicubes h{4,3,3,3,3,3} and the alternated vertices create 7-orthoplex {3,3,3,3,3,4} facets.
D7 lattice
The vertex arrangement of the 7-demicubic honeycomb is the D7 lattice.[1] The 84 vertices of the rectified 7-orthoplex vertex figure of the 7-demicubic honeycomb reflect the kissing number 84 of this lattice.[2] The best known is 126, from the E7 lattice and the 331 honeycomb.
The D+
7
packing (also called D2
7
) can be constructed by the union of two D7 lattices. The D+
n
packings form lattices only in even dimensions. The kissing number is 26=64 (2n-1 for n<8, 240 for n=8, and 2n(n-1) for n>8).[3]
∪
The D*
7
lattice (also called D4
7
and C2
7
) can be constructed by the union of all four 7-demicubic lattices:[4] It is also the 7-dimensional body centered cubic, the union of two 7-cube honeycombs in dual positions.
∪ ∪ ∪ = ∪ .
The kissing number of the D*
7
lattice is 14 (2n for n≥5) and its Voronoi tessellation is a quadritruncated 7-cubic honeycomb, , containing all with tritruncated 7-orthoplex, Voronoi cells.[5]
Symmetry constructions
There are three uniform construction symmetries of this tessellation. Each symmetry can be represented by arrangements of different colors on the 128 7-demicube facets around each vertex.
Coxeter group Schläfli symbol Coxeter-Dynkin diagram Vertex figure
Symmetry
Facets/verf
${\tilde {B}}_{7}$ = [31,1,3,3,3,3,4]
= [1+,4,3,3,3,3,3,4]
h{4,3,3,3,3,3,4} =
[3,3,3,3,3,4]
128: 7-demicube
14: 7-orthoplex
${\tilde {D}}_{7}$ = [31,1,3,3,31,1]
= [1+,4,3,3,3,31,1]
h{4,3,3,3,3,31,1} =
[35,1,1]
64+64: 7-demicube
14: 7-orthoplex
2×½${\tilde {C}}_{7}$ = [[(4,3,3,3,3,4,2+)]]ht0,7{4,3,3,3,3,3,4} 64+32+32: 7-demicube
14: 7-orthoplex
See also
• 7-cubic honeycomb
References
• Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8
• pp. 154–156: Partial truncation or alternation, represented by h prefix: h{4,4}={4,4}; h{4,3,4}={31,1,4}, h{4,3,3,4}={3,3,4,3}, ...
• Kaleidoscopes: Selected Writings of H. S. M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6
• (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
• Conway JH, Sloane NJH (1998). Sphere Packings, Lattices and Groups (3rd ed.). ISBN 0-387-98585-9.
Notes
1. "The Lattice D7".
2. Sphere packings, lattices, and groups, by John Horton Conway, Neil James Alexander Sloane, Eiichi Bannai
3. Conway (1998), p. 119
4. "The Lattice D7".
5. Conway (1998), p. 466
External links
Fundamental convex regular and uniform honeycombs in dimensions 2–9
Space Family ${\tilde {A}}_{n-1}$ ${\tilde {C}}_{n-1}$ ${\tilde {B}}_{n-1}$ ${\tilde {D}}_{n-1}$ ${\tilde {G}}_{2}$ / ${\tilde {F}}_{4}$ / Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\tilde {E}}_{n-1}
E2 Uniform tiling {3[3]} δ3 hδ3 qδ3 Hexagonal
E3 Uniform convex honeycomb {3[4]} δ4 hδ4 qδ4
E4 Uniform 4-honeycomb {3[5]} δ5 hδ5 qδ5 24-cell honeycomb
E5 Uniform 5-honeycomb {3[6]} δ6 hδ6 qδ6
E6 Uniform 6-honeycomb {3[7]} δ7 hδ7 qδ7 222
E7 Uniform 7-honeycomb {3[8]} δ8 hδ8 qδ8 133 • 331
E8 Uniform 8-honeycomb {3[9]} δ9 hδ9 qδ9 152 • 251 • 521
E9 Uniform 9-honeycomb {3[10]} δ10 hδ10 qδ10
E10 Uniform 10-honeycomb {3[11]} δ11 hδ11 qδ11
En-1 Uniform (n-1)-honeycomb {3[n]} δn hδn qδn 1k2 • 2k1 • k21
| Wikipedia |
7-cube
In geometry, a 7-cube is a seven-dimensional hypercube with 128 vertices, 448 edges, 672 square faces, 560 cubic cells, 280 tesseract 4-faces, 84 penteract 5-faces, and 14 hexeract 6-faces.
7-cube
Hepteract
Orthogonal projection
inside Petrie polygon
The central orange vertex is doubled
TypeRegular 7-polytope
Familyhypercube
Schläfli symbol{4,35}
Coxeter-Dynkin diagrams
6-faces14 {4,34}
5-faces84 {4,33}
4-faces280 {4,3,3}
Cells560 {4,3}
Faces672 {4}
Edges448
Vertices128
Vertex figure6-simplex
Petrie polygontetradecagon
Coxeter groupC7, [35,4]
Dual7-orthoplex
Propertiesconvex, Hanner polytope
It can be named by its Schläfli symbol {4,35}, being composed of 3 6-cubes around each 5-face. It can be called a hepteract, a portmanteau of tesseract (the 4-cube) and hepta for seven (dimensions) in Greek. It can also be called a regular tetradeca-7-tope or tetradecaexon, being a 7 dimensional polytope constructed from 14 regular facets.
Related polytopes
The 7-cube is 7th in a series of hypercube:
Petrie polygon orthographic projections
Line segment Square Cube 4-cube 5-cube 6-cube 7-cube 8-cube
The dual of a 7-cube is called a 7-orthoplex, and is a part of the infinite family of cross-polytopes.
Applying an alternation operation, deleting alternating vertices of the hepteract, creates another uniform polytope, called a demihepteract, (part of an infinite family called demihypercubes), which has 14 demihexeractic and 64 6-simplex 6-faces.
As a configuration
This configuration matrix represents the 7-cube. The rows and columns correspond to vertices, edges, faces, cells, 4-faces, 5-faces and 6-faces. The diagonal numbers say how many of each element occur in the whole 7-cube. The nondiagonal numbers say how many of the column's element occur in or at the row's element.[1][2]
${\begin{bmatrix}{\begin{matrix}128&7&21&35&35&21&7\\2&448&6&15&20&15&6\\4&4&672&5&10&10&5\\8&12&6&560&4&6&4\\16&32&24&8&280&3&3\\32&80&80&40&10&84&2\\64&192&240&160&60&12&14\end{matrix}}\end{bmatrix}}$
Cartesian coordinates
Cartesian coordinates for the vertices of a hepteract centered at the origin and edge length 2 are
(±1,±1,±1,±1,±1,±1,±1)
while the interior of the same consists of all points (x0, x1, x2, x3, x4, x5, x6) with -1 < xi < 1.
Projections
This hypercube graph is an orthogonal projection. This orientation shows columns of vertices positioned a vertex-edge-vertex distance from one vertex on the left to one vertex on the right, and edges attaching adjacent columns of vertices. The number of vertices in each column represents rows in Pascal's triangle, being 1:7:21:35:35:21:7:1.
orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph
Dihedral symmetry [14] [12] [10]
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph
Dihedral symmetry [8] [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]
References
1. Coxeter, Regular Polytopes, sec 1.8 Configurations
2. Coxeter, Complex Regular Polytopes, p.117
• H.S.M. Coxeter:
• Coxeter, Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8, p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
• H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973, p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
• Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6
• (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
• (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
• (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
• Norman Johnson Uniform Polytopes, Manuscript (1991)
• N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. (1966)
• Klitzing, Richard. "7D uniform polytopes (polyexa) o3o3o3o3o3o4x - hept".
External links
• Weisstein, Eric W. "Hypercube". MathWorld.
• Weisstein, Eric W. "Hypercube graph". MathWorld.
• Olshevsky, George. "Measure polytope". Glossary for Hyperspace. Archived from the original on 4 February 2007.
• Multi-dimensional Glossary: hypercube Garrett Jones
• Rotation of 7D-Cube www.4d-screen.de
Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron
Uniform polychoron Pentachoron 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell
Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221
Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321
Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421
Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds
| Wikipedia |
3-7 kisrhombille
In geometry, the 3-7 kisrhombille tiling is a semiregular dual tiling of the hyperbolic plane. It is constructed by congruent right triangles with 4, 6, and 14 triangles meeting at each vertex.
3-7 kisrhombille
TypeDual semiregular hyperbolic tiling
FacesRight triangle
EdgesInfinite
VerticesInfinite
Coxeter diagram
Symmetry group[7,3], (*732)
Rotation group[7,3]+, (732)
Dual polyhedronTruncated triheptagonal tiling
Face configurationV4.6.14
Propertiesface-transitive
Wikimedia Commons has media related to Uniform dual tiling V 4-6-14.
The image shows a Poincaré disk model projection of the hyperbolic plane.
It is labeled V4.6.14 because each right triangle face has three types of vertices: one with 4 triangles, one with 6 triangles, and one with 14 triangles. It is the dual tessellation of the truncated triheptagonal tiling which has one square and one heptagon and one tetrakaidecagon at each vertex.
Naming
The name 3-7 kisrhombille is given by Conway, seeing it as a 3-7 rhombic tiling, divided by a kis operator, adding a center point to each rhombus, and dividing into four triangles.
Symmetry
There are no mirror removal subgroups of [7,3]. The only small index subgroup is the alternation, [7,3]+, (732).
Small index subgroups of [7,3], (*732)
Type Reflectional Rotational
index 1 2
Diagram
Coxeter
(orbifold)
[7,3] =
(*732)
[7,3]+ =
(732)
Related polyhedra and tilings
Three isohedral (regular or quasiregular) tilings can be constructed from this tiling by combining triangles:
Projections centered on different triangle points
Poincaré
disk
model
Center Heptagon Triangle Rhombic
Klein
disk
model
Related
tiling
Heptagonal tiling Triangular tiling Rhombic tiling
Uniform heptagonal/triangular tilings
Symmetry: [7,3], (*732) [7,3]+, (732)
{7,3} t{7,3} r{7,3} t{3,7} {3,7} rr{7,3} tr{7,3} sr{7,3}
Uniform duals
V73 V3.14.14 V3.7.3.7 V6.6.7 V37 V3.4.7.4 V4.6.14 V3.3.3.3.7
It is topologically related to a polyhedra sequence; see discussion. This group is special for having all even number of edges per vertex and form bisecting planes through the polyhedra and infinite lines in the plane, and are the reflection domains for the (2,3,n) triangle groups – for the heptagonal tiling, the important (2,3,7) triangle group.
See also the uniform tilings of the hyperbolic plane with (2,3,7) symmetry.
The kisrhombille tilings can be seen as from the sequence of rhombille tilings, starting with the cube, with faces divided or kissed at the corners by a face central point.
*n32 symmetry mutation of omnitruncated tilings: 4.6.2n
Sym.
*n32
[n,3]
Spherical Euclid. Compact hyperb. Paraco. Noncompact hyperbolic
*232
[2,3]
*332
[3,3]
*432
[4,3]
*532
[5,3]
*632
[6,3]
*732
[7,3]
*832
[8,3]
*∞32
[∞,3]
[12i,3]
[9i,3]
[6i,3]
[3i,3]
Figures
Config. 4.6.4 4.6.6 4.6.8 4.6.10 4.6.12 4.6.14 4.6.16 4.6.∞ 4.6.24i 4.6.18i 4.6.12i 4.6.6i
Duals
Config. V4.6.4 V4.6.6 V4.6.8 V4.6.10 V4.6.12 V4.6.14 V4.6.16 V4.6.∞ V4.6.24i V4.6.18i V4.6.12i V4.6.6i
Just as the (2,3,7) triangle group is a quotient of the modular group (2,3,∞), the associated tiling is the quotient of the modular tiling, as depicted in the video at right.
References
1. Platonic tilings of Riemann surfaces: The Modular Group, Gerard Westendorp
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 19, The Hyperbolic Archimedean Tessellations)
See also
• Hexakis triangular tiling
• Tilings of regular polygons
• List of uniform tilings
• Uniform tilings in hyperbolic plane
Tessellation
Periodic
• Pythagorean
• Rhombille
• Schwarz triangle
• Rectangle
• Domino
• Uniform tiling and honeycomb
• Coloring
• Convex
• Kisrhombille
• Wallpaper group
• Wythoff
Aperiodic
• Ammann–Beenker
• Aperiodic set of prototiles
• List
• Einstein problem
• Socolar–Taylor
• Gilbert
• Penrose
• Pentagonal
• Pinwheel
• Quaquaversal
• Rep-tile and Self-tiling
• Sphinx
• Socolar
• Truchet
Other
• Anisohedral and Isohedral
• Architectonic and catoptric
• Circle Limit III
• Computer graphics
• Honeycomb
• Isotoxal
• List
• Packing
• Problems
• Domino
• Wang
• Heesch's
• Squaring
• Dividing a square into similar rectangles
• Prototile
• Conway criterion
• Girih
• Regular Division of the Plane
• Regular grid
• Substitution
• Voronoi
• Voderberg
By vertex type
Spherical
• 2n
• 33.n
• V33.n
• 42.n
• V42.n
Regular
• 2∞
• 36
• 44
• 63
Semi-
regular
• 32.4.3.4
• V32.4.3.4
• 33.42
• 33.∞
• 34.6
• V34.6
• 3.4.6.4
• (3.6)2
• 3.122
• 42.∞
• 4.6.12
• 4.82
Hyper-
bolic
• 32.4.3.5
• 32.4.3.6
• 32.4.3.7
• 32.4.3.8
• 32.4.3.∞
• 32.5.3.5
• 32.5.3.6
• 32.6.3.6
• 32.6.3.8
• 32.7.3.7
• 32.8.3.8
• 33.4.3.4
• 32.∞.3.∞
• 34.7
• 34.8
• 34.∞
• 35.4
• 37
• 38
• 3∞
• (3.4)3
• (3.4)4
• 3.4.62.4
• 3.4.7.4
• 3.4.8.4
• 3.4.∞.4
• 3.6.4.6
• (3.7)2
• (3.8)2
• 3.142
• 3.162
• (3.∞)2
• 3.∞2
• 42.5.4
• 42.6.4
• 42.7.4
• 42.8.4
• 42.∞.4
• 45
• 46
• 47
• 48
• 4∞
• (4.5)2
• (4.6)2
• 4.6.12
• 4.6.14
• V4.6.14
• 4.6.16
• V4.6.16
• 4.6.∞
• (4.7)2
• (4.8)2
• 4.8.10
• V4.8.10
• 4.8.12
• 4.8.14
• 4.8.16
• 4.8.∞
• 4.102
• 4.10.12
• 4.122
• 4.12.16
• 4.142
• 4.162
• 4.∞2
• (4.∞)2
• 54
• 55
• 56
• 5∞
• 5.4.6.4
• (5.6)2
• 5.82
• 5.102
• 5.122
• (5.∞)2
• 64
• 65
• 66
• 68
• 6.4.8.4
• (6.8)2
• 6.82
• 6.102
• 6.122
• 6.162
• 73
• 74
• 77
• 7.62
• 7.82
• 7.142
• 83
• 84
• 86
• 88
• 8.62
• 8.122
• 8.162
• ∞3
• ∞4
• ∞5
• ∞∞
• ∞.62
• ∞.82
| Wikipedia |
Truncatable prime
In number theory, a left-truncatable prime is a prime number which, in a given base, contains no 0, and if the leading ("left") digit is successively removed, then all resulting numbers are prime. For example, 9137, since 9137, 137, 37 and 7 are all prime. Decimal representation is often assumed and always used in this article.
A right-truncatable prime is a prime which remains prime when the last ("right") digit is successively removed. 7393 is an example of a right-truncatable prime, since 7393, 739, 73, and 7 are all prime.
A left-and-right-truncatable prime is a prime which remains prime if the leading ("left") and last ("right") digits are simultaneously successively removed down to a one- or two-digit prime. 1825711 is an example of a left-and-right-truncatable prime, since 1825711, 82571, 257, and 5 are all prime.
In base 10, there are exactly 4260 left-truncatable primes, 83 right-truncatable primes, and 920,720,315 left-and-right-truncatable primes.
History
An author named Leslie E. Card in early volumes of the Journal of Recreational Mathematics (which started its run in 1968) considered a topic close to that of right-truncatable primes, calling sequences that by adding digits to the right in sequence to an initial number not necessarily prime snowball primes.
Discussion of the topic dates to at least November 1969 issue of Mathematics Magazine, where truncatable primes were called prime primes by two co-authors (Murray Berg and John E. Walstrom).
Decimal truncatable primes
There are 4260 left-truncatable primes:
2, 3, 5, 7, 13, 17, 23, 37, 43, 47, 53, 67, 73, 83, 97, 113, 137, 167, 173, 197, 223, 283, 313, 317, 337, 347, 353, 367, 373, 383, 397, 443, 467, 523, 547, 613, 617, 643, 647, 653, 673, 683, 743, 773, 797, 823, 853, 883, 937, 947, 953, 967, 983, 997, ... (sequence A024785 in the OEIS)
The largest is the 24-digit 357686312646216567629137.
There are 83 right-truncatable primes. The complete list:
2, 3, 5, 7, 23, 29, 31, 37, 53, 59, 71, 73, 79, 233, 239, 293, 311, 313, 317, 373, 379, 593, 599, 719, 733, 739, 797, 2333, 2339, 2393, 2399, 2939, 3119, 3137, 3733, 3739, 3793, 3797, 5939, 7193, 7331, 7333, 7393, 23333, 23339, 23399, 23993, 29399, 31193, 31379, 37337, 37339, 37397, 59393, 59399, 71933, 73331, 73939, 233993, 239933, 293999, 373379, 373393, 593933, 593993, 719333, 739391, 739393, 739397, 739399, 2339933, 2399333, 2939999, 3733799, 5939333, 7393913, 7393931, 7393933, 23399339, 29399999, 37337999, 59393339, 73939133 (sequence A024770 in the OEIS)
The largest is the 8-digit 73939133. All primes above 5 end with digit 1, 3, 7 or 9, so a right-truncatable prime can only contain those digits after the leading digit.
There are 920,720,315 left-and-right-truncatable primes:[1]
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 127, 131, 137, 139, 151, 157, 173, 179, 223, 227, 229, 233, 239, 251, 257, 271, 277, 331, 337, 353, 359, 373, 379, 421, 431, 433, 439, 457, 479, 521, 523, 557, 571, 577, 631, 653, 659, 673, 677, 727, 733, 739, 751, 757, 773, 821, 823, 827, 829, 839, 853, 857, 859, 877, 929, 937, 953, 971, 977, 1117, 1171, 1193, 1231, 1237, 1291, 1297, 1319, 1373, 1433, 1439, 1471, 1531, 1597, 1613, 1619, ... (sequence A077390 in the OEIS)
There are 331,780,864 left-and-right-truncatable primes with an odd number of digits. The largest is the 97-digit prime 7228828176786792552781668926755667258635743361825711373791931117197999133917737137399993737111177.
There are 588,939,451 left-and-right-truncatable primes with an even number of digits. The largest is the 104-digit prime 91617596742869619884432721391145374777686825634291523771171391111313737919133977331737137933773713713973.
There are 15 primes which are both left-truncatable and right-truncatable. They have been called two-sided primes. The complete list:
2, 3, 5, 7, 23, 37, 53, 73, 313, 317, 373, 797, 3137, 3797, 739397 (sequence A020994 in the OEIS)
A left-truncatable prime is called restricted if all of its left extensions are composite i.e. there is no other left-truncatable prime of which this prime is the left-truncated "tail". Thus 7937 is a restricted left-truncatable prime because the nine 5-digit numbers ending in 7937 are all composite, whereas 3797 is a left-truncatable prime that is not restricted because 33797 is also prime.
There are 1442 restricted left-truncatable primes:
2, 5, 773, 3373, 3947, 4643, 5113, 6397, 6967, 7937, 15647, 16823, 24373, 33547, 34337, 37643, 56983, 57853, 59743, 62383, 63347, 63617, 69337, 72467, 72617, 75653, 76367, 87643, 92683, 97883, 98317, ... (sequence A240768 in the OEIS)
Similarly, a right-truncatable prime is called restricted if all of its right extensions are composite. There are 27 restricted right-truncatable primes:
53, 317, 599, 797, 2393, 3793, 3797, 7331, 23333, 23339, 31193, 31379, 37397, 73331, 373393, 593993, 719333, 739397, 739399, 2399333, 7393931, 7393933, 23399339, 29399999, 37337999, 59393339, 73939133 (sequence A239747 in the OEIS)
Other bases
While the primality of a number does not depend on the numeral system used, truncatable primes are defined only in relation with a given base. A variation involves removing 2 or more decimal digits at a time. This is mathematically equivalent to using base 100 or a larger power of 10, with the restriction that base 10n digits must be at least 10n−1, in order to match a decimal n-digit number with no leading 0.
See also
• Permutable prime
References
1. Sloane, N. J. A. (ed.). "Sequence A077390". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
• Weisstein, Eric W. "Truncatable Prime". MathWorld.
• Caldwell, Chris, left-truncatable prime and right-truncatable primes, at the Prime Pages glossary.
• Rivera, Carlos, Problems & Puzzles: Puzzle 2.- Prime strings and Puzzle 131.- Growing primes
External links
• Grime, Dr. James. "357686312646216567629137" (video). YouTube. Brady Haran. Archived from the original on 2021-12-21. Retrieved 27 July 2018.
Prime number classes
By formula
• Fermat (22n + 1)
• Mersenne (2p − 1)
• Double Mersenne (22p−1 − 1)
• Wagstaff (2p + 1)/3
• Proth (k·2n + 1)
• Factorial (n! ± 1)
• Primorial (pn# ± 1)
• Euclid (pn# + 1)
• Pythagorean (4n + 1)
• Pierpont (2m·3n + 1)
• Quartan (x4 + y4)
• Solinas (2m ± 2n ± 1)
• Cullen (n·2n + 1)
• Woodall (n·2n − 1)
• Cuban (x3 − y3)/(x − y)
• Leyland (xy + yx)
• Thabit (3·2n − 1)
• Williams ((b−1)·bn − 1)
• Mills (⌊A3n⌋)
By integer sequence
• Fibonacci
• Lucas
• Pell
• Newman–Shanks–Williams
• Perrin
• Partitions
• Bell
• Motzkin
By property
• Wieferich (pair)
• Wall–Sun–Sun
• Wolstenholme
• Wilson
• Lucky
• Fortunate
• Ramanujan
• Pillai
• Regular
• Strong
• Stern
• Supersingular (elliptic curve)
• Supersingular (moonshine theory)
• Good
• Super
• Higgs
• Highly cototient
• Unique
Base-dependent
• Palindromic
• Emirp
• Repunit (10n − 1)/9
• Permutable
• Circular
• Truncatable
• Minimal
• Delicate
• Primeval
• Full reptend
• Unique
• Happy
• Self
• Smarandache–Wellin
• Strobogrammatic
• Dihedral
• Tetradic
Patterns
• Twin (p, p + 2)
• Bi-twin chain (n ± 1, 2n ± 1, 4n ± 1, …)
• Triplet (p, p + 2 or p + 4, p + 6)
• Quadruplet (p, p + 2, p + 6, p + 8)
• k-tuple
• Cousin (p, p + 4)
• Sexy (p, p + 6)
• Chen
• Sophie Germain/Safe (p, 2p + 1)
• Cunningham (p, 2p ± 1, 4p ± 3, 8p ± 7, ...)
• Arithmetic progression (p + a·n, n = 0, 1, 2, 3, ...)
• Balanced (consecutive p − n, p, p + n)
By size
• Mega (1,000,000+ digits)
• Largest known
• list
Complex numbers
• Eisenstein prime
• Gaussian prime
Composite numbers
• Pseudoprime
• Catalan
• Elliptic
• Euler
• Euler–Jacobi
• Fermat
• Frobenius
• Lucas
• Somer–Lucas
• Strong
• Carmichael number
• Almost prime
• Semiprime
• Sphenic number
• Interprime
• Pernicious
Related topics
• Probable prime
• Industrial-grade prime
• Illegal prime
• Formula for primes
• Prime gap
First 60 primes
• 2
• 3
• 5
• 7
• 11
• 13
• 17
• 19
• 23
• 29
• 31
• 37
• 41
• 43
• 47
• 53
• 59
• 61
• 67
• 71
• 73
• 79
• 83
• 89
• 97
• 101
• 103
• 107
• 109
• 113
• 127
• 131
• 137
• 139
• 149
• 151
• 157
• 163
• 167
• 173
• 179
• 181
• 191
• 193
• 197
• 199
• 211
• 223
• 227
• 229
• 233
• 239
• 241
• 251
• 257
• 263
• 269
• 271
• 277
• 281
List of prime numbers
| Wikipedia |
71 knot
In knot theory, the 71 knot, also known as the septoil knot, the septafoil knot, or the (7, 2)-torus knot, is one of seven prime knots with crossing number seven. It is the simplest torus knot after the trefoil and cinquefoil.
71 knot
Arf invariant0
Braid length7
Braid no.2
Bridge no.2
Crosscap no.1
Crossing no.7
Genus3
Hyperbolic volume0
Stick no.9
Unknotting no.3
Conway notation[7]
A–B notation71
Dowker notation8, 10, 12, 14, 2, 4, 6
Last /Next63 / 72
Other
alternating, torus, fibered, prime, reversible
Properties
The 71 knot is invertible but not amphichiral. Its Alexander polynomial is
$\Delta (t)=t^{3}-t^{2}+t-1+t^{-1}-t^{-2}+t^{-3},\,$
its Conway polynomial is
$\nabla (z)=z^{6}+5z^{4}+6z^{2}+1,\,$
and its Jones polynomial is
$V(q)=q^{-3}+q^{-5}-q^{-6}+q^{-7}-q^{-8}+q^{-9}-q^{-10}.\,$[1]
Example
See also
• Heptagram
References
1. "7_1", The Knot Atlas.
Knot theory (knots and links)
Hyperbolic
• Figure-eight (41)
• Three-twist (52)
• Stevedore (61)
• 62
• 63
• Endless (74)
• Carrick mat (818)
• Perko pair (10161)
• (−2,3,7) pretzel (12n242)
• Whitehead (52
1
)
• Borromean rings (63
2
)
• L10a140
• Conway knot (11n34)
Satellite
• Composite knots
• Granny
• Square
• Knot sum
Torus
• Unknot (01)
• Trefoil (31)
• Cinquefoil (51)
• Septafoil (71)
• Unlink (02
1
)
• Hopf (22
1
)
• Solomon's (42
1
)
Invariants
• Alternating
• Arf invariant
• Bridge no.
• 2-bridge
• Brunnian
• Chirality
• Invertible
• Crosscap no.
• Crossing no.
• Finite type invariant
• Hyperbolic volume
• Khovanov homology
• Genus
• Knot group
• Link group
• Linking no.
• Polynomial
• Alexander
• Bracket
• HOMFLY
• Jones
• Kauffman
• Pretzel
• Prime
• list
• Stick no.
• Tricolorability
• Unknotting no. and problem
Notation
and operations
• Alexander–Briggs notation
• Conway notation
• Dowker–Thistlethwaite notation
• Flype
• Mutation
• Reidemeister move
• Skein relation
• Tabulation
Other
• Alexander's theorem
• Berge
• Braid theory
• Conway sphere
• Complement
• Double torus
• Fibered
• Knot
• List of knots and links
• Ribbon
• Slice
• Sum
• Tait conjectures
• Twist
• Wild
• Writhe
• Surgery theory
• Category
• Commons
| Wikipedia |
74 knot
In mathematical knot theory, 74 is the name of a 7-crossing knot which can be visually depicted in a highly-symmetric form, and so appears in the symbolism and/or artistic ornamentation of various cultures.
74
Arf invariant0
Braid length9
Braid no.4
Bridge no.2
Crosscap no.3
Crossing no.7
Genus1
Hyperbolic volume5.13794
Stick no.9
Unknotting no.2
Conway notation[313]
A–B notation74
Dowker notation6, 10, 12, 14, 4, 2, 8
Last /Next73 / 75
Other
alternating, hyperbolic, prime, reversible, tricolorable
Visual representations
The interlaced version of the simplest form of the Endless knot symbol of Buddhism is topologically equivalent to the 74 knot (though it appears to have nine crossings), as is the interlaced version of the unicursal hexagram of occultism.[1] (However, the endless knot symbol has more complex forms not equivalent to 74, and both the endless knot and unicursal hexagram can appear in non-interlaced versions, in which case they are not knots at all.)
• One form of the Endless knot of Buddhism
• Interwoven unicursal hexagram.
• 74 knot in Celtic artistic form, also found in some Hausa embroideries.[2]
• A 74 knot combined with the Syrian flag is used as a logo by the National Coordination Committee for Democratic Change.
Wikimedia Commons has media related to 7 4 knots.
Example
Sources
1. "7_4", The Knot Atlas.
2. Celtic Art: The Methods of Construction by George Bain, p. 27 (ISBN 0-486-22923-8)
Knot theory (knots and links)
Hyperbolic
• Figure-eight (41)
• Three-twist (52)
• Stevedore (61)
• 62
• 63
• Endless (74)
• Carrick mat (818)
• Perko pair (10161)
• (−2,3,7) pretzel (12n242)
• Whitehead (52
1
)
• Borromean rings (63
2
)
• L10a140
• Conway knot (11n34)
Satellite
• Composite knots
• Granny
• Square
• Knot sum
Torus
• Unknot (01)
• Trefoil (31)
• Cinquefoil (51)
• Septafoil (71)
• Unlink (02
1
)
• Hopf (22
1
)
• Solomon's (42
1
)
Invariants
• Alternating
• Arf invariant
• Bridge no.
• 2-bridge
• Brunnian
• Chirality
• Invertible
• Crosscap no.
• Crossing no.
• Finite type invariant
• Hyperbolic volume
• Khovanov homology
• Genus
• Knot group
• Link group
• Linking no.
• Polynomial
• Alexander
• Bracket
• HOMFLY
• Jones
• Kauffman
• Pretzel
• Prime
• list
• Stick no.
• Tricolorability
• Unknotting no. and problem
Notation
and operations
• Alexander–Briggs notation
• Conway notation
• Dowker–Thistlethwaite notation
• Flype
• Mutation
• Reidemeister move
• Skein relation
• Tabulation
Other
• Alexander's theorem
• Berge
• Braid theory
• Conway sphere
• Complement
• Double torus
• Fibered
• Knot
• List of knots and links
• Ribbon
• Slice
• Sum
• Tait conjectures
• Twist
• Wild
• Writhe
• Surgery theory
• Category
• Commons
| Wikipedia |
Pixel connectivity
In image processing, pixel connectivity is the way in which pixels in 2-dimensional (or hypervoxels in n-dimensional) images relate to their neighbors.
Relevant topics on
Graph connectivity
• Connectivity
• Algebraic connectivity
• Cycle rank
• Rank (graph theory)
• SPQR tree
• St-connectivity
• K-connectivity certificate
• Pixel connectivity
• Vertex separator
• Strongly connected component
• Biconnected graph
• Bridge
Formulation
In order to specify a set of connectivities, the dimension $N$ and the width of the neighborhood $n$, must be specified. The dimension of a neighborhood is valid for any dimension $n\geq 1$. A common width is 3, which means along each dimension, the central cell will be adjacent to 1 cell on either side for all dimensions.
Let $M_{N}^{n}$ represent a N-dimensional hypercubic neighborhood with size on each dimension of $n=2k+1,k\in \mathbb {Z} $
Let ${\vec {q}}$ represent a discrete vector in the first orthant from the center structuring element to a point on the boundary of $M_{N}^{n}$. This implies that each element $q_{i}\in \{0,1,...,k\},\forall i\in \{1,2,...,N\}$ and that at least one component $q_{i}=k$
Let $S_{N}^{d}$ represent a N-dimensional hypersphere with radius of $d=\left\Vert {\vec {q}}\right\Vert $.
Define the amount of elements on the hypersphere $S_{N}^{d}$ within the neighborhood $M_{N}^{n}$ as $E$. For a given ${\vec {q}}$, $E$ will be equal to the amount of permutations of ${\vec {q}}$ multiplied by the number of orthants.
Let $n_{j}$ represent the amount of elements in vector ${\vec {q}}$ which take the value $j$. $n_{j}=\sum _{i=1}^{N}(q_{i}=j)$
The total number of permutation of ${\vec {q}}$ can be represented by a multinomial as ${\frac {N!}{\prod _{j=0}^{k}n_{j}!}}$
If any $q_{i}=0$, then the vector ${\vec {q}}$ is shared in common between orthants. Because of this, the multiplying factor on the permutation must be adjusted from $2^{N}$ to be $2^{N-n_{0}}$
Multiplying the number of amount of permutations by the adjusted amount of orthants yields,
$E={\frac {N!}{\prod _{j=0}^{k}n_{j}!}}2^{N-n_{0}}$
Let $V$ represent the number of elements inside of the hypersphere $S_{N}^{d}$ within the neighborhood $M_{N}^{n}$. $V$ will be equal to the number of elements on the hypersphere plus all of the elements on the inner shells. The shells must be ordered by increasing order of $\left\Vert {\vec {q}}\right\Vert =r$. Assume the ordered vectors ${\vec {q}}$ are assigned a coefficient $p$ representing its place in order. Then an ordered vector ${\vec {q_{p}}},p\in \{1,2,...,\sum _{x=1}^{k}(x+1)\}$ if all $r$ are unique. Therefore $V$ can be defined iteratively as
$V_{\vec {q_{p}}}=V_{\vec {q_{p-1}}}+E_{\vec {q_{p}}},V_{\vec {q_{0}}}=0$,
or
$V_{\vec {q_{p}}}=\sum _{x=1}^{p}E_{\vec {q_{x}}}$
If some $\left\Vert {\vec {q_{x}}}\right\Vert =\left\Vert {\vec {q_{y}}}\right\Vert $, then both vectors should be considered as the same $p$ such that
$V_{\vec {q_{p}}}=V_{\vec {q_{p-1}}}+E_{\vec {q_{p,1}}}+E_{\vec {q_{p,2}}},V_{\vec {q_{0}}}=0$
Note that each neighborhood will need to have the values from the next smallest neighborhood added. Ex. $V_{{\vec {q}}=(0,2)}=V_{{\vec {q}}=(1,1)}+E_{{\vec {q}}=(0,2)}$
$V$ includes the center hypervoxel, which is not included in the connectivity. Subtracting 1 yields the neighborhood connectivity, $G$
$G=V-1$[1]
Table of Selected Connectivities
Neighborhood Size:
$M_{N}^{n}$
Connectivity Type Typical Vector:
${\vec {q}}$
Sphere Radius
$d$
Elements on Sphere:
$E$
Elements in Sphere:
$V$
Neighborhood Connectivity:
$G$
1 edge (0) √0 1*1=1 1 0
3 point (1) √1 1*2=2 3 2
5 point-point (2) √4 1*2=2 5 4
...
1x1 face (0,0) √0 1*1=1 1 0
3x3 edge (0,1) √1 2*2=4 5 4
point (1,1) √2 1*4=4 9 8
5x5 edge-edge (0,2) √4 2*2=4 13 12
point-edge (1,2) √5 2*4=8 21 20
point-point (2,2) √8 1*4=4 25 24
...
1x1x1 volume (0,0,0) √0 1*1=1 1 0
3x3x3 face (0,0,1) √1 3*2=6 7 6
edge (0,1,1) √2 3*4=12 19 18
point (1,1,1) √3 1*8=8 27 26
5x5x5 face-face (0,0,2) √4 3*2=6 33 32
edge-face (0,1,2) √5 6*4=24 57 56
point-face (1,1,2) √6 3*8=24 81 80
edge-edge (0,2,2) √8 3*4=12 93 92
point-edge (1,2,2) √9 3*8=24 117 116
point-point (2,2,2) √12 1*8=8 125 124
...
1x1x1x1 hypervolume (0,0,0,0) √0 1*1=1 1 0
3x3x3x3 volume (0,0,0,1) √1 4*2=8 9 8
face (0,0,1,1) √2 6*4=24 33 32
edge (0,1,1,1) √3 4*8=32 65 64
point (1,1,1,1) √4 1*16=16 81 80
...
Example
Consider solving for $G|{\vec {q}}=(0,1,1)$
In this scenario, $N=3$ since the vector is 3-dimensional. $n_{0}=1$ since there is one $q_{i}=0$. Likewise, $n_{1}=2$. $k=1,n=3$ since $\max q_{i}=1$. $d={\sqrt {0^{2}+1^{2}+1^{2}}}={\sqrt {2}}$. The neighborhood is $M_{3}^{3}$ and the hypersphere is $S_{3}^{\sqrt {2}}$
$E={\frac {3!}{1!*2!*0!}}2^{3-1}={\frac {6}{2}}4=12$
The basic ${\vec {q}}$ in the neighborhood $N_{3}^{3}$, ${\vec {q_{1}}}=(0,0,0)$. The Manhattan Distance between our vector and the basic vector is $\left\Vert {\vec {q}}-{\vec {q_{0}}}\right\Vert _{1}=2$, so ${\vec {q}}={\vec {q_{3}}}$. Therefore,
$G_{\vec {q_{3}}}=V_{\vec {q_{3}}}-1=E_{\vec {q_{1}}}+E_{\vec {q_{2}}}+E_{\vec {q_{3}}}-1=E_{{\vec {q}}=(0,0,0)}+E_{{\vec {q}}=(0,0,1)}+E_{{\vec {q}}=(0,1,1)}$
$E_{{\vec {q}}=(0,0,0)}={\frac {3!}{3!*0!*0!}}2^{3-3}={\frac {6}{6}}1=1$
$E_{{\vec {q}}=(0,0,1)}={\frac {3!}{2!*1!}}2^{3-2}={\frac {6}{2}}2=6$
$G=1+6+12-1=18$
Which matches the supplied table
Higher values of k & N
The assumption that all $\left\Vert {\vec {q_{p}}}\right\Vert =r$ are unique does not hold for higher values of k & N. Consider $N=2,k=5$, and the vectors ${\vec {q_{A}}}=(0,5),{\vec {q_{B}}}=(3,4)$. Although ${\vec {q_{A}}}$ is located in $M_{2}^{5}$, the value for $r=25$, whereas ${\vec {q_{B}}}$ is in the smaller space $M_{2}^{4}$ but has an equivalent value $r=25$. ${\vec {q_{C}}}=(4,4)\in M_{2}^{4}$ but has a higher value of $r=32$ than the minimum vector in $M_{2}^{5}$.
For the this assumption to hold, ${\begin{cases}N=2,k\leq 4\\N=3,k\leq 2\\N=4,k\leq 1\end{cases}}$
At higher values of $k$ & $N$, Values of $d$ will become ambiguous. This means that specification of a given $d$ could refer to multiple ${\vec {q_{p}}}\in M_{n}^{N}$.
Types of connectivity
2-dimensional
4-connected
4-connected pixels are neighbors to every pixel that touches one of their edges. These pixels are connected horizontally and vertically. In terms of pixel coordinates, every pixel that has the coordinates
$\textstyle (x\pm 1,y)$ or $\textstyle (x,y\pm 1)$
is connected to the pixel at $\textstyle (x,y)$.
See also: Von Neumann neighborhood
6-connected
6-connected pixels are neighbors to every pixel that touches one of their corners (which includes pixels that touch one of their edges) in a hexagonal grid or stretcher bond rectangular grid.
There are several ways to map hexagonal tiles to integer pixel coordinates. With one method, in addition to the 4-connected pixels, the two pixels at coordinates $\textstyle (x+1,y+1)$ and $\textstyle (x-1,y-1)$ are connected to the pixel at $\textstyle (x,y)$.
8-connected
8-connected pixels are neighbors to every pixel that touches one of their edges or corners. These pixels are connected horizontally, vertically, and diagonally. In addition to 4-connected pixels, each pixel with coordinates $\textstyle (x\pm 1,y\pm 1)$ is connected to the pixel at $\textstyle (x,y)$.
See also: Moore neighborhood
6-connected
6-connected pixels are neighbors to every pixel that touches one of their faces. These pixels are connected along one of the primary axes. Each pixel with coordinates $\textstyle (x\pm 1,y,z)$, $\textstyle (x,y\pm 1,z)$, or $\textstyle (x,y,z\pm 1)$ is connected to the pixel at $\textstyle (x,y,z)$.
18-connected
18-connected pixels are neighbors to every pixel that touches one of their faces or edges. These pixels are connected along either one or two of the primary axes. In addition to 6-connected pixels, each pixel with coordinates $\textstyle (x\pm 1,y\pm 1,z)$, $\textstyle (x\pm 1,y\mp 1,z)$, $\textstyle (x\pm 1,y,z\pm 1)$, $\textstyle (x\pm 1,y,z\mp 1)$, $\textstyle (x,y\pm 1,z\pm 1)$, or $\textstyle (x,y\pm 1,z\mp 1)$ is connected to the pixel at $\textstyle (x,y,z)$.
26-connected
26-connected pixels are neighbors to every pixel that touches one of their faces, edges, or corners. These pixels are connected along either one, two, or all three of the primary axes. In addition to 18-connected pixels, each pixel with coordinates $\textstyle (x\pm 1,y\pm 1,z\pm 1)$, $\textstyle (x\pm 1,y\pm 1,z\mp 1)$, $\textstyle (x\pm 1,y\mp 1,z\pm 1)$, or $\textstyle (x\mp 1,y\pm 1,z\pm 1)$ is connected to the pixel at $\textstyle (x,y,z)$.
See also
• Grid cell topology
• Moore neighborhood
References
1. Jonker, Pieter (1992). Morphological Image Processing: Architecture and VLSI design. Kluwer Technische Boeken B.V. pp. 92–96. ISBN 978-1-4615-2804-3.
• A. Rosenfeld , A. C. Kak (1982), Digital Picture Processing, Academic Press, Inc., ISBN 0-12-597302-0
• Cheng, CC; Peng, GJ; Hwang, WL (2009), "Subband Weighting With Pixel Connectivity for 3-D Wavelet Coding", IEEE Transactions on Image Processing, 18 (1): 52–62, doi:10.1109/TIP.2008.2007067, PMID 19095518, retrieved 2009-02-16
• Cheng, CC; Peng, GJ; Hwang, WL (2009), "Subband Weighting With Pixel Connectivity for 3-D Wavelet Coding", IEEE Transactions on Image Processing, 18 (1): 52–62, doi:10.1109/TIP.2008.2007067, PMID 19095518, retrieved 2009-02-16
| Wikipedia |
8-demicubic honeycomb
The 8-demicubic honeycomb, or demiocteractic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 8-space. It is constructed as an alternation of the regular 8-cubic honeycomb.
8-demicubic honeycomb
(No image)
TypeUniform 8-honeycomb
FamilyAlternated hypercube honeycomb
Schläfli symbolh{4,3,3,3,3,3,3,4}
Coxeter diagrams =
=
Facets{3,3,3,3,3,3,4}
h{4,3,3,3,3,3,3}
Vertex figureRectified 8-orthoplex
Coxeter group${\tilde {B}}_{8}$ [4,3,3,3,3,3,31,1]
${\tilde {D}}_{8}$ [31,1,3,3,3,3,31,1]
It is composed of two different types of facets. The 8-cubes become alternated into 8-demicubes h{4,3,3,3,3,3,3} and the alternated vertices create 8-orthoplex {3,3,3,3,3,3,4} facets .
D8 lattice
The vertex arrangement of the 8-demicubic honeycomb is the D8 lattice.[1] The 112 vertices of the rectified 8-orthoplex vertex figure of the 8-demicubic honeycomb reflect the kissing number 112 of this lattice.[2] The best known is 240, from the E8 lattice and the 521 honeycomb.
${\tilde {E}}_{8}$ contains ${\tilde {D}}_{8}$ as a subgroup of index 270.[3] Both ${\tilde {E}}_{8}$ and ${\tilde {D}}_{8}$ can be seen as affine extensions of $D_{8}$ from different nodes:
The D+
8
lattice (also called D2
8
) can be constructed by the union of two D8 lattices.[4] This packing is only a lattice for even dimensions. The kissing number is 240. (2n-1 for n<8, 240 for n=8, and 2n(n-1) for n>8).[5] It is identical to the E8 lattice. At 8-dimensions, the 240 contacts contain both the 27=128 from lower dimension contact progression (2n-1), and 16*7=112 from higher dimensions (2n(n-1)).
∪ = .
The D*
8
lattice (also called D4
8
and C2
8
) can be constructed by the union of all four D8 lattices:[6] It is also the 7-dimensional body centered cubic, the union of two 7-cube honeycombs in dual positions.
∪ ∪ ∪ = ∪ .
The kissing number of the D*
8
lattice is 16 (2n for n≥5).[7] and its Voronoi tessellation is a quadrirectified 8-cubic honeycomb, , containing all trirectified 8-orthoplex Voronoi cell, .[8]
Symmetry constructions
There are three uniform construction symmetries of this tessellation. Each symmetry can be represented by arrangements of different colors on the 256 8-demicube facets around each vertex.
Coxeter group Schläfli symbol Coxeter-Dynkin diagram Vertex figure
Symmetry
Facets/verf
${\tilde {B}}_{8}$ = [31,1,3,3,3,3,3,4]
= [1+,4,3,3,3,3,3,3,4]
h{4,3,3,3,3,3,3,4} =
[3,3,3,3,3,3,4]
256: 8-demicube
16: 8-orthoplex
${\tilde {D}}_{8}$ = [31,1,3,3,3,31,1]
= [1+,4,3,3,3,3,31,1]
h{4,3,3,3,3,3,31,1} =
[36,1,1]
128+128: 8-demicube
16: 8-orthoplex
2×½${\tilde {C}}_{8}$ = [[(4,3,3,3,3,3,4,2+)]]ht0,8{4,3,3,3,3,3,3,4} 128+64+64: 8-demicube
16: 8-orthoplex
See also
• 8-cubic honeycomb
• Uniform polytope
Notes
1. "The Lattice D8".
2. Sphere packings, lattices, and groups, by John Horton Conway, Neil James Alexander Sloane, Eiichi Bannai
3. Johnson (2015) p.177
4. Kaleidoscopes: Selected Writings of H. S. M. Coxeter, Paper 18, "Extreme forms" (1950)
5. Conway (1998), p. 119
6. "The Lattice D8".
7. Conway (1998), p. 120
8. Conway (1998), p. 466
References
• Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8
• pp. 154–156: Partial truncation or alternation, represented by h prefix: h{4,4}={4,4}; h{4,3,4}={31,1,4}, h{4,3,3,4}={3,3,4,3}, ...
• Kaleidoscopes: Selected Writings of H. S. M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6
• (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
• N.W. Johnson: Geometries and Transformations, (2018)
• Conway JH, Sloane NJH (1998). Sphere Packings, Lattices and Groups (3rd ed.). ISBN 0-387-98585-9.
External links
Fundamental convex regular and uniform honeycombs in dimensions 2–9
Space Family ${\tilde {A}}_{n-1}$ ${\tilde {C}}_{n-1}$ ${\tilde {B}}_{n-1}$ ${\tilde {D}}_{n-1}$ ${\tilde {G}}_{2}$ / ${\tilde {F}}_{4}$ / ${\tilde {E}}_{n-1}$
E2 Uniform tiling {3[3]} δ3 hδ3 qδ3 Hexagonal
E3 Uniform convex honeycomb {3[4]} δ4 hδ4 qδ4
E4 Uniform 4-honeycomb {3[5]} δ5 hδ5 qδ5 24-cell honeycomb
E5 Uniform 5-honeycomb {3[6]} δ6 hδ6 qδ6
E6 Uniform 6-honeycomb {3[7]} δ7 hδ7 qδ7 222
E7 Uniform 7-honeycomb {3[8]} δ8 hδ8 qδ8 133 • 331
E8 Uniform 8-honeycomb {3[9]} δ9 hδ9 qδ9 152 • 251 • 521
E9 Uniform 9-honeycomb {3[10]} δ10 hδ10 qδ10
E10 Uniform 10-honeycomb {3[11]} δ11 hδ11 qδ11
En-1 Uniform (n-1)-honeycomb {3[n]} δn hδn qδn 1k2 • 2k1 • k21
| Wikipedia |
8-cube
In geometry, an 8-cube is an eight-dimensional hypercube. It has 256 vertices, 1024 edges, 1792 square faces, 1792 cubic cells, 1120 tesseract 4-faces, 448 5-cube 5-faces, 112 6-cube 6-faces, and 16 7-cube 7-faces.
8-cube
Octeract
Orthogonal projection
inside Petrie polygon
TypeRegular 8-polytope
Familyhypercube
Schläfli symbol{4,36}
Coxeter-Dynkin diagrams
7-faces16 {4,35}
6-faces112 {4,34}
5-faces448 {4,33}
4-faces1120 {4,32}
Cells1792 {4,3}
Faces1792 {4}
Edges1024
Vertices256
Vertex figure7-simplex
Petrie polygonhexadecagon
Coxeter groupC8, [36,4]
Dual8-orthoplex
Propertiesconvex, Hanner polytope
It is represented by Schläfli symbol {4,36}, being composed of 3 7-cubes around each 6-face. It is called an octeract, a portmanteau of tesseract (the 4-cube) and oct for eight (dimensions) in Greek. It can also be called a regular hexdeca-8-tope or hexadecazetton, being an 8-dimensional polytope constructed from 16 regular facets.
It is a part of an infinite family of polytopes, called hypercubes. The dual of an 8-cube can be called an 8-orthoplex and is a part of the infinite family of cross-polytopes.
Cartesian coordinates
Cartesian coordinates for the vertices of an 8-cube centered at the origin and edge length 2 are
(±1,±1,±1,±1,±1,±1,±1,±1)
while the interior of the same consists of all points (x0, x1, x2, x3, x4, x5, x6, x7) with -1 < xi < 1.
As a configuration
This configuration matrix represents the 8-cube. The rows and columns correspond to vertices, edges, faces, cells, 4-faces, 5-faces, 6-faces, and 7-faces. The diagonal numbers say how many of each element occur in the whole 8-cube. The nondiagonal numbers say how many of the column's element occur in or at the row's element.[1][2]
${\begin{bmatrix}{\begin{matrix}256&8&28&56&70&56&28&8\\2&1024&7&21&35&35&21&7\\4&4&1792&6&15&20&15&6\\8&12&6&1792&5&10&10&5\\16&32&24&8&1120&4&6&4\\32&80&80&40&10&448&3&3\\64&192&240&160&60&12&112&2\\128&448&672&560&280&84&14&16\end{matrix}}\end{bmatrix}}$
The diagonal f-vector numbers are derived through the Wythoff construction, dividing the full group order of a subgroup order by removing one mirror at a time.[3]
B8k-facefkf0f1f2f3f4f5f6f7k-figurenotes
A7( ) f0 256828567056288{3,3,3,3,3,3}B8/A7 = 2^8*8!/8! = 256
A6A1{ } f1 210247213535217{3,3,3,3,3}B8/A6A1 = 2^8*8!/7!/2 = 1024
A5B2{4} f2 44179261520156{3,3,3,3}B8/A5B2 = 2^8*8!/6!/4/2 = 1792
A4B3{4,3} f3 81261792510105{3,3,3}B8/A4B3 = 2^8*8!/5!/8/3! = 1792
A3B4{4,3,3} f4 16322481120464{3,3}B8/A3B4 = 2^8*8!/4!/2^4/4! = 1120
A2B5{4,3,3,3} f5 328080401044833{3}B8/A2B5 = 2^8*8!/3!/2^5/5! = 448
A1B6{4,3,3,3,3} f6 6419224016060121122{ }B8/A1B6 = 2^8*8!/2/2^6/6!= 112
B7{4,3,3,3,3,3} f7 128448672560280841416( )B8/B7 = 2^8*8!/2^7/7! = 16
Projections
This 8-cube graph is an orthogonal projection. This orientation shows columns of vertices positioned a vertex-edge-vertex distance from one vertex on the left to one vertex on the right, and edges attaching adjacent columns of vertices. The number of vertices in each column represents rows in Pascal's triangle, being 1:8:28:56:70:56:28:8:1.
orthographic projections
B8 B7
[16] [14]
B6 B5
[12] [10]
B4 B3 B2
[8] [6] [4]
A7 A5 A3
[8] [6] [4]
Derived polytopes
Applying an alternation operation, deleting alternating vertices of the octeract, creates another uniform polytope, called a 8-demicube, (part of an infinite family called demihypercubes), which has 16 demihepteractic and 128 8-simplex facets.
Related polytopes
The 8-cube is 8th in an infinite series of hypercube:
Petrie polygon orthographic projections
Line segment Square Cube 4-cube 5-cube 6-cube 7-cube 8-cube
References
1. Coxeter, Regular Polytopes, sec 1.8 Configurations
2. Coxeter, Complex Regular Polytopes, p.117
3. Klitzing, Richard. "o3o3o3o3o3o3o4x - octo".
• H.S.M. Coxeter:
• Coxeter, Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8, p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
• Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6
• (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
• (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
• (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
• Norman Johnson Uniform Polytopes, Manuscript (1991)
• N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. (1966)
• Klitzing, Richard. "8D uniform polytopes (polyzetta) o3o3o3o3o3o3o4x - octo".
External links
• Weisstein, Eric W. "Hypercube". MathWorld.
• Olshevsky, George. "Measure polytope". Glossary for Hyperspace. Archived from the original on 4 February 2007.
• Multi-dimensional Glossary: hypercube Garrett Jones
Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron
Uniform polychoron Pentachoron 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell
Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221
Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321
Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421
Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds
| Wikipedia |
8-simplex
In geometry, an 8-simplex is a self-dual regular 8-polytope. It has 9 vertices, 36 edges, 84 triangle faces, 126 tetrahedral cells, 126 5-cell 4-faces, 84 5-simplex 5-faces, 36 6-simplex 6-faces, and 9 7-simplex 7-faces. Its dihedral angle is cos−1(1/8), or approximately 82.82°.
Regular enneazetton
(8-simplex)
Orthogonal projection
inside Petrie polygon
TypeRegular 8-polytope
Familysimplex
Schläfli symbol{3,3,3,3,3,3,3}
Coxeter-Dynkin diagram
7-faces9 7-simplex
6-faces36 6-simplex
5-faces84 5-simplex
4-faces126 5-cell
Cells126 tetrahedron
Faces84 triangle
Edges36
Vertices9
Vertex figure7-simplex
Petrie polygonenneagon
Coxeter groupA8 [3,3,3,3,3,3,3]
DualSelf-dual
Propertiesconvex
It can also be called an enneazetton, or ennea-8-tope, as a 9-facetted polytope in eight-dimensions. The name enneazetton is derived from ennea for nine facets in Greek and -zetta for having seven-dimensional facets, and -on.
As a configuration
This configuration matrix represents the 8-simplex. The rows and columns correspond to vertices, edges, faces, cells, 4-faces, 5-faces, 6-faces and 7-faces. The diagonal numbers say how many of each element occur in the whole 8-simplex. The nondiagonal numbers say how many of the column's element occur in or at the row's element. This self-dual simplex's matrix is identical to its 180 degree rotation.[1][2]
${\begin{bmatrix}{\begin{matrix}9&8&28&56&70&56&28&8\\2&36&7&21&35&35&21&7\\3&3&84&6&15&20&15&6\\4&6&4&126&5&10&10&5\\5&10&10&5&126&4&6&4\\6&15&20&15&6&84&3&3\\7&21&35&35&21&7&36&2\\8&28&56&70&56&28&8&9\end{matrix}}\end{bmatrix}}$
Coordinates
The Cartesian coordinates of the vertices of an origin-centered regular enneazetton having edge length 2 are:
$\left(1/6,\ {\sqrt {1/28}},\ {\sqrt {1/21}},\ {\sqrt {1/15}},\ {\sqrt {1/10}},\ {\sqrt {1/6}},\ {\sqrt {1/3}},\ \pm 1\right)$
$\left(1/6,\ {\sqrt {1/28}},\ {\sqrt {1/21}},\ {\sqrt {1/15}},\ {\sqrt {1/10}},\ {\sqrt {1/6}},\ -2{\sqrt {1/3}},\ 0\right)$
$\left(1/6,\ {\sqrt {1/28}},\ {\sqrt {1/21}},\ {\sqrt {1/15}},\ {\sqrt {1/10}},\ -{\sqrt {3/2}},\ 0,\ 0\right)$
$\left(1/6,\ {\sqrt {1/28}},\ {\sqrt {1/21}},\ {\sqrt {1/15}},\ -2{\sqrt {2/5}},\ 0,\ 0,\ 0\right)$
$\left(1/6,\ {\sqrt {1/28}},\ {\sqrt {1/21}},\ -{\sqrt {5/3}},\ 0,\ 0,\ 0,\ 0\right)$
$\left(1/6,\ {\sqrt {1/28}},\ -{\sqrt {12/7}},\ 0,\ 0,\ 0,\ 0,\ 0\right)$
$\left(1/6,\ -{\sqrt {7/4}},\ 0,\ 0,\ 0,\ 0,\ 0,\ 0\right)$
$\left(-4/3,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0\right)$
More simply, the vertices of the 8-simplex can be positioned in 9-space as permutations of (0,0,0,0,0,0,0,0,1). This construction is based on facets of the 9-orthoplex.
Another origin-centered construction uses (1,1,1,1,1,1,1,1)/3 and permutations of (1,1,1,1,1,1,1,-11)/12 for edge length √2.
Images
orthographic projections
Ak Coxeter plane A8 A7 A6 A5
Graph
Dihedral symmetry [9] [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [5] [4] [3]
Related polytopes and honeycombs
This polytope is a facet in the uniform tessellations: 251, and 521 with respective Coxeter-Dynkin diagrams:
,
This polytope is one of 135 uniform 8-polytopes with A8 symmetry.
A8 polytopes
t0
t1
t2
t3
t01
t02
t12
t03
t13
t23
t04
t14
t24
t34
t05
t15
t25
t06
t16
t07
t012
t013
t023
t123
t014
t024
t124
t034
t134
t234
t015
t025
t125
t035
t135
t235
t045
t145
t016
t026
t126
t036
t136
t046
t056
t017
t027
t037
t0123
t0124
t0134
t0234
t1234
t0125
t0135
t0235
t1235
t0145
t0245
t1245
t0345
t1345
t2345
t0126
t0136
t0236
t1236
t0146
t0246
t1246
t0346
t1346
t0156
t0256
t1256
t0356
t0456
t0127
t0137
t0237
t0147
t0247
t0347
t0157
t0257
t0167
t01234
t01235
t01245
t01345
t02345
t12345
t01236
t01246
t01346
t02346
t12346
t01256
t01356
t02356
t12356
t01456
t02456
t03456
t01237
t01247
t01347
t02347
t01257
t01357
t02357
t01457
t01267
t01367
t012345
t012346
t012356
t012456
t013456
t023456
t123456
t012347
t012357
t012457
t013457
t023457
t012367
t012467
t013467
t012567
t0123456
t0123457
t0123467
t0123567
t01234567
References
1. Coxeter 1973, §1.8 Configurations
2. Coxeter, H.S.M. (1991). Regular Complex Polytopes (2nd ed.). Cambridge University Press. p. 117. ISBN 9780521394901.
• Coxeter, H.S.M.:
• — (1973). "Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)". Regular Polytopes (3rd ed.). Dover. pp. 296. ISBN 0-486-61480-8.
• Sherk, F. Arthur; McMullen, Peter; Thompson, Anthony C.; Weiss, Asia Ivic, eds. (1995). Kaleidoscopes: Selected Writings of H.S.M. Coxeter. Wiley. ISBN 978-0-471-01003-6.
• (Paper 22) — (1940). "Regular and Semi Regular Polytopes I". Math. Zeit. 46: 380–407. doi:10.1007/BF01181449. S2CID 186237114.
• (Paper 23) — (1985). "Regular and Semi-Regular Polytopes II". Math. Zeit. 188 (4): 559–591. doi:10.1007/BF01161657. S2CID 120429557.
• (Paper 24) — (1988). "Regular and Semi-Regular Polytopes III". Math. Zeit. 200: 3–45. doi:10.1007/BF01161745. S2CID 186237142.
• Conway, John H.; Burgiel, Heidi; Goodman-Strass, Chaim (2008). "26. Hemicubes: 1n1". The Symmetries of Things. p. 409. ISBN 978-1-56881-220-5.
• Johnson, Norman (1991). "Uniform Polytopes" (Manuscript). {{cite document}}: Cite document requires |publisher= (help)
• Johnson, N.W. (1966). The Theory of Uniform Polytopes and Honeycombs (PhD). University of Toronto. OCLC 258527038.
• Klitzing, Richard. "8D uniform polytopes (polyzetta) x3o3o3o3o3o3o3o — ene".
External links
• Glossary for hyperspace, George Olshevsky.
• Polytopes of Various Dimensions
• Multi-dimensional Glossary
Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron
Uniform polychoron Pentachoron 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell
Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221
Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321
Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421
Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds
| Wikipedia |
Eight queens puzzle
The eight queens puzzle is the problem of placing eight chess queens on an 8×8 chessboard so that no two queens threaten each other; thus, a solution requires that no two queens share the same row, column, or diagonal. There are 92 solutions. The problem was first posed in the mid-19th century. In the modern era, it is often used as an example problem for various computer programming techniques.
abcdefgh
8
8
77
66
55
44
33
22
11
abcdefgh
The only symmetrical solution to the eight queens puzzle (up to rotation and reflection)
The eight queens puzzle is a special case of the more general n queens problem of placing n non-attacking queens on an n×n chessboard. Solutions exist for all natural numbers n with the exception of n = 2 and n = 3. Although the exact number of solutions is only known for n ≤ 27, the asymptotic growth rate of the number of solutions is approximately (0.143 n)n.
History
Chess composer Max Bezzel published the eight queens puzzle in 1848. Franz Nauck published the first solutions in 1850.[1] Nauck also extended the puzzle to the n queens problem, with n queens on a chessboard of n×n squares.
Since then, many mathematicians, including Carl Friedrich Gauss, have worked on both the eight queens puzzle and its generalized n-queens version. In 1874, S. Gunther proposed a method using determinants to find solutions.[1] J.W.L. Glaisher refined Gunther's approach.
In 1972, Edsger Dijkstra used this problem to illustrate the power of what he called structured programming. He published a highly detailed description of a depth-first backtracking algorithm.[2]
Constructing and counting solutions when n = 8
The problem of finding all solutions to the 8-queens problem can be quite computationally expensive, as there are 4,426,165,368 possible arrangements of eight queens on an 8×8 board,[lower-alpha 1] but only 92 solutions. It is possible to use shortcuts that reduce computational requirements or rules of thumb that avoids brute-force computational techniques. For example, by applying a simple rule that chooses one queen from each column, it is possible to reduce the number of possibilities to 16,777,216 (that is, 88) possible combinations. Generating permutations further reduces the possibilities to just 40,320 (that is, 8!), which can then be checked for diagonal attacks.
The eight queens puzzle has 92 distinct solutions. If solutions that differ only by the symmetry operations of rotation and reflection of the board are counted as one, the puzzle has 12 solutions. These are called fundamental solutions; representatives of each are shown below.
A fundamental solution usually has eight variants (including its original form) obtained by rotating 90, 180, or 270° and then reflecting each of the four rotational variants in a mirror in a fixed position. However, one of the 12 fundamental solutions (solution 12 below) is identical to its own 180° rotation, so has only four variants (itself and its reflection, its 90° rotation and the reflection of that).[lower-alpha 2] Thus, the total number of distinct solutions is 11×8 + 1×4 = 92.
All fundamental solutions are presented below:
abcdefgh
8
8
77
66
55
44
33
22
11
abcdefgh
Solution 1
abcdefgh
8
8
77
66
55
44
33
22
11
abcdefgh
Solution 2
abcdefgh
8
8
77
66
55
44
33
22
11
abcdefgh
Solution 3
abcdefgh
8
8
77
66
55
44
33
22
11
abcdefgh
Solution 4
abcdefgh
8
8
77
66
55
44
33
22
11
abcdefgh
Solution 5
abcdefgh
8
8
77
66
55
44
33
22
11
abcdefgh
Solution 6
abcdefgh
8
8
77
66
55
44
33
22
11
abcdefgh
Solution 7
abcdefgh
8
8
77
66
55
44
33
22
11
abcdefgh
Solution 8
abcdefgh
8
8
77
66
55
44
33
22
11
abcdefgh
Solution 9
abcdefgh
8
8
77
66
55
44
33
22
11
abcdefgh
Solution 10
abcdefgh
8
8
77
66
55
44
33
22
11
abcdefgh
Solution 11
abcdefgh
8
8
77
66
55
44
33
22
11
abcdefgh
Solution 12
Solution 10 has the additional property that no three queens are in a straight line.
Existence of solutions
Brute-force algorithms to count the number of solutions are computationally manageable for n = 8, but would be intractable for problems of n ≥ 20, as 20! = 2.433 × 1018. If the goal is to find a single solution, one can show solutions exist for all n ≥ 4 with no search whatsoever.[3][4] These solutions exhibit stair-stepped patterns, as in the following examples for n = 8, 9 and 10:
abcdefgh
8
8
77
66
55
44
33
22
11
abcdefgh
Staircase solution for 8 queens
abcdefghi
99
88
77
66
55
44
33
22
11
abcdefghi
Staircase solution for 9 queens
abcdefghij
1010
99
88
77
66
55
44
33
22
11
abcdefghij
Staircase solution for 10 queens
The examples above can be obtained with the following formulas.[3] Let (i, j) be the square in column i and row j on the n × n chessboard, k an integer.
One approach[3] is
1. If the remainder from dividing n by 6 is not 2 or 3 then the list is simply all even numbers followed by all odd numbers not greater than n.
2. Otherwise, write separate lists of even and odd numbers (2, 4, 6, 8 – 1, 3, 5, 7).
3. If the remainder is 2, swap 1 and 3 in odd list and move 5 to the end (3, 1, 7, 5).
4. If the remainder is 3, move 2 to the end of even list and 1,3 to the end of odd list (4, 6, 8, 2 – 5, 7, 9, 1, 3).
5. Append odd list to the even list and place queens in the rows given by these numbers, from left to right (a2, b4, c6, d8, e3, f1, g7, h5).
For n = 8 this results in fundamental solution 1 above. A few more examples follow.
• 14 queens (remainder 2): 2, 4, 6, 8, 10, 12, 14, 3, 1, 7, 9, 11, 13, 5.
• 15 queens (remainder 3): 4, 6, 8, 10, 12, 14, 2, 5, 7, 9, 11, 13, 15, 1, 3.
• 20 queens (remainder 2): 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 3, 1, 7, 9, 11, 13, 15, 17, 19, 5.
Counting solutions for other sizes n
Exact enumeration
There is no known formula for the exact number of solutions for placing n queens on an n × n board i.e. the number of independent sets of size n in an n × n queen's graph. The 27×27 board is the highest-order board that has been completely enumerated.[5] The following tables give the number of solutions to the n queens problem, both fundamental (sequence A002562 in the OEIS) and all (sequence A000170 in the OEIS), for all known cases.
n fundamental all
1 1 1
2 0 0
3 0 0
4 1 2
5 2 10
6 1 4
7 6 40
8 12 92
9 46 352
10 92 724
11 341 2,680
12 1,787 14,200
13 9,233 73,712
14 45,752 365,596
15 285,053 2,279,184
16 1,846,955 14,772,512
17 11,977,939 95,815,104
18 83,263,591 666,090,624
19 621,012,754 4,968,057,848
20 4,878,666,808 39,029,188,884
21 39,333,324,973 314,666,222,712
22 336,376,244,042 2,691,008,701,644
23 3,029,242,658,210 24,233,937,684,440
24 28,439,272,956,934 227,514,171,973,736
25 275,986,683,743,434 2,207,893,435,808,352
26 2,789,712,466,510,289 22,317,699,616,364,044
27 29,363,495,934,315,694 234,907,967,154,122,528
Asymptotic enumeration
In 2021, Michael Simkin proved that for large numbers n, the number of solutions of the n queens problem is approximately $(0.143n)^{n}$.[6] More precisely, the number ${\mathcal {Q}}(n)$ of solutions has asymptotic growth
${\mathcal {Q}}(n)=((1\pm o(1))ne^{-\alpha })^{n}$
where $\alpha $ is a constant that lies between 1.939 and 1.945.[7] (Here o(1) represents little o notation.)
If one instead considers a toroidal chessboard (where diagonals "wrap around" from the top edge to the bottom and from the left edge to the right), it is only possible to place n queens on an $n\times n$ board if $n\equiv 1,5\mod 6.$ In this case, the asymptotic number of solutions is[8][9]
$T(n)=((1+o(1))ne^{-3})^{n}.$
Related problems
Higher dimensions
Find the number of non-attacking queens that can be placed in a d-dimensional chess space of size n. More than n queens can be placed in some higher dimensions (the smallest example is four non-attacking queens in a 3×3×3 chess space), and it is in fact known that for any k, there are higher dimensions where nk queens do not suffice to attack all spaces.[10][11]
Using pieces other than queens
On an 8×8 board one can place 32 knights, or 14 bishops, 16 kings or eight rooks, so that no two pieces attack each other. In the case of knights, an easy solution is to place one on each square of a given color, since they move only to the opposite color. The solution is also easy for rooks and kings. Sixteen kings can be placed on the board by dividing it into 2-by-2 squares and placing the kings at equivalent points on each square. Placements of n rooks on an n×n board are in direct correspondence with order-n permutation matrices.
Chess variations
Related problems can be asked for chess variations such as shogi. For instance, the n+k dragon kings problem asks to place k shogi pawns and n+k mutually nonattacking dragon kings on an n×n shogi board.[12]
Nonstandard boards
Pólya studied the n queens problem on a toroidal ("donut-shaped") board and showed that there is a solution on an n×n board if and only if n is not divisible by 2 or 3.[13] In 2009 Pearson and Pearson algorithmically populated three-dimensional boards (n×n×n) with n2 queens, and proposed that multiples of these can yield solutions for a four-dimensional version of the puzzle.[14]
Domination
Given an n×n board, the domination number is the minimum number of queens (or other pieces) needed to attack or occupy every square. For n = 8 the queen's domination number is 5.[15][16]
Queens and other pieces
Variants include mixing queens with other pieces; for example, placing m queens and m knights on an n×n board so that no piece attacks another[17] or placing queens and pawns so that no two queens attack each other.[18]
Magic squares
In 1992, Demirörs, Rafraf, and Tanik published a method for converting some magic squares into n-queens solutions, and vice versa.[19]
Latin squares
In an n×n matrix, place each digit 1 through n in n locations in the matrix so that no two instances of the same digit are in the same row or column.
Exact cover
Consider a matrix with one primary column for each of the n ranks of the board, one primary column for each of the n files, and one secondary column for each of the 4n − 6 nontrivial diagonals of the board. The matrix has n2 rows: one for each possible queen placement, and each row has a 1 in the columns corresponding to that square's rank, file, and diagonals and a 0 in all the other columns. Then the n queens problem is equivalent to choosing a subset of the rows of this matrix such that every primary column has a 1 in precisely one of the chosen rows and every secondary column has a 1 in at most one of the chosen rows; this is an example of a generalized exact cover problem, of which sudoku is another example.
n-queens completion
The completion problem asks whether, given an n×n chessboard on which some queens are already placed, it is possible to place a queen in every remaining row so that no two queens attack each other. This and related questions are NP-complete and #P-complete.[20] Any placement of at most n/60 queens can be completed, while there are partial configurations of roughly n/4 queens that cannot be completed.[21]
Exercise in algorithm design
Finding all solutions to the eight queens puzzle is a good example of a simple but nontrivial problem. For this reason, it is often used as an example problem for various programming techniques, including nontraditional approaches such as constraint programming, logic programming or genetic algorithms. Most often, it is used as an example of a problem that can be solved with a recursive algorithm, by phrasing the n queens problem inductively in terms of adding a single queen to any solution to the problem of placing n−1 queens on an n×n chessboard. The induction bottoms out with the solution to the 'problem' of placing 0 queens on the chessboard, which is the empty chessboard.
This technique can be used in a way that is much more efficient than the naïve brute-force search algorithm, which considers all 648 = 248 = 281,474,976,710,656 possible blind placements of eight queens, and then filters these to remove all placements that place two queens either on the same square (leaving only 64!/56! = 178,462,987,637,760 possible placements) or in mutually attacking positions. This very poor algorithm will, among other things, produce the same results over and over again in all the different permutations of the assignments of the eight queens, as well as repeating the same computations over and over again for the different sub-sets of each solution. A better brute-force algorithm places a single queen on each row, leading to only 88 = 224 = 16,777,216 blind placements.
It is possible to do much better than this. One algorithm solves the eight rooks puzzle by generating the permutations of the numbers 1 through 8 (of which there are 8! = 40,320), and uses the elements of each permutation as indices to place a queen on each row. Then it rejects those boards with diagonal attacking positions.
The backtracking depth-first search program, a slight improvement on the permutation method, constructs the search tree by considering one row of the board at a time, eliminating most nonsolution board positions at a very early stage in their construction. Because it rejects rook and diagonal attacks even on incomplete boards, it examines only 15,720 possible queen placements. A further improvement, which examines only 5,508 possible queen placements, is to combine the permutation based method with the early pruning method: the permutations are generated depth-first, and the search space is pruned if the partial permutation produces a diagonal attack. Constraint programming can also be very effective on this problem.
An alternative to exhaustive search is an 'iterative repair' algorithm, which typically starts with all queens on the board, for example with one queen per column.[22] It then counts the number of conflicts (attacks), and uses a heuristic to determine how to improve the placement of the queens. The 'minimum-conflicts' heuristic – moving the piece with the largest number of conflicts to the square in the same column where the number of conflicts is smallest – is particularly effective: it easily finds a solution to even the 1,000,000 queens problem.[23][24]
Unlike the backtracking search outlined above, iterative repair does not guarantee a solution: like all greedy procedures, it may get stuck on a local optimum. (In such a case, the algorithm may be restarted with a different initial configuration.) On the other hand, it can solve problem sizes that are several orders of magnitude beyond the scope of a depth-first search.
As an alternative to backtracking, solutions can be counted by recursively enumerating valid partial solutions, one row at a time. Rather than constructing entire board positions, blocked diagonals and columns are tracked with bitwise operations. This does not allow the recovery of individual solutions.[25][26]
Sample program
The following program is a translation of Niklaus Wirth's solution into the Python programming language, but does without the index arithmetic found in the original and instead uses lists to keep the program code as simple as possible. By using a coroutine in the form of a generator function, both versions of the original can be unified to compute either one or all of the solutions. Only 15,720 possible queen placements are examined.[27][28]
def queens(n, i, a, b, c):
if i < n:
for j in range(n):
if j not in a and i+j not in b and i-j not in c:
yield from queens(n, i+1, a+[j], b+[i+j], c+[i-j])
else:
yield a
for solution in queens(8, 0, [], [], []):
print(solution)
In popular culture
• In the game The 7th Guest, the 8th Puzzle: "The Queen's Dilemma" in the game room of the Stauf mansion is the de facto eight queens puzzle.[29]: 48–49, 289–290
• In the game Professor Layton and the Curious Village, the 130th puzzle: "Too Many Queens 5" (クイーンの問題5) is an eight queens puzzle.[30]
See also
• Mathematical game
• Mathematical puzzle
• No-three-in-line problem
• Rook polynomial
• Costas array
Notes
1. The number of combinations of 8 squares from 64 is the binomial coefficient 64C8.
2. Other symmetries are possible for other values of n. For example, there is a placement of five nonattacking queens on a 5×5 board that is identical to its own 90° rotation. Such solutions have only two variants (itself and its reflection). If n > 1, it is not possible for a solution to be equal to its own reflection because that would require two queens to be facing each other.
References
1. W. W. Rouse Ball (1960) "The Eight Queens Problem", in Mathematical Recreations and Essays, Macmillan, New York, pp. 165–171.
2. O.-J. Dahl, E. W. Dijkstra, C. A. R. Hoare Structured Programming, Academic Press, London, 1972 ISBN 0-12-200550-3, pp. 72–82.
3. Bo Bernhardsson (1991). "Explicit Solutions to the N-Queens Problem for All N". SIGART Bull. 2 (2): 7. doi:10.1145/122319.122322. S2CID 10644706.
4. E. J. Hoffman et al., "Construction for the Solutions of the m Queens Problem". Mathematics Magazine, Vol. XX (1969), pp. 66–72.
5. The Q27 Project
6. Sloman, Leila (21 September 2021). "Mathematician Answers Chess Problem About Attacking Queens". Quanta Magazine. Retrieved 22 September 2021.
7. Simkin, Michael (28 July 2021). "The number of $n$-queens configurations". arXiv:2107.13460v2 [math.CO].
8. Luria, Zur (15 May 2017). "New bounds on the number of n-queens configurations". arXiv:1705.05225v2 [math.CO].
9. Bowtell, Candida; Keevash, Peter (16 September 2021). "The $n$-queens problem". arXiv:2109.08083v1 [math.CO].
10. J. Barr and S. Rao (2006), The n-Queens Problem in Higher Dimensions, Elemente der Mathematik, vol 61 (4), pp. 133–137.
11. Martin S. Pearson. "Queens On A Chessboard – Beyond The 2nd Dimension" (php). Retrieved 27 January 2020.
12. Chatham, Doug (1 December 2018). "Reflections on the n +k dragon kings problem". Recreational Mathematics Magazine. 5 (10): 39–55. doi:10.2478/rmm-2018-0007.
13. G. Pólya, Uber die "doppelt-periodischen" Losungen des n-Damen-Problems, George Pólya: Collected papers Vol. IV, G-C. Rota, ed., MIT Press, Cambridge, London, 1984, pp. 237–247
14. "Queens on a Chessboard - Beyond the 2nd Dimension".
15. Burger, A. P.; Cockayne, E. J.; Mynhardt, C. M. (1997), "Domination and irredundance in the queens' graph", Discrete Mathematics, 163 (1–3): 47–66, doi:10.1016/0012-365X(95)00327-S, hdl:1828/2670, MR 1428557
16. Weakley, William D. (2018), "Queens around the world in twenty-five years", in Gera, Ralucca; Haynes, Teresa W.; Hedetniemi, Stephen T. (eds.), Graph Theory: Favorite Conjectures and Open Problems – 2, Problem Books in Mathematics, Cham: Springer, pp. 43–54, doi:10.1007/978-3-319-97686-0_5, MR 3889146
17. "Queens and knights problem". Archived from the original on 16 October 2005. Retrieved 20 September 2005.
18. Bell, Jordan; Stevens, Brett (2009). "A survey of known results and research areas for n-queens". Discrete Mathematics. 309 (1): 1–31. doi:10.1016/j.disc.2007.12.043.
19. O. Demirörs, N. Rafraf, and M.M. Tanik. Obtaining n-queens solutions from magic squares and constructing magic squares from n-queens solutions. Journal of Recreational Mathematics, 24:272–280, 1992
20. Gent, Ian P.; Jefferson, Christopher; Nightingale, Peter (August 2017). "Complexity of n-Queens Completion". Journal of Artificial Intelligence Research. 59: 815–848. doi:10.1613/jair.5512. ISSN 1076-9757. Retrieved 7 September 2017.
21. Glock, Stefan; Correia, David Munhá; Sudakov, Benny (6 July 2022). "The n-queens completion problem". Research in the Mathematical Sciences. 9 (41): 41. doi:10.1007/s40687-022-00335-1. PMC 9259550. PMID 35815227. S2CID 244478527.
22. A Polynomial Time Algorithm for the N-Queen Problem by Rok Sosic and Jun Gu, 1990. Describes run time for up to 500,000 Queens which was the max they could run due to memory constraints.
23. Minton, Steven; Johnston, Mark D.; Philips, Andrew B.; Laird, Philip (1 December 1992). "Minimizing conflicts: a heuristic repair method for constraint satisfaction and scheduling problems". Artificial Intelligence. 58 (1): 161–205. doi:10.1016/0004-3702(92)90007-K. ISSN 0004-3702. S2CID 14830518.
24. Sosic, R.; Gu, Jun (October 1994). "Efficient local search with conflict minimization: a case study of the n-queens problem". IEEE Transactions on Knowledge and Data Engineering. 6 (5): 661–668. doi:10.1109/69.317698. ISSN 1558-2191.
25. Qiu, Zongyan (February 2002). "Bit-vector encoding of n-queen problem". ACM SIGPLAN Notices. 37 (2): 68–70. doi:10.1145/568600.568613.
26. Richards, Martin (1997). Backtracking Algorithms in MCPL using Bit Patterns and Recursion (PDF) (Technical report). University of Cambridge Computer Laboratory. UCAM-CL-TR-433.
27. Wirth, Niklaus (1976). Algorithms + Data Structures = Programs. Bibcode:1976adsp.book.....W. ISBN 978-0-13-022418-7. {{cite book}}: |journal= ignored (help) p. 145
28. Wirth, Niklaus (2012) [orig. 2004]. "The Eight Queens Problem". Algorithms and Data Structures (PDF). Oberon version with corrections and authorized modifications. pp. 114–118.
29. DeMaria, Rusel (15 November 1993). The 7th Guest: The Official Strategy Guide (PDF). Prima Games. ISBN 978-1-5595-8468-5. Retrieved 22 April 2021.
30. "ナゾ130 クイーンの問題5". ゲームの匠 (in Japanese). Retrieved 17 September 2021.
Further reading
• Bell, Jordan; Stevens, Brett (2009). "A survey of known results and research areas for n-queens". Discrete Mathematics. 309 (1): 1–31. doi:10.1016/j.disc.2007.12.043.
• Watkins, John J. (2004). Across the Board: The Mathematics of Chess Problems. Princeton: Princeton University Press. ISBN 978-0-691-11503-0.
• Allison, L.; Yee, C.N.; McGaughey, M. (1988). "Three Dimensional NxN-Queens Problems". Department of Computer Science, Monash University, Australia.
• Nudelman, S. (1995). "The Modular N-Queens Problem in Higher Dimensions". Discrete Mathematics. 146 (1–3): 159–167. doi:10.1016/0012-365X(94)00161-5.
• Engelhardt, M. (August 2010). "Der Stammbaum der Lösungen des Damenproblems (in German, means The pedigree chart of solutions to the 8-queens problem". Spektrum der Wissenschaft: 68–71.
• On The Modular N-Queen Problem in Higher Dimensions, Ricardo Gomez, Juan Jose Montellano and Ricardo Strausz (2004), Instituto de Matematicas, Area de la Investigacion Cientifica, Circuito Exterior, Ciudad Universitaria, Mexico.
• Budd, Timothy (2002). "A Case Study: The Eight Queens Puzzle" (PDF). An Introduction to Object-Oriented Programming (3rd ed.). Addison Wesley Longman. pp. 125–145. ISBN 0-201-76031-2.
• Wirth, Niklaus (2004) [updated 2012]. "The Eight Queens Problem". Algorithms and Data Structures (PDF). Oberon version with corrections and authorized modifications. pp. 114–118.
External links
The Wikibook Algorithm Implementation has a page on the topic of: N-queens problem
• Weisstein, Eric W. "Queens Problem". MathWorld.
• queens-cpm on GitHub Eight Queens Puzzle in Turbo Pascal for CP/M
• eight-queens.py on GitHub Eight Queens Puzzle one line solution in Python
• Solutions in more than 100 different programming languages (on Rosetta Code)
Magic polygons
Types
• Magic circle
• Magic hexagon
• Magic hexagram
• Magic square
• Magic star
• Magic triangle
Related shapes
• Alphamagic square
• Antimagic square
• Geomagic square
• Heterosquare
• Pandiagonal magic square
• Most-perfect magic square
Higher dimensional shapes
• Magic cube
• classes
• Magic hypercube
• Magic hyperbeam
Classification
• Associative magic square
• Pandiagonal magic square
• Multimagic square
Related concepts
• Latin square
• Word square
• Number Scrabble
• Eight queens puzzle
• Magic constant
• Magic graph
• Magic series
| Wikipedia |
Octagon
In geometry, an octagon (from the Greek ὀκτάγωνον oktágōnon, "eight angles") is an eight-sided polygon or 8-gon.
Regular octagon
A regular octagon
TypeRegular polygon
Edges and vertices8
Schläfli symbol{8}, t{4}
Coxeter–Dynkin diagrams
Symmetry groupDihedral (D8), order 2×8
Internal angle (degrees)135°
PropertiesConvex, cyclic, equilateral, isogonal, isotoxal
Dual polygonSelf
A regular octagon has Schläfli symbol {8} [1] and can also be constructed as a quasiregular truncated square, t{4}, which alternates two types of edges. A truncated octagon, t{8} is a hexadecagon, {16}. A 3D analog of the octagon can be the rhombicuboctahedron with the triangular faces on it like the replaced edges, if one considers the octagon to be a truncated square.
Properties
The sum of all the internal angles of any octagon is 1080°. As with all polygons, the external angles total 360°.
If squares are constructed all internally or all externally on the sides of an octagon, then the midpoints of the segments connecting the centers of opposite squares form a quadrilateral that is both equidiagonal and orthodiagonal (that is, whose diagonals are equal in length and at right angles to each other).[2]: Prop. 9
The midpoint octagon of a reference octagon has its eight vertices at the midpoints of the sides of the reference octagon. If squares are constructed all internally or all externally on the sides of the midpoint octagon, then the midpoints of the segments connecting the centers of opposite squares themselves form the vertices of a square.[2]: Prop. 10
Regularity
A regular octagon is a closed figure with sides of the same length and internal angles of the same size. It has eight lines of reflective symmetry and rotational symmetry of order 8. A regular octagon is represented by the Schläfli symbol {8}. The internal angle at each vertex of a regular octagon is 135° ($\scriptstyle {\frac {3\pi }{4}}$ radians). The central angle is 45° ($\scriptstyle {\frac {\pi }{4}}$ radians).
Area
The area of a regular octagon of side length a is given by
$A=2\cot {\frac {\pi }{8}}a^{2}=2(1+{\sqrt {2}})a^{2}\approx 4.828\,a^{2}.$
In terms of the circumradius R, the area is
$A=4\sin {\frac {\pi }{4}}R^{2}=2{\sqrt {2}}R^{2}\approx 2.828\,R^{2}.$
In terms of the apothem r (see also inscribed figure), the area is
$A=8\tan {\frac {\pi }{8}}r^{2}=8({\sqrt {2}}-1)r^{2}\approx 3.314\,r^{2}.$
These last two coefficients bracket the value of pi, the area of the unit circle.
The area can also be expressed as
$\,\!A=S^{2}-a^{2},$
where S is the span of the octagon, or the second-shortest diagonal; and a is the length of one of the sides, or bases. This is easily proven if one takes an octagon, draws a square around the outside (making sure that four of the eight sides overlap with the four sides of the square) and then takes the corner triangles (these are 45–45–90 triangles) and places them with right angles pointed inward, forming a square. The edges of this square are each the length of the base.
Given the length of a side a, the span S is
$S={\frac {a}{\sqrt {2}}}+a+{\frac {a}{\sqrt {2}}}=(1+{\sqrt {2}})a\approx 2.414a.$
The span, then, is equal to the silver ratio times the side, a.
The area is then as above:
$A=((1+{\sqrt {2}})a)^{2}-a^{2}=2(1+{\sqrt {2}})a^{2}\approx 4.828a^{2}.$
Expressed in terms of the span, the area is
$A=2({\sqrt {2}}-1)S^{2}\approx 0.828S^{2}.$
Another simple formula for the area is
$\ A=2aS.$
More often the span S is known, and the length of the sides, a, is to be determined, as when cutting a square piece of material into a regular octagon. From the above,
$a\approx S/2.414.$
The two end lengths e on each side (the leg lengths of the triangles (green in the image) truncated from the square), as well as being $e=a/{\sqrt {2}},$ may be calculated as
$\,\!e=(S-a)/2.$
Circumradius and inradius
The circumradius of the regular octagon in terms of the side length a is[3]
$R=\left({\frac {\sqrt {4+2{\sqrt {2}}}}{2}}\right)a\approx 1.307a,$
and the inradius is
$r=\left({\frac {1+{\sqrt {2}}}{2}}\right)a\approx 1.207a.$
(that is one-half the silver ratio times the side, a, or one-half the span, S)
The inradius can be calculated from the circumradius as
$r=R\cos {\frac {\pi }{8}}$
Diagonality
The regular octagon, in terms of the side length a, has three different types of diagonals:
• Short diagonal;
• Medium diagonal (also called span or height), which is twice the length of the inradius;
• Long diagonal, which is twice the length of the circumradius.
The formula for each of them follows from the basic principles of geometry. Here are the formulas for their length:
• Short diagonal: $a{\sqrt {2+{\sqrt {2}}}}$ ;
• Medium diagonal: $(1+{\sqrt {2}})a$ ; (silver ratio times a)
• Long diagonal: $a{\sqrt {4+2{\sqrt {2}}}}$ .
Construction
A regular octagon at a given circumcircle may be constructed as follows:
1. Draw a circle and a diameter AOE, where O is the center and A, E are points on the circumcircle.
2. Draw another diameter GOC, perpendicular to AOE.
3. (Note in passing that A,C,E,G are vertices of a square).
4. Draw the bisectors of the right angles GOA and EOG, making two more diameters HOD and FOB.
5. A,B,C,D,E,F,G,H are the vertices of the octagon.
Octagon at a given circumcircle
Octagon at a given side length, animation
(The construction is very similar to that of hexadecagon at a given side length.)
A regular octagon can be constructed using a straightedge and a compass, as 8 = 23, a power of two:
The regular octagon can be constructed with meccano bars. Twelve bars of size 4, three bars of size 5 and two bars of size 6 are required.
Each side of a regular octagon subtends half a right angle at the centre of the circle which connects its vertices. Its area can thus be computed as the sum of eight isosceles triangles, leading to the result:
${\text{Area}}=2a^{2}({\sqrt {2}}+1)$
for an octagon of side a.
Standard coordinates
The coordinates for the vertices of a regular octagon centered at the origin and with side length 2 are:
• (±1, ±(1+√2))
• (±(1+√2), ±1).
Dissectibility
8-cube projection 24 rhomb dissection
Regular
Isotoxal
Coxeter states that every zonogon (a 2m-gon whose opposite sides are parallel and of equal length) can be dissected into m(m-1)/2 parallelograms.[4] In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the regular octagon, m=4, and it can be divided into 6 rhombs, with one example shown below. This decomposition can be seen as 6 of 24 faces in a Petrie polygon projection plane of the tesseract. The list (sequence A006245 in the OEIS) defines the number of solutions as eight, by the eight orientations of this one dissection. These squares and rhombs are used in the Ammann–Beenker tilings.
Regular octagon dissected
Tesseract
4 rhombs and 2 square
Skew
A skew octagon is a skew polygon with eight vertices and edges but not existing on the same plane. The interior of such an octagon is not generally defined. A skew zig-zag octagon has vertices alternating between two parallel planes.
A regular skew octagon is vertex-transitive with equal edge lengths. In three dimensions it is a zig-zag skew octagon and can be seen in the vertices and side edges of a square antiprism with the same D4d, [2+,8] symmetry, order 16.
Petrie polygons
The regular skew octagon is the Petrie polygon for these higher-dimensional regular and uniform polytopes, shown in these skew orthogonal projections of in A7, B4, and D5 Coxeter planes.
A7 D5 B4
7-simplex
5-demicube
16-cell
Tesseract
Symmetry
Symmetry
The eleven symmetries of a regular octagon. Lines of reflections are blue through vertices, purple through edges, and gyration orders are given in the center. Vertices are colored by their symmetry position.
The regular octagon has Dih8 symmetry, order 16. There are three dihedral subgroups: Dih4, Dih2, and Dih1, and four cyclic subgroups: Z8, Z4, Z2, and Z1, the last implying no symmetry.
Example octagons by symmetry
r16
d8
g8
p8
d4
g4
p4
d2
g2
p2
a1
On the regular octagon, there are eleven distinct symmetries. John Conway labels full symmetry as r16.[5] The dihedral symmetries are divided depending on whether they pass through vertices (d for diagonal) or edges (p for perpendiculars) Cyclic symmetries in the middle column are labeled as g for their central gyration orders. Full symmetry of the regular form is r16 and no symmetry is labeled a1.
The most common high symmetry octagons are p8, an isogonal octagon constructed by four mirrors can alternate long and short edges, and d8, an isotoxal octagon constructed with equal edge lengths, but vertices alternating two different internal angles. These two forms are duals of each other and have half the symmetry order of the regular octagon.
Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the g8 subgroup has no degrees of freedom but can seen as directed edges.
Use
The octagonal shape is used as a design element in architecture. The Dome of the Rock has a characteristic octagonal plan. The Tower of the Winds in Athens is another example of an octagonal structure. The octagonal plan has also been in church architecture such as St. George's Cathedral, Addis Ababa, Basilica of San Vitale (in Ravenna, Italia), Castel del Monte (Apulia, Italia), Florence Baptistery, Zum Friedefürsten Church (Germany) and a number of octagonal churches in Norway. The central space in the Aachen Cathedral, the Carolingian Palatine Chapel, has a regular octagonal floorplan. Uses of octagons in churches also include lesser design elements, such as the octagonal apse of Nidaros Cathedral.
Architects such as John Andrews have used octagonal floor layouts in buildings for functionally separating office areas from building services, such as in the Intelsat Headquarters of Washington or Callam Offices in Canberra.
• Umbrellas often have an octagonal outline.
• The famous Bukhara rug design incorporates an octagonal "elephant's foot" motif.
• The street & block layout of Barcelona's Eixample district is based on non-regular octagons
• Janggi uses octagonal pieces.
• Japanese lottery machines often have octagonal shape.
• Stop sign used in English-speaking countries, as well as in most European countries
• An icon of a stop sign with a hand at the middle.
• The trigrams of the Taoist bagua are often arranged octagonally
• Famous octagonal gold cup from the Belitung shipwreck
• Classes at Shimer College are traditionally held around octagonal tables
• The Labyrinth of the Reims Cathedral with a quasi-octagonal shape.
• The movement of the analog stick(s) of the Nintendo 64 controller, the GameCube controller, the Wii Nunchuk and the Classic Controller is restricted by a rotated octagonal area, allowing the stick to move in only eight different directions.
Derived figures
• The truncated square tiling has 2 octagons around every vertex.
• An octagonal prism contains two octagonal faces.
• An octagonal antiprism contains two octagonal faces.
• The truncated cuboctahedron contains 6 octagonal faces.
• The omnitruncated cubic honeycomb
Related polytopes
The octagon, as a truncated square, is first in a sequence of truncated hypercubes:
Truncated hypercubes
Image ...
Name Octagon Truncated cube Truncated tesseract Truncated 5-cube Truncated 6-cube Truncated 7-cube Truncated 8-cube
Coxeter diagram
Vertex figure ( )v( )
( )v{ }
( )v{3}
( )v{3,3}
( )v{3,3,3} ( )v{3,3,3,3} ( )v{3,3,3,3,3}
As an expanded square, it is also first in a sequence of expanded hypercubes:
Expanded hypercubes
...
Octagon Rhombicuboctahedron Runcinated tesseract Stericated 5-cube Pentellated 6-cube Hexicated 7-cube Heptellated 8-cube
See also
• Bumper pool
• Hansen's small octagon
• Octagon house
• Octagonal number
• Octagram
• Octahedron, 3D shape with eight faces.
• Oktogon, a major intersection in Budapest, Hungary
• Rub el Hizb (also known as Al Quds Star and as Octa Star)
• Smoothed octagon
References
1. Wenninger, Magnus J. (1974), Polyhedron Models, Cambridge University Press, p. 9, ISBN 9780521098595.
2. Dao Thanh Oai (2015), "Equilateral triangles and Kiepert perspectors in complex numbers", Forum Geometricorum 15, 105--114. http://forumgeom.fau.edu/FG2015volume15/FG201509index.html
3. Weisstein, Eric. "Octagon." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Octagon.html
4. Coxeter, Mathematical recreations and Essays, Thirteenth edition, p.141
5. John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, (2008) The Symmetries of Things, ISBN 978-1-56881-220-5 (Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275-278)
External links
Look up octagon in Wiktionary, the free dictionary.
• Octagon Calculator
• Definition and properties of an octagon With interactive animation
Polygons (List)
Triangles
• Acute
• Equilateral
• Ideal
• Isosceles
• Kepler
• Obtuse
• Right
Quadrilaterals
• Antiparallelogram
• Bicentric
• Crossed
• Cyclic
• Equidiagonal
• Ex-tangential
• Harmonic
• Isosceles trapezoid
• Kite
• Orthodiagonal
• Parallelogram
• Rectangle
• Right kite
• Right trapezoid
• Rhombus
• Square
• Tangential
• Tangential trapezoid
• Trapezoid
By number
of sides
1–10 sides
• Monogon (1)
• Digon (2)
• Triangle (3)
• Quadrilateral (4)
• Pentagon (5)
• Hexagon (6)
• Heptagon (7)
• Octagon (8)
• Nonagon (Enneagon, 9)
• Decagon (10)
11–20 sides
• Hendecagon (11)
• Dodecagon (12)
• Tridecagon (13)
• Tetradecagon (14)
• Pentadecagon (15)
• Hexadecagon (16)
• Heptadecagon (17)
• Octadecagon (18)
• Icosagon (20)
>20 sides
• Icositrigon (23)
• Icositetragon (24)
• Triacontagon (30)
• 257-gon
• Chiliagon (1000)
• Myriagon (10,000)
• 65537-gon
• Megagon (1,000,000)
• Apeirogon (∞)
Star polygons
• Pentagram
• Hexagram
• Heptagram
• Octagram
• Enneagram
• Decagram
• Hendecagram
• Dodecagram
Classes
• Concave
• Convex
• Cyclic
• Equiangular
• Equilateral
• Infinite skew
• Isogonal
• Isotoxal
• Magic
• Pseudotriangle
• Rectilinear
• Regular
• Reinhardt
• Simple
• Skew
• Star-shaped
• Tangential
• Weakly simple
| Wikipedia |
9-orthoplex
In geometry, a 9-orthoplex or 9-cross polytope, is a regular 9-polytope with 18 vertices, 144 edges, 672 triangle faces, 2016 tetrahedron cells, 4032 5-cells 4-faces, 5376 5-simplex 5-faces, 4608 6-simplex 6-faces, 2304 7-simplex 7-faces, and 512 8-simplex 8-faces.
Regular 9-orthoplex
Ennecross
Orthogonal projection
inside Petrie polygon
TypeRegular 9-polytope
Familyorthoplex
Schläfli symbol{37,4}
{36,31,1}
Coxeter-Dynkin diagrams
8-faces512 {37}
7-faces2304 {36}
6-faces4608 {35}
5-faces5376 {34}
4-faces4032 {33}
Cells2016 {3,3}
Faces672 {3}
Edges144
Vertices18
Vertex figureOctacross
Petrie polygonOctadecagon
Coxeter groupsC9, [37,4]
D9, [36,1,1]
Dual9-cube
Propertiesconvex, Hanner polytope
It has two constructed forms, the first being regular with Schläfli symbol {37,4}, and the second with alternately labeled (checkerboarded) facets, with Schläfli symbol {36,31,1} or Coxeter symbol 611.
It is one of an infinite family of polytopes, called cross-polytopes or orthoplexes. The dual polytope is the 9-hypercube or enneract.
Alternate names
• Enneacross, derived from combining the family name cross polytope with ennea for nine (dimensions) in Greek
• Pentacosidodecayotton as a 512-facetted 9-polytope (polyyotton)
Construction
There are two Coxeter groups associated with the 9-orthoplex, one regular, dual of the enneract with the C9 or [4,37] symmetry group, and a lower symmetry with two copies of 8-simplex facets, alternating, with the D9 or [36,1,1] symmetry group.
Cartesian coordinates
Cartesian coordinates for the vertices of a 9-orthoplex, centered at the origin, are
(±1,0,0,0,0,0,0,0,0), (0,±1,0,0,0,0,0,0,0), (0,0,±1,0,0,0,0,0,0), (0,0,0,±1,0,0,0,0,0), (0,0,0,0,±1,0,0,0,0), (0,0,0,0,0,±1,0,0,0), (0,0,0,0,0,0,±1,0,0), (0,0,0,0,0,0,0,±1,0), (0,0,0,0,0,0,0,0,±1)
Every vertex pair is connected by an edge, except opposites.
Images
orthographic projections
B9 B8 B7
[18] [16] [14]
B6 B5
[12] [10]
B4 B3 B2
[8] [6] [4]
A7 A5 A3
— — —
[8] [6] [4]
References
• H.S.M. Coxeter:
• H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
• Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6
• (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
• (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
• (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
• Norman Johnson Uniform Polytopes, Manuscript (1991)
• N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D.
• Klitzing, Richard. "9D uniform polytopes (polyyotta) x3o3o3o3o3o3o3o4o - vee".
External links
• Olshevsky, George. "Cross polytope". Glossary for Hyperspace. Archived from the original on 4 February 2007.
• Polytopes of Various Dimensions
• Multi-dimensional Glossary
Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron
Uniform polychoron Pentachoron 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell
Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221
Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321
Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421
Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds
| Wikipedia |
97.5th percentile point
In probability and statistics, the 97.5th percentile point of the standard normal distribution is a number commonly used for statistical calculations. The approximate value of this number is 1.96, meaning that 95% of the area under a normal curve lies within approximately 1.96 standard deviations of the mean. Because of the central limit theorem, this number is used in the construction of approximate 95% confidence intervals. Its ubiquity is due to the arbitrary but common convention of using confidence intervals with 95% probability in science and frequentist statistics, though other probabilities (90%, 99%, etc.) are sometimes used.[1][2][3][4] This convention seems particularly common in medical statistics,[5][6][7] but is also common in other areas of application, such as earth sciences,[8] social sciences and business research.[9]
There is no single accepted name for this number; it is also commonly referred to as the "standard normal deviate", "normal score" or "Z score" for the 97.5 percentile point, the .975 point, or just its approximate value, 1.96.
If X has a standard normal distribution, i.e. X ~ N(0,1),
$\mathrm {P} (X>1.96)\approx 0.025,\,$
$\mathrm {P} (X<1.96)\approx 0.975,\,$
and as the normal distribution is symmetric,
$\mathrm {P} (-1.96<X<1.96)\approx 0.95.\,$
One notation for this number is z.975.[10] From the probability density function of the standard normal distribution, the exact value of z.975 is determined by
${\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{z_{.975}}e^{-x^{2}/2}\,\mathrm {d} x=0.975.$
History
The use of this number in applied statistics can be traced to the influence of Ronald Fisher's classic textbook, Statistical Methods for Research Workers, first published in 1925:
"The value for which P = .05, or 1 in 20, is 1.96 or nearly 2 ; it is convenient to take this point as a limit in judging whether a deviation is to be considered significant or not."[11]
In Table 1 of the same work, he gave the more precise value 1.959964.[12] In 1970, the value truncated to 20 decimal places was calculated to be
1.95996 39845 40054 23552...[13][14]
The commonly used approximate value of 1.96 is therefore accurate to better than one part in 50,000, which is more than adequate for applied work.
Some people even use the value of 2 in the place of 1.96, reporting a 95.4% confidence interval as a 95% confidence interval. This is not recommended but is occasionally seen.[15]
Software functions
The inverse of the standard normal CDF can be used to compute the value. The following is a table of function calls that return 1.96 in some commonly used applications:
Application Function call
Excel NORM.S.INV(0.975)
MATLAB norminv(0.975)
R qnorm(0.975)
Python (SciPy) scipy.stats.norm.ppf(0.975)
SAS probit(0.025);
SPSS x = COMPUTE IDF.NORMAL(0.975,0,1).
Stata invnormal(0.975)
Wolfram Language (Mathematica) InverseCDF[NormalDistribution[0, 1], 0.975][16][17]
See also
• Margin of error
• Probit
• Reference range
• Standard error (statistics)
• 68–95–99.7 rule
References
1. Rees, DG (1987), Foundations of Statistics, CRC Press, p. 246, ISBN 0-412-28560-6, Why 95% confidence? Why not some other confidence level? The use of 95% is partly convention, but levels such as 90%, 98% and sometimes 99.9% are also used.
2. "Engineering Statistics Handbook: Confidence Limits for the Mean". National Institute of Standards and Technology. Archived from the original on 5 February 2008. Retrieved 4 February 2008. Although the choice of confidence coefficient is somewhat arbitrary, in practice 90%, 95%, and 99% intervals are often used, with 95% being the most commonly used.
3. Olson, Eric T; Olson, Tammy Perry (2000), Real-Life Math: Statistics, Walch Publishing, p. 66, ISBN 0-8251-3863-9, While other stricter, or looser, limits may be chosen, the 95 percent interval is very often preferred by statisticians.
4. Swift, MB (2009). "Comparison of Confidence Intervals for a Poisson Mean - Further Considerations". Communications in Statistics - Theory and Methods. 38 (5): 748–759. doi:10.1080/03610920802255856. S2CID 120748700. In modern applied practice, almost all confidence intervals are stated at the 95% level.
5. Simon, Steve (2002), Why 95% confidence limits?, archived from the original on 28 January 2008, retrieved 1 February 2008
6. Moher, D; Schulz, KF; Altman, DG (2001), "The CONSORT statement: revised recommendations for improving the quality of reports of parallel-group randomised trials.", Lancet, 357 (9263): 1191–1194, doi:10.1016/S0140-6736(00)04337-3, PMID 11323066, S2CID 52871971, retrieved 4 February 2008
7. "Resources for Authors: Research". BMJ Publishing Group Ltd. Archived from the original on 18 July 2009. Retrieved 2008-02-04. For standard original research articles please provide the following headings and information: [...] results - main results with (for quantitative studies) 95% confidence intervals and, where appropriate, the exact level of statistical significance and the number need to treat/harm
8. Borradaile, Graham J. (2003), Statistics of Earth Science Data, Springer, p. 79, ISBN 3-540-43603-0, For simplicity, we adopt the common earth sciences convention of a 95% confidence interval.
9. Cook, Sarah (2004), Measuring Customer Service Effectiveness, Gower Publishing, p. 24, ISBN 0-566-08538-0, Most researchers use a 95 per cent confidence interval
10. Gosling, J. (1995), Introductory Statistics, Pascal Press, pp. 78–9, ISBN 1-86441-015-9
11. Fisher, Ronald (1925), Statistical Methods for Research Workers, Edinburgh: Oliver and Boyd, p. 47, ISBN 0-05-002170-2
12. Fisher, Ronald (1925), Statistical Methods for Research Workers, Edinburgh: Oliver and Boyd, ISBN 0-05-002170-2, Table 1
13. White, John S. (June 1970), "Tables of Normal Percentile Points", Journal of the American Statistical Association, American Statistical Association, 65 (330): 635–638, doi:10.2307/2284575, JSTOR 2284575
14. Sloane, N. J. A. (ed.). "Sequence A220510". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
15. "Estimating the Population Mean Using Intervals". stat.wmich.edu. Statistical Computation Lab. Archived from the original on 4 July 2018. Retrieved 7 August 2018.
16. InverseCDF, Wolfram Language Documentation Center.
17. NormalDistribution, Wolfram Language Documentation Center.
Further reading
• Gardner, Martin J; Altman, Douglas G, eds. (1989), Statistics with confidence, BMJ Books, ISBN 978-0-7279-0222-1
| Wikipedia |
99 Points of Intersection
99 Points of Intersection: Examples—Pictures—Proofs is a book on constructions in Euclidean plane geometry in which three or more lines or curves meet in a single point of intersection. This book was originally written in German by Hans Walser as 99 Schnittpunkte (Eagle / Ed. am Gutenbergplatz, 2004),[1] translated into English by Peter Hilton and Jean Pedersen, and published by the Mathematical Association of America in 2006 in their MAA Spectrum series (ISBN 978-0-88385-553-9).[2]
99 Points of Intersection
First edition (German)
AuthorHans Walser
Original title99 Schnittpunkte
Translators
• Peter Hilton
• Jean Pedersen
LanguageGerman
SubjectEuclidean plane geometry
Publisher
• Eagle / Ed. am Gutenbergplatz
• Mathematical Association of America
Publication date
2004, 2006
Topics and organization
The book is organized into three sections.[2][3] The first section provides introductory material, describing different mathematical situations in which multiple curves might meet, and providing different possible explanations for this phenomenon, including symmetry, geometric transformations, and membership of the curves in a pencil of curves.[4] The second section shows the 99 points of intersection of the title. Each is given on its own page, as a large figure with three smaller figures showing its construction, with a one-line caption but no explanatory text.[2][3] The third section provides background material and proofs for some of these points of intersection,[3] as well as extending and generalizing some of these results.[5]
Some of these points of intersection are standard; for instance, these include the construction of the centroid of a triangle as the point where its three median lines meet, the construction of the orthocenter as the point where the three altitudes meet, and the construction of the circumcenter as the point where the three perpendicular bisectors of the sides meet,[6] as well as two versions of Ceva's theorem.[4] However, others are new to this book,[2] and include intersections related to silver rectangles, tangent circles, the Pythagorean theorem, and the nine-point hyperbola.[4]
Audience
John Jensen writes that "the clear and uncluttered illustrations of intersection make for a rich source for geometric investigation by high school geometry students".[5] And although Gerry Leversha calls the book "eccentric" and states that it "is clearly nothing to do with any syllabus anywhere",[4] Jensen suggests that its examples would make a good complement to coursework both in exploratory geometry using interactive geometry software and in a geometry course focused on the formal proof of geometry propositions. He adds that the book itself is a proof of the possibility of presenting geometry without detailed explanations, and of introducing students to the beauty of the subject.[5]
References
1. Werner, Dirk, Review of 99 Schnittpunkte (in German), German Mathematical Society
2. Poplicher, Mihaela (September 2006), "Review of 99 Points of Intersection", MAA Reviews
3. Ashbacher, Charles (2004–2005), "Review of 99 Points of Intersection", Journal of Recreational Mathematics, 33 (3): 215–216
4. Leversha, Gerry (November 2008), "Review of 99 Points of Intersection", The Mathematical Gazette, 92 (525): 588–589, doi:10.1017/S0025557200184074, JSTOR 27821873, S2CID 185487968
5. Jensen, John (March 2007), "Review of 99 Points of Intersection", The Mathematics Teacher, 100 (7): 511–512, JSTOR 27972312
6. Coupland, Mary (2006), "Review of 99 Points of Intersection", Australian Mathematics Teacher, 62 (3): 32
External links
• Schnittpunkte, web site with a larger collection of points of intersection, by Hans Walser
| Wikipedia |
A/B testing
A/B testing (also known as bucket testing, split-run testing, or split testing) is a user experience research methodology.[1] A/B tests consist of a randomized experiment that usually involves two variants (A and B),[2][3][4] although the concept can be also extended to multiple variants of the same variable. It includes application of statistical hypothesis testing or "two-sample hypothesis testing" as used in the field of statistics. A/B testing is a way to compare multiple versions of a single variable, for example by testing a subject's response to variant A against variant B, and determining which of the variants is more effective.[5]
Overview
"A/B testing" is a shorthand for a simple randomized controlled experiment, in which a number of samples (e.g. A and B) of a single vector-variable are compared.[1] These values are similar except for one variation which might affect a user's behavior. A/B tests are widely considered the simplest form of controlled experiment, especially when they only involve two variants. However, by adding more variants to the test, its complexity grows.[6]
A/B tests are useful for understanding user engagement and satisfaction of online features like a new feature or product.[7] Large social media sites like LinkedIn, Facebook, and Instagram use A/B testing to make user experiences more successful and as a way to streamline their services.[7]
Today, A/B tests are being used also for conducting complex experiments on subjects such as network effects when users are offline, how online services affect user actions, and how users influence one another.[7] A/B testing is used by data engineers, marketers, designers, software engineers, and entrepreneurs, among others.[8] Many positions rely on the data from A/B tests, as they allow companies to understand growth, increase revenue, and optimize customer satisfaction.[8]
Version A might be used at present (thus forming the control group), while version B is modified in some respect vs. A (thus forming the treatment group). For instance, on an e-commerce website the purchase funnel is typically a good candidate for A/B testing, since even marginal-decreases in drop-off rates can represent a significant gain in sales. Significant improvements can be sometimes seen through testing elements like copy text, layouts, images and colors,[9] but not always. In these tests, users only see one of two versions, since the goal is to discover which of the two versions is preferable.[10]
Multivariate testing or multinomial testing is similar to A/B testing, but may test more than two versions at the same time or use more controls. Simple A/B tests are not valid for observational, quasi-experimental or other non-experimental situations - commonplace with survey data, offline data, and other, more complex phenomena.
A/B testing is claimed by some to be a change in philosophy and business-strategy in certain niches, though the approach is identical to a between-subjects design, which is commonly used in a variety of research traditions.[11][12][13] A/B testing as a philosophy of web development brings the field into line with a broader movement toward evidence-based practice. The benefits of A/B testing are considered to be that it can be performed continuously on almost anything, especially since most marketing automation software now typically comes with the ability to run A/B tests on an ongoing basis.
Common test statistics
"Two-sample hypothesis tests" are appropriate for comparing the two samples where the samples are divided by the two control cases in the experiment. Z-tests are appropriate for comparing means under stringent conditions regarding normality and a known standard deviation. Student's t-tests are appropriate for comparing means under relaxed conditions when less is assumed. Welch's t test assumes the least and is therefore the most commonly used test in a two-sample hypothesis test where the mean of a metric is to be optimized. While the mean of the variable to be optimized is the most common choice of estimator, others are regularly used.
For a comparison of two binomial distributions such as a click-through rate one would use Fisher's exact test.
Assumed distributionExample caseStandard testAlternative test
GaussianAverage revenue per userWelch's t-test (Unpaired t-test)Student's t-test
BinomialClick-through rateFisher's exact testBarnard's test
PoissonTransactions per paying userE-test[14]C-test
MultinomialNumber of each product purchasedChi-squared testG-test
UnknownMann–Whitney U testGibbs sampling
Challenges
When conducting A/B testing, the user should evaluate the pros and cons of it to see if it aligns best with the results that they're hoping for.
Pros: Through A/B testing, it is easy to get a clear idea of what users prefer, since it is directly testing one thing over the other. It is based on real user behavior so the data can be very helpful especially when determining what works better between two options. In addition, it can also provide answers to very specific design questions. One example of this is Google's A/B testing with hyperlink colors. In order to optimize revenue, they tested dozens of different hyperlink hues to see which color the users tend to click more on.
Cons: There are, however, a couple of cons to A/B testing. Like mentioned above, A/B testing is good for specific design questions but it can also be a downside since it is mostly only good for specific design problems with very measurable outcomes. It could also be a very costly and timely process. Depending on the size of the company and/or team, there could be a lot of meetings and discussions about what exactly to test and what the impact of the A/B test is. If there's not a significant impact, it could end up as a waste of time and resources.
In December 2018, representatives with experience in large-scale A/B testing from thirteen different organizations (Airbnb, Amazon, Booking.com, Facebook, Google, LinkedIn, Lyft, Microsoft, Netflix, Twitter, Uber, and Stanford University) attended a summit and summarized the top challenges in a SIGKDD Explorations paper.[15] The challenges can be grouped into four areas: Analysis, Engineering and Culture, Deviations from Traditional A/B tests, and Data quality.
History
It is difficult to definitively establish when A/B testing was first used. The first randomized double-blind trial, to assess the effectiveness of a homeopathic drug, occurred in 1835.[16] Experimentation with advertising campaigns, which has been compared to modern A/B testing, began in the early twentieth century.[17] The advertising pioneer Claude Hopkins used promotional coupons to test the effectiveness of his campaigns. However, this process, which Hopkins described in his Scientific Advertising, did not incorporate concepts such as statistical significance and the null hypothesis, which are used in statistical hypothesis testing.[18] Modern statistical methods for assessing the significance of sample data were developed separately in the same period. This work was done in 1908 by William Sealy Gosset when he altered the Z-test to create Student's t-test.[19][20]
With the growth of the internet, new ways to sample populations have become available. Google engineers ran their first A/B test in the year 2000 in an attempt to determine what the optimum number of results to display on its search engine results page would be.[5] The first test was unsuccessful due to glitches that resulted from slow loading times. Later A/B testing research would be more advanced, but the foundation and underlying principles generally remain the same, and in 2011, 11 years after Google's first test, Google ran over 7,000 different A/B tests.[5]
In 2012, a Microsoft employee working on the search engine Microsoft Bing created an experiment to test different ways of displaying advertising headlines. Within hours, the alternative format produced a revenue increase of 12% with no impact on user-experience metrics.[4] Today, companies like Microsoft and Google each conduct over 10,000 A/B tests annually.[4]
Many companies now use the "designed experiment" approach to making marketing decisions, with the expectation that relevant sample results can improve positive conversion results. It is an increasingly common practice as the tools and expertise grow in this area.[21]
Examples
Email marketing
A company with a customer database of 2,000 people decides to create an email campaign with a discount code in order to generate sales through its website. It creates two versions of the email with different call to action (the part of the copy which encourages customers to do something — in the case of a sales campaign, make a purchase) and identifying promotional code.
• To 1,000 people it sends the email with the call to action stating, "Offer ends this Saturday! Use code A1",
• and to another 1,000 people it sends the email with the call to action stating, "Offer ends soon! Use code B1".
All other elements of the emails' copy and layout are identical. The company then monitors which campaign has the higher success rate by analyzing the use of the promotional codes. The email using the code A1 has a 5% response rate (50 of the 1,000 people emailed used the code to buy a product), and the email using the code B1 has a 3% response rate (30 of the recipients used the code to buy a product). The company therefore determines that in this instance, the first Call To Action is more effective and will use it in future sales. A more nuanced approach would involve applying statistical testing to determine if the differences in response rates between A1 and B1 were statistically significant (that is, highly likely that the differences are real, repeatable, and not due to random chance).[22]
In the example above, the purpose of the test is to determine which is the more effective way to encourage customers to make a purchase. If, however, the aim of the test had been to see which email would generate the higher click-rate – that is, the number of people who actually click onto the website after receiving the email – then the results might have been different.
For example, even though more of the customers receiving the code B1 accessed the website, because the Call To Action didn't state the end-date of the promotion many of them may feel no urgency to make an immediate purchase. Consequently, if the purpose of the test had been simply to see which email would bring more traffic to the website, then the email containing code B1 might well have been more successful. An A/B test should have a defined outcome that is measurable such as number of sales made, click-rate conversion, or number of people signing up/registering.[23]
A/B testing for product pricing
A/B testing can be used to determine the right price for the product, as this is perhaps one of the most difficult tasks when a new product or service is launched. A/B testing (especially valid for digital goods) is an excellent way to find out which price-point and offering maximize the total revenue.
Political A/B testing
A/B tests have also been used by political campaigns. In 2007, Barack Obama's presidential campaign used A/B testing as a way to garner online attraction and understand what voters wanted to see from the presidential candidate.[24] For example, Obama's team tested four distinct buttons on their website that led users to sign up for newsletters. Additionally, the team used six different accompanying images to draw in users. Through A/B testing, staffers were able to determine how to effectively draw in voters and garner additional interest.[24]
HTTP Routing and API feature testing
A/B testing is very common when deploying a newer version of an API.[25] For real-time user experience testing, an HTTP Layer-7 Reverse proxy is configured in such a way that, N% of the HTTP traffic goes into the newer version of the backend instance, while the remaining 100-N% of HTTP traffic hits the (stable) older version of the backend HTTP application service.[25] This is usually done for limiting the exposure of customers to a newer backend instance such that, if there is a bug on the newer version, only N% of the total user agents or clients get affected while others get routed to a stable backend, which is a common ingress control mechanism.[25]
Segmentation and targeting
A/B tests most commonly apply the same variant (e.g., user interface element) with equal probability to all users. However, in some circumstances, responses to variants may be heterogeneous. That is, while a variant A might have a higher response rate overall, variant B may have an even higher response rate within a specific segment of the customer base.[26]
For instance, in the above example, the breakdown of the response rates by gender could have been:
GenderOverallMenWomen
Total sends 2,0001,0001,000
Total responses 803545
Variant A 50/ 1,000 (5%)10/ 500 (2%)40/ 500 (8%)
Variant B 30/ 1,000 (3%)25/ 500 (5%)5/ 500 (1%)
In this case, we can see that while variant A had a higher response rate overall, variant B actually had a higher response rate with men.
As a result, the company might select a segmented strategy as a result of the A/B test, sending variant B to men and variant A to women in the future. In this example, a segmented strategy would yield an increase in expected response rates from $ 5\%={\frac {40+10}{500+500}}$ to $ 6.5\%={\frac {40+25}{500+500}}$ – constituting a 30% increase.
If segmented results are expected from the A/B test, the test should be properly designed at the outset to be evenly distributed across key customer attributes, such as gender. That is, the test should both (a) contain a representative sample of men vs. women, and (b) assign men and women randomly to each “variant” (variant A vs. variant B). Failure to do so could lead to experiment bias and inaccurate conclusions to be drawn from the test.[27]
This segmentation and targeting approach can be further generalized to include multiple customer attributes rather than a single customer attribute – for example, customers' age and gender – to identify more nuanced patterns that may exist in the test results.
See also
• Adaptive control
• Between-group design experiment
• Choice modelling
• Multi-armed bandit
• Multivariate testing
• Randomized controlled trial
• Scientific control
• Test statistic
References
1. Young, Scott W. H. (August 2014). "Improving Library User Experience with A/B Testing: Principles and Process". Weave: Journal of Library User Experience. 1 (1). doi:10.3998/weave.12535642.0001.101. hdl:2027/spo.12535642.0001.101.
2. Kohavi, Ron; Xu, Ya; Tang, Diane (2000). Trustworthy Online Controlled Experiments: A Practical Guide to A/B Testing. Cambridge University Press.
3. Kohavi, Ron; Longbotham, Roger (2023). "Online Controlled Experiments and A/B Tests". In Phung, Dinh; Webb, Geoff; Sammut, Claude (eds.). Encyclopedia of Machine Learning and Data Science. Springer.
4. Kohavi, Ron; Thomke, Stefan (September 2017). "The Surprising Power of Online Experiments". Harvard Business Review: 74–82.
5. "The ABCs of A/B Testing - Pardot". Pardot. 12 July 2012. Retrieved 2016-02-21.
6. Kohavi, Ron; Longbotham, Roger (2017). "Online Controlled Experiments and A/B Testing". Encyclopedia of Machine Learning and Data Mining. pp. 922–929. doi:10.1007/978-1-4899-7687-1_891. ISBN 978-1-4899-7685-7.
7. Xu, Ya; Chen, Nanyu; Fernandez, Addrian; Sinno, Omar; Bhasin, Anmol (10 August 2015). "From Infrastructure to Culture: A/B Testing Challenges in Large Scale Social Networks". Proceedings of the 21th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining. pp. 2227–2236. doi:10.1145/2783258.2788602. ISBN 9781450336642. S2CID 15847833.
8. Siroker, Dan; Koomen, Pete (2013-08-07). A / B Testing: The Most Powerful Way to Turn Clicks Into Customers. John Wiley & Sons. ISBN 978-1-118-65920-5.
9. "Split Testing Guide for Online Stores". webics.com.au. August 27, 2012. Retrieved 2012-08-28.
10. Kaufman, Emilie (2014). "On the Complexity of A/B Testing" (PDF). 35. arXiv:1405.3224. Bibcode:2014arXiv1405.3224K – via JMLR: Workshop and Conference Proceedings. {{cite journal}}: Cite journal requires |journal= (help)
11. Christian, Brian (2000-02-27). "The A/B Test: Inside the Technology That's Changing the Rules of Business | Wired Business". Wired.com. Retrieved 2014-03-18.
12. Christian, Brian. "Test Everything: Notes on the A/B Revolution | Wired Enterprise". Wired. Retrieved 2014-03-18.
13. Cory Doctorow (2012-04-26). "A/B testing: the secret engine of creation and refinement for the 21st century". Boing Boing. Retrieved 2014-03-18.
14. Krishnamoorthy, K.; Thomson, Jessica (2004). "A more powerful test for comparing two Poisson means". Journal of Statistical Planning and Inference. 119: 23–35. doi:10.1016/S0378-3758(02)00408-1. S2CID 26753532.
15. Gupta, Somit; Kohavi, Ronny; Tang, Diane; Xu, Ya; Andersen, Reid; Bakshy, Eytan; Cardin, Niall; Chandran, Sumitha; Chen, Nanyu; Coey, Dominic; Curtis, Mike; Deng, Alex; Duan, Weitao; Forbes, Peter; Frasca, Brian; Guy, Tommy; Imbens, Guido W.; Saint Jacques, Guillaume; Kantawala, Pranav; Katsev, Ilya; Katzwer, Moshe; Konutgan, Mikael; Kunakova, Elena; Lee, Minyong; Lee, MJ; Liu, Joseph; McQueen, James; Najmi, Amir; Smith, Brent; Trehan, Vivek; Vermeer, Lukas; Walker, Toby; Wong, Jeffrey; Yashkov, Igor (June 2019). "Top Challenges from the first Practical Online Controlled Experiments Summit". SIGKDD Explorations. 21 (1): 20–35. doi:10.1145/3331651.3331655. S2CID 153314606.
16. Stolberg, M (December 2006). "Inventing the randomized double-blind trial: the Nuremberg salt test of 1835". Journal of the Royal Society of Medicine. 99 (12): 642–643. doi:10.1177/014107680609901216. PMC 1676327. PMID 17139070.
17. "What is A/B Testing." Convertize. Retrieved 2020-01-28.
18. "Claude Hopkins Turned Advertising Into A Science." Retrieved 2019-11-01.
19. "Brief history and background for the one sample t-test". 20 June 2007.
20. Box, Joan Fisher (1987). "Guinness, Gosset, Fisher, and Small Samples". Statistical Science. 2 (1): 45–52. doi:10.1214/ss/1177013437.
21. "A/B Testing: The ABCs of Paid Social Media". Anyword. 2020-01-17. Retrieved 2022-04-08.
22. Amazon.com. "The Math Behind A/B Testing". Archived from the original on 2015-09-21. Retrieved 2015-04-12.
23. Kohavi, Ron; Longbotham, Roger; Sommerfield, Dan; Henne, Randal M. (February 2009). "Controlled experiments on the web: survey and practical guide". Data Mining and Knowledge Discovery. 18 (1): 140–181. doi:10.1007/s10618-008-0114-1. S2CID 17165746.
24. Siroker, Dan; Koomen, Pete (2013-08-07). A / B Testing: The Most Powerful Way to Turn Clicks Into Customers. John Wiley & Sons. ISBN 978-1-118-65920-5.
25. Szucs, Sandor (2018). "Modern HTTP Routing" (PDF). Usenix.org.
26. "Advanced A/B Testing Tactics That You Should Know | Testing & Usability". Online-behavior.com. Archived from the original on 2014-03-19. Retrieved 2014-03-18.
27. "Eight Ways You've Misconfigured Your A/B Test". Dr. Jason Davis. 2013-09-12. Retrieved 2014-03-18.
Software testing
The "box" approach
• Black-box testing
• All-pairs testing
• Exploratory testing
• Fuzz testing
• Model-based testing
• Scenario testing
• Grey-box testing
• White-box testing
• API testing
• Mutation testing
• Static testing
Testing levels
• Acceptance testing
• Integration testing
• System testing
• Unit testing
Testing types, techniques,
and tactics
• A/B testing
• Benchmark
• Compatibility testing
• Concolic testing
• Concurrent testing
• Conformance testing
• Continuous testing
• Destructive testing
• Development testing
• Dynamic program analysis
• Installation testing
• Random testing
• Regression testing
• Security testing
• Smoke testing (software)
• Software performance testing
• Symbolic execution
• Test automation
• Usability testing
See also
• Graphical user interface testing
• Manual testing
• Orthogonal array testing
• Pair testing
• Soak testing
• Software reliability testing
• Stress testing
• Web testing
| Wikipedia |
A-equivalence
In mathematics, ${\mathcal {A}}$-equivalence, sometimes called right-left equivalence, is an equivalence relation between map germs.
Let $M$ and $N$ be two manifolds, and let $f,g:(M,x)\to (N,y)$ be two smooth map germs. We say that $f$ and $g$ are ${\mathcal {A}}$-equivalent if there exist diffeomorphism germs $\phi :(M,x)\to (M,x)$ :(M,x)\to (M,x)} and $\psi :(N,y)\to (N,y)$ :(N,y)\to (N,y)} such that $\psi \circ f=g\circ \phi .$
In other words, two map germs are ${\mathcal {A}}$-equivalent if one can be taken onto the other by a diffeomorphic change of co-ordinates in the source (i.e. $M$) and the target (i.e. $N$).
Let $\Omega (M_{x},N_{y})$ denote the space of smooth map germs $(M,x)\to (N,y).$ Let ${\mbox{diff}}(M_{x})$ be the group of diffeomorphism germs $(M,x)\to (M,x)$ and ${\mbox{diff}}(N_{y})$ be the group of diffeomorphism germs $(N,y)\to (N,y).$ The group $G:={\mbox{diff}}(M_{x})\times {\mbox{diff}}(N_{y})$ acts on $\Omega (M_{x},N_{y})$ in the natural way: $(\phi ,\psi )\cdot f=\psi ^{-1}\circ f\circ \phi .$ Under this action we see that the map germs $f,g:(M,x)\to (N,y)$ are ${\mathcal {A}}$-equivalent if, and only if, $g$ lies in the orbit of $f$, i.e. $g\in {\mbox{orb}}_{G}(f)$ (or vice versa).
A map germ is called stable if its orbit under the action of $G:={\mbox{diff}}(M_{x})\times {\mbox{diff}}(N_{y})$ is open relative to the Whitney topology. Since $\Omega (M_{x},N_{y})$ is an infinite dimensional space metric topology is no longer trivial. Whitney topology compares the differences in successive derivatives and gives a notion of proximity within the infinite dimensional space. A base for the open sets of the topology in question is given by taking $k$-jets for every $k$ and taking open neighbourhoods in the ordinary Euclidean sense. Open sets in the topology are then unions of these base sets.
Consider the orbit of some map germ $orb_{G}(f).$ The map germ $f$ is called simple if there are only finitely many other orbits in a neighbourhood of each of its points. Vladimir Arnold has shown that the only simple singular map germs $(\mathbb {R} ^{n},0)\to (\mathbb {R} ,0)$ for $1\leq n\leq 3$ are the infinite sequence $A_{k}$ ($k\in \mathbb {N} $), the infinite sequence $D_{4+k}$ ($k\in \mathbb {N} $), $E_{6},$ $E_{7},$ and $E_{8}.$
See also
• K-equivalence (contact equivalence)
References
• M. Golubitsky and V. Guillemin, Stable Mappings and Their Singularities. Graduate Texts in Mathematics, Springer.
| Wikipedia |
Advanced level mathematics
Advanced Level (A-Level) Mathematics is a qualification of further education taken in the United Kingdom (and occasionally other countries as well). In the UK, A-Level exams are traditionally taken by 17-18 year-olds after a two-year course at a sixth form or college. Advanced Level Further Mathematics is often taken by students who wish to study a mathematics-based degree at university, or related degree courses such as physics or computer science.
Like other A-level subjects, mathematics has been assessed in a modular system since the introduction of Curriculum 2000, whereby each candidate must take six modules, with the best achieved score in each of these modules (after any retake) contributing to the final grade.[1] Most students will complete three modules in one year, which will create an AS-level qualification in their own right and will complete the A-level course the following year—with three more modules.
The system in which mathematics is assessed is changing for students starting courses in 2017 (as part of the A-level reforms first introduced in 2015), where the reformed specifications have reverted to a linear structure with exams taken only at the end of the course in a single sitting.
In addition, while schools could choose freely between taking Statistics, Mechanics or Discrete Mathematics (also known as Decision Mathematics) modules with the ability to specialise in one branch of applied Mathematics in the older modular specification, in the new specifications, both Mechanics and Statistics were made compulsory, with Discrete Mathematics being made exclusive as an option to students pursuing a Further Mathematics course. The first assessment opportunity for the new specification is 2018 and 2019 for A-levels in Mathematics and Further Mathematics, respectively.
2000s specification
Prior to the 2017 reform, the basic A-Level course consisted of six modules, four pure modules (C1, C2, C3, and C4) and two applied modules in Statistics, Mechanics and/or Decision Mathematics. The C1 through C4 modules are referred to by A-level textbooks as "Core" modules, encompassing the major topics of mathematics such as logarithms, differentiation/integration and geometric/arithmetic progressions.
The two chosen modules for the final two parts of the A-Level are determined either by a student's personal choices, or the course choice of their school/college, though it commonly took the form of S1 (Statistics) and M1 (Mechanics).
Further mathematics
Main article: Further Mathematics
Students that were studying for (or had completed) an A-level in Mathematics had the opportunity to study an A-level in Further Mathematics, which required taking a further 6 modules to give a second qualification. The grades of the two A-levels will be independent of each other, with Further Mathematics requiring students to take a minimum of two Further Pure modules, one of which must be FP1, and the other either FP2 or FP3, which are simply extensions of the four Core modules from the normal Maths A-Level. Four more modules need to be taken; those available vary with different specifications.[2]
Not all schools are able to offer Further Mathematics, due to a low student number (meaning that the course is not financially viable) or a lack of suitably experienced teachers. To fulfil the demand, extra tutoring is available, with providers such as the Further Mathematics Support Programme.[3]
Some students had the opportunity to take a third maths qualification, "Additional Further Mathematics", which added more modules from those not used for Mathematics or Further Mathematics. Schools that offer this qualification usually only took this to AS-level, taking three modules, although some students went further, taking the extra six modules to gain another full A-Level qualification. Additional Further Mathematics is offered by Edexcel only, and a Pure Mathematics A-level is available for students who—on the Edexcel exam board—take the modules C1, C2, C3, C4, FP1 and either FP2 or FP3. No comparable qualification has been available since the 2017 reforms.
Results and statistics
Each module carried a maximum of 100 UMS points towards the total grade, and each module is also given a separate grade depending on its score. The number of points required for different grades were defined as follows:
Grade Module (Out of 100) AS level (Out of 300) A level (Out of 600)
A* - - 480 (With 180/200 from C3 + C4)
A 80 240 480
B 70 210 420
C 60 180 360
D 50 150 300
E 40 120 240
The proportion of candidates acquiring these grades in 2007 are below:
Mathematics
Male Female Combined
Entries 60093
Grade A 42.6% 45.5% 43.7%
Grade B 20.8% 22.2% 21.4%
Grade C 15.8% 15.1% 15.6%
Grade D 10.9% 10.0% 10.4%
Grade E 6.4% 5.0% 5.9%
Grade U 3.5% 2.2% 3.0%
Further mathematics
Male Female Combined
Entries 7972
Grade A 57.1% 56.2% 56.8%
Grade B 19.4% 20.2% 20.3%
Grade C 11.6% 10.9% 11.4%
Grade D 6.6% 6.2% 6.5%
Grade E 3.6% 3.0% 3.4%
Grade U 1.7% 1.5% 1.6%
2017 specification
A new specification was introduced in 2017 for first examination in summer 2019. Under this specification, there are three papers which must all be taken in the same year. There are three overarching themes - “Argument, language and proof”, “Problem solving” and “Modelling” throughout the assessment.[4]
Each board structures the three papers as follows:
AQA
• Paper 1: Pure Mathematics
• Paper 2: Content on Paper 1 plus Mechanics
• Paper 3: Content on Paper 1 plus Statistics
[5]
Edexcel
• Paper 1: Pure Mathematics 1
• Paper 2: Pure Mathematics 2
• Paper 3: Statistics and Mechanics
[6]
OCR
• Paper 1: Pure Mathematics
• Paper 2: Pure Mathematics and Statistics
• Paper 3: Pure Mathematics and Mechanics
[7]
Grading
It was suggested by the Department for Education that the high proportion of candidates who obtain grade A makes it difficult for universities to distinguish between the most able candidates. As a result, the 2010 exam session introduced the grade A*—which serves to distinguish between the better candidates.[8]
Prior to the 2017 reforms, the A* grade in maths was awarded to candidates who achieve an A (480/600) in their overall A Level, as well as achieving a combined score of 180/200 in modules Core 3 and Core 4. For the reformed specification, the A* is given by a more traditional grade boundary based on the raw mark achieved by the candidate over their papers.
The A* grade in further maths was awarded slightly differently. The same minimum score of 480/600 was required across all six modules. However, a 90% average (or a score of 270/300) had to be obtained across the candidate's best 'A2' modules.[9] A2 modules included any modules other than those with a '1' (FP1, S1, M1 and D1 are not A2 modules, whereas FP2, FP3, FP4 (from AQA only), S2, S3, S4, M2, M3 and D2 are).
List of Subjects
```List of subjects in A Level Mathematics```
1. Core Mathematics:
Covers foundational topics like algebra, calculus, trigonometry, and coordinate geometry.
2. Further Mathematics:
Expands upon Core Mathematics with additional areas such as complex numbers, matrices, differential equations, and numerical methods.
3. Pure Mathematics:
Explores advanced topics in algebra, calculus, and mathematical proofs.
4. Applied Mathematics:
Focuses on practical applications of mathematical concepts to solve real-world problems in various fields.
5. Mechanics:
Focuses on the study of motion, forces, and vectors, particularly relevant for physics or engineering interests.
6. Statistics:
Involves collecting, analyzing, and interpreting data, including topics like probability, hypothesis testing, regression analysis, and sampling.
7. Discrete Mathematics:
Deals with separate and distinct mathematical structures, including topics such as combinatorics, graph theory, and algorithms.
8. Decision Mathematics:
Applies mathematical techniques to solve real-world problems related to optimization, networks, and decision-making.
9. Financial Mathematics:
Applies mathematical concepts to analyze financial markets, investments, and risk management.
References
1. "A level retake courses at CIFE colleges". CIFE. Retrieved 2020-01-22.
2. "The Further Mathematics Support Programme". furthermaths.org.uk. Retrieved 2020-01-22.
3. "Supporting and Promoting Advanced and Further Mathematics". The Further Mathematics Support Programme. Archived from the original on 11 June 2011. Retrieved 25 May 2011.
4. STEM Learning https://www.stem.org.uk/cpd/ondemand/75722/teaching-level-mathematics-proof-problem-solving-and-modelling
5. "AQA: Specification at a glance". aqa.org.uk. Archived from the original on 2016-07-21. Retrieved 2020-01-22.
6. "Pearson Edexcel AS and A level Mathematics (2017) | Pearson qualifications". qualifications.pearson.com. Retrieved 2020-01-22.
7. "OCR - AS and A Level - Mathematics A - H230, H240 (from 2017) - Specification at a glance". www.ocr.org.uk. Archived from the original on 2019-06-03. Retrieved 2020-01-22.
8. "OCR: A Level A* frequently asked questions" (PDF). ocr.org.uk. Archived (PDF) from the original on 2013-02-23. Retrieved 2020-01-21.
9. "Understanding A level marks and grades | Pearson qualifications". qualifications.pearson.com. Retrieved 2020-01-22.
External links
• Underground Mathematics (Resources on A-level mathematics)
| Wikipedia |
a-paracompact space
In mathematics, in the field of topology, a topological space is said to be a-paracompact if every open cover of the space has a locally finite refinement. In contrast to the definition of paracompactness, the refinement is not required to be open.
Every paracompact space is a-paracompact, and in regular spaces the two notions coincide.
References
• Willard, Stephen (2004). General Topology. Dover Publications. ISBN 0-486-43479-6.
| Wikipedia |
Alfred Frölicher
Alfred Frölicher (often misspelled Fröhlicher) was a Swiss mathematician (8 October 8 1927 – 1 July 2010). He was a full professor at the Université de Fribourg (1962-1965), and then at the Université de Genève (1966-1993). He introduced the Frölicher spectral sequence and the Frölicher–Nijenhuis bracket and Frölicher spaces and Frölicher groups.
He received his Ph.D. from ETH Zurich in 1954, with thesis Zur Differentialgeometrie der komplexe Strukturen written under the direction of Beno Eckmann and Heinz Hopf.[1]
Publications
• Bucher, W.; Frölicher, Alfred (1966), Calculus in vector spaces without norm, Lecture Notes in Mathematics, vol. 30, Berlin, New York: Springer-Verlag, ISBN 978-1-399-86684-2, MR 0213869
• Faure, Claude-Alain; Frölicher, Alfred (2000), Modern projective geometry, Mathematics and its Applications, vol. 521, Boston: Kluwer Academic Publishers, ISBN 978-0-7923-6525-9, MR 1783451
• Frölicher, Alfred; Kriegl, Andreas (1988), Linear spaces and differentiation theory, Pure and Applied Mathematics (New York), John Wiley & Sons, ISBN 978-0-471-91786-1, MR 0961256
References
1. Alfred Frölicher at the Mathematics Genealogy Project
Authority control
International
• ISNI
• VIAF
National
• Norway
• France
• BnF data
• Germany
• Israel
• Belgium
• United States
• Netherlands
Academics
• MathSciNet
• Mathematics Genealogy Project
• zbMATH
Other
• IdRef
| Wikipedia |
Alfred North Whitehead
Alfred North Whitehead OM FRS FBA (15 February 1861 – 30 December 1947) was an English mathematician and philosopher. He created the philosophical school known as process philosophy,[21] which has been applied in a wide variety of disciplines, including ecology, theology, education, physics, biology, economics, and psychology.
Alfred North Whitehead
OM FRS FBA
Whitehead in 1936
Born(1861-02-15)15 February 1861
Ramsgate, England
Died30 December 1947(1947-12-30) (aged 86)
Cambridge, Massachusetts, U.S.
EducationTrinity College, Cambridge
(B.A., 1884)
Era20th-century philosophy
RegionWestern philosophy
School
• Analytic philosophy (early)
• Logicism (early)
• Process philosophy
• Process theology
Institutions
• Imperial College London
• Harvard University
Academic advisorsEdward Routh[1]
Doctoral students
• Raphael Demos
• Charles Hartshorne
• Susanne Langer
• W. V. O. Quine
• Bertrand Russell
• Gregory Vlastos
• Paul Weiss
Main interests
• Metaphysics
• mathematics
Notable ideas
Process philosophy
Process theology
Influences
• Aristotle[2]
• Henri Bergson[3]
• Francis Herbert Bradley[4]
• John Dewey[3]
• David Hume[2]
• William James[3]
• Immanuel Kant[5]
• Gottfried Wilhelm Leibniz[6]
• John Locke[5]
• Isaac Newton[6]
• Plato[2]
• George Santayana[6]
Influenced
• James Luther Adams
• Wilfred Eade Agar[7]
• David Bohm[7]
• C. D. Broad[8]
• Milič Čapek[7]
• Donald Davidson[9]
• Gilles Deleuze[10]
• Susanne Langer[11]
• Ervin László[12]
• Maurice Merleau-Ponty[8]
• F. S. C. Northrop[11]
• Talcott Parsons,[13]
• Ilya Prigogine,[8]
• W. V. O. Quine[14]
• Bertrand Russell[6]
• B. F. Skinner[15]
• Wolfgang Smith[16]
• John Lighton Synge[7]
• Jules Vuillemin[8]
• Conrad Hal Waddington[7]
• Michel Weber[17]
• Sewall Wright[18]
• Eric Voegelin[19]
• Ken Wilber[20]
Signature
In his early career Whitehead wrote primarily on mathematics, logic, and physics. He wrote the three-volume Principia Mathematica (1910–1913), with his former student Bertrand Russell. Principia Mathematica is considered one of the twentieth century's most important works in mathematical logic, and placed 23rd in a list of the top 100 English-language nonfiction books of the twentieth century by Modern Library.[22]
Beginning in the late 1910s and early 1920s, Whitehead gradually turned his attention from mathematics to philosophy of science, and finally to metaphysics. He developed a comprehensive metaphysical system which radically departed from most of Western philosophy. Whitehead argued that reality consists of processes rather than material objects, and that processes are best defined by their relations with other processes, thus rejecting the theory that reality is fundamentally constructed by bits of matter that exist independently of one another.[23] Whitehead's philosophical works – particularly Process and Reality – are regarded as the foundational texts of process philosophy.
Whitehead's process philosophy argues that "there is urgency in coming to see the world as a web of interrelated processes of which we are integral parts, so that all of our choices and actions have consequences for the world around us."[23] For this reason, one of the most promising applications of Whitehead's thought in recent years has been in the area of ecological civilization and environmental ethics pioneered by John B. Cobb.[24][25]
Life
Childhood and education
Alfred North Whitehead was born in Ramsgate, Kent, England, in 1861.[26] His father, Alfred Whitehead, became an Anglican minister after being headmaster of Chatham House Academy, a school for boys previously headed by Alfred's father, Thomas Whitehead.[27] Whitehead himself recalled both of them as being very successful school masters, with his grandfather being the more "remarkable" man.[27]
Whitehead's mother was Maria Sarah Buckmaster. Her maternal great-grandmother was Jane North (1776-1847), whose maiden surname was given to Whitehead, and several other members of his family over time. His mother, Maria Buckmaster had eleven siblings. The son of her brother Thomas, Walter Selby Buckmaster, was twice an Olympics silver medal winner for Polo (1900, 1908) for Britain, and is said to be "one of the finest polo players England has ever produced".[28] Whitehead does not appear to have been close to his mother, although he and Evelyn (full name: Evelyn Ada Maud Rice Willoughby Wade), whom he married in 1890, are recorded in the English Census of 1891 as living with Alfred's mother and father. Lowe notes that there appears to have been mutual dislike between Whitehead's wife, Evelyn, and his mother, Maria.
Griffin relates how Bertrand Russell, a colleague and collaborator of Whitehead, was a very close friend of Whitehead and of his wife, Evelyn. Griffin retells Russell's story of how, one evening in 1901, "they found Evelyn Whitehead in the middle of what appeared to be a dangerous and acutely painful angina attack. ... [but] It seems that she suffered from a psychosomatic disorder ... [and] the danger was illusory." Griffin posits that Russell exaggerated the drama of her illness, and that both Evelyn and Russell were habitually given to melodrama.[29] Intensity of emotion was encourgaged by their avant garde associates in the turbulent Bloomsbury Group which "discussed aesthetic and philosophical questions in a spirit of agnosticism and were strongly influenced by G.E. Moore's Principia Ethica (1903) and by A. N. Whitehead's and Bertrand Russell's Principia Mathematica (1910–13), in the light of which they searched for definitions of the good, the true, and the beautiful".[30]
Alfred's brother Henry became Bishop of Madras and wrote the closely observed ethnographic account Village Gods of South-India (Calcutta: Association Press, 1921).
Whitehead was educated at Sherborne,[31] a prominent English public school, where he excelled in sports and mathematics[32] and was head prefect of his class.[33]
In 1880, he began attending Trinity College, Cambridge, and studied mathematics.[34] His academic advisor was Edward Routh.[1] He earned his B.A. from Trinity in 1884, writing his dissertation on James Clerk Maxwell's A Treatise on Electricity and Magnetism, and graduated as fourth wrangler.[35]
Career
Elected a fellow of Trinity in 1884, Whitehead would teach and write on mathematics and physics at the college until 1910, spending the 1890s writing his Treatise on Universal Algebra (1898), and the 1900s collaborating with his former pupil, Bertrand Russell, on the first edition of Principia Mathematica.[36] He was a Cambridge Apostle.[37]
In 1910, Whitehead resigned his senior lectureship in mathematics at Trinity and moved to London without first obtaining another job.[38] After being unemployed for a year, he accepted a position as lecturer in applied mathematics and mechanics at University College London, but was passed over a year later for the Goldsmid Chair of Applied Mathematics and Mechanics, a position for which he had hoped to be seriously considered.[39]
In 1914, Whitehead accepted a position as professor of applied mathematics at the newly chartered Imperial College London, where his old friend Andrew Forsyth had recently been appointed chief professor of mathematics.[40]
In 1918, Whitehead's academic responsibilities began to seriously expand as he accepted a number of high administrative positions within the University of London system, of which Imperial College London was a member at the time. He was elected dean of the Faculty of Science at the University of London in late 1918 (a post he held for four years), a member of the University of London's Senate in 1919, and chairman of the Senate's Academic (leadership) Council in 1920, a post which he held until he departed for America in 1924.[40] Whitehead was able to exert his newfound influence to successfully lobby for a new history of science department, help establish a Bachelor of Science degree (previously only Bachelor of Arts degrees had been offered), and make the school more accessible to less wealthy students.[41]
Toward the end of his time in England, Whitehead turned his attention to philosophy. Though he had no advanced training in philosophy, his philosophical work soon became highly regarded. After publishing The Concept of Nature in 1920, he served as president of the Aristotelian Society from 1922 to 1923.[42]
Move to the United States, 1924
In 1924, Henry Osborn Taylor invited the 63-year-old Whitehead to join the faculty at Harvard University as a professor of philosophy.[43] The Whiteheads would spend the rest of their lives in the United States.
During his time at Harvard, Whitehead produced his most important philosophical contributions. In 1925, he wrote Science and the Modern World, which was immediately hailed as an alternative to the Cartesian dualism then prevalent in popular science.[44] He was elected to the American Academy of Arts and Sciences that same year.[45] He was elected to the American Philosophical Society in 1926.[46] Lectures from 1927 to 1928, were published in 1929 as a book named Process and Reality, which has been compared to Immanuel Kant's Critique of Pure Reason.[24]
Family and death
In 1890, Whitehead married Evelyn Wade, an Irishwoman raised in France; they had a daughter, Jessie, and two sons, Thomas and Eric.[33] Thomas followed his father to Harvard in 1931, to teach at the Business School. Eric died in action at the age of 19, while serving in the Royal Flying Corps during World War I.[47]
From 1910, the Whiteheads had a cottage in the village of Lockeridge, near Marlborough, Wiltshire; from there he completed Principia Mathematica.[48][49]
The Whiteheads remained in the United States after moving to Harvard in 1924. Alfred retired from Harvard in 1937 and remained in Cambridge, Massachusetts, until his death on 30 December 1947.[50]
Legacy
The two-volume biography of Whitehead by Victor Lowe[51] is the most definitive presentation of the life of Whitehead. However, many details of Whitehead's life remain obscure because he left no Nachlass (personal archive); his family carried out his instructions that all of his papers be destroyed after his death.[52] Additionally, Whitehead was known for his "almost fanatical belief in the right to privacy," and for writing very few personal letters of the kind that would help to gain insight on his life.[52] Wrote Lowe in his preface, "No professional biographer in his right mind would touch him."[26]
Led by Executive Editor Brian G. Henning and General Editor George R. Lucas Jr., the Whitehead Research Project of the Center for Process Studies is currently working on a critical edition of Whitehead's published and unpublished works.[53] The first volume of the Edinburgh Critical Edition of the Complete Works of Alfred North Whitehead was published in 2017 by Paul A. Bogaard and Jason Bell as The Harvard Lectures of Alfred North Whitehead, 1924–1925: The Philosophical Presuppositions of Science.[54]
Mathematics and logic
In addition to numerous articles on mathematics, Whitehead wrote three major books on the subject: A Treatise on Universal Algebra (1898), Principia Mathematica (co-written with Bertrand Russell and published in three volumes between 1910 and 1913), and An Introduction to Mathematics (1911). The former two books were aimed exclusively at professional mathematicians, while the latter book was intended for a larger audience, covering the history of mathematics and its philosophical foundations.[55] Principia Mathematica in particular is regarded as one of the most important works in mathematical logic of the 20th century.
In addition to his legacy as a co-writer of Principia Mathematica, Whitehead's theory of "extensive abstraction" is considered foundational for the branch of ontology and computer science known as "mereotopology," a theory describing spatial relations among wholes, parts, parts of parts, and the boundaries between parts.[56]
A Treatise on Universal Algebra
In A Treatise on Universal Algebra (1898), the term universal algebra had essentially the same meaning that it has today: the study of algebraic structures themselves, rather than examples ("models") of algebraic structures.[57] Whitehead credits William Rowan Hamilton and Augustus De Morgan as originators of the subject matter, and James Joseph Sylvester with coining the term itself.[57][58]
At the time, structures such as Lie algebras and hyperbolic quaternions drew attention to the need to expand algebraic structures beyond the associatively multiplicative class. In a review Alexander Macfarlane wrote: "The main idea of the work is not unification of the several methods, nor generalization of ordinary algebra so as to include them, but rather the comparative study of their several structures."[59] In a separate review, G. B. Mathews wrote, "It possesses a unity of design which is really remarkable, considering the variety of its themes."[60]
A Treatise on Universal Algebra sought to examine Hermann Grassmann's theory of extension ("Ausdehnungslehre"), Boole's algebra of logic, and Hamilton's quaternions (this last number system was to be taken up in Volume II, which was never finished due to Whitehead's work on Principia Mathematica).[61] Whitehead wrote in the preface:
Such algebras have an intrinsic value for separate detailed study; also they are worthy of comparative study, for the sake of the light thereby thrown on the general theory of symbolic reasoning, and on algebraic symbolism in particular... The idea of a generalized conception of space has been made prominent, in the belief that the properties and operations involved in it can be made to form a uniform method of interpretation of the various algebras.[62]
Whitehead, however, had no results of a general nature.[57] His hope of "form[ing] a uniform method of interpretation of the various algebras" presumably would have been developed in Volume II, had Whitehead completed it. Further work on the subject was minimal until the early 1930s, when Garrett Birkhoff and Øystein Ore began publishing on universal algebras.[63]
Principia Mathematica
Principia Mathematica (1910–1913) is Whitehead's most famous mathematical work. Written with former student Bertrand Russell, Principia Mathematica is considered one of the twentieth century's most important works in mathematics, and placed 23rd in a list of the top 100 English-language nonfiction books of the twentieth century by Modern Library.[22]
Principia Mathematica's purpose was to describe a set of axioms and inference rules in symbolic logic from which all mathematical truths could in principle be proven. Whitehead and Russell were working on such a foundational level of mathematics and logic that it took them until page 86 of Volume II to prove that 1+1=2, a proof humorously accompanied by the comment, "The above proposition is occasionally useful."[64]
Whitehead and Russell had thought originally that Principia Mathematica would take a year to complete; it ended up taking them ten years.[65] When it came time for publication, the three-volume work was so long (more than 2,000 pages) and its audience so narrow (professional mathematicians) that it was initially published at a loss of 600 pounds, 300 of which was paid by Cambridge University Press, 200 by the Royal Society of London, and 50 apiece by Whitehead and Russell themselves.[65] Despite the initial loss, today there is likely no major academic library in the world which does not hold a copy of Principia Mathematica.[66]
The ultimate substantive legacy of Principia Mathematica is mixed. It is generally accepted that Kurt Gödel's incompleteness theorem of 1931 definitively demonstrated that for any set of axioms and inference rules proposed to encapsulate mathematics, there would in fact be some truths of mathematics which could not be deduced from them, and hence that Principia Mathematica could never achieve its aims.[67] However, Gödel could not have come to this conclusion without Whitehead and Russell's book. In this way, Principia Mathematica's legacy might be described as its key role in disproving the possibility of achieving its own stated goals.[68] But beyond this somewhat ironic legacy, the book popularized modern mathematical logic and drew important connections between logic, epistemology, and metaphysics.[69]
An Introduction to Mathematics
Unlike Whitehead's previous two books on mathematics, An Introduction to Mathematics (1911) was not aimed exclusively at professional mathematicians but was intended for a larger audience. The book covered the nature of mathematics, its unity and internal structure, and its applicability to nature.[55] Whitehead wrote in the opening chapter:
The object of the following Chapters is not to teach mathematics, but to enable students from the very beginning of their course to know what the science is about, and why it is necessarily the foundation of exact thought as applied to natural phenomena.[70]
The book can be seen as an attempt to understand the growth in unity and interconnection of mathematics as a whole, as well as an examination of the mutual influence of mathematics and philosophy, language, and physics.[71] Although the book is little-read, in some ways it prefigures certain points of Whitehead's later work in philosophy and metaphysics.[72]
Views on education
Whitehead showed a deep concern for educational reform at all levels. In addition to his numerous individually written works on the subject, Whitehead was appointed by Britain's Prime Minister David Lloyd George as part of a 20-person committee to investigate the educational systems and practices of the UK in 1921 and recommend reform.[73]
Whitehead's most complete work on education is the 1929 book The Aims of Education and Other Essays, which collected numerous essays and addresses by Whitehead on the subject published between 1912 and 1927. The essay from which Aims of Education derived its name was delivered as an address in 1916 when Whitehead was president of the London Branch of the Mathematical Association. In it, he cautioned against the teaching of what he called "inert ideas" – ideas that are disconnected scraps of information, with no application to real life or culture. He opined that "education with inert ideas is not only useless: it is, above all things, harmful."[74]
Rather than teach small parts of a large number of subjects, Whitehead advocated teaching a relatively few important concepts that the student could organically link to many different areas of knowledge, discovering their application in actual life.[75] For Whitehead, education should be the exact opposite of the multidisciplinary, value-free school model[74][76] – it should be transdisciplinary, and laden with values and general principles that provide students with a bedrock of wisdom and help them to make connections between areas of knowledge that are usually regarded as separate.
In order to make this sort of teaching a reality, however, Whitehead pointed to the need to minimize the importance of (or radically alter) standard examinations for school entrance. Whitehead writes:
Every school is bound on pain of extinction to train its boys for a small set of definite examinations. No headmaster has a free hand to develop his general education or his specialist studies in accordance with the opportunities of his school, which are created by its staff, its environment, its class of boys, and its endowments. I suggest that no system of external tests which aims primarily at examining individual scholars can result in anything but educational waste.[77]
Whitehead argued that curriculum should be developed specifically for its own students by its own staff, or else risk total stagnation, interrupted only by occasional movements from one group of inert ideas to another.
Above all else in his educational writings, Whitehead emphasized the importance of imagination and the free play of ideas. In his essay "Universities and Their Function", Whitehead writes provocatively on imagination:
Imagination is not to be divorced from the facts: it is a way of illuminating the facts. It works by eliciting the general principles which apply to the facts, as they exist, and then by an intellectual survey of alternative possibilities which are consistent with those principles. It enables men to construct an intellectual vision of a new world.[78]
Whitehead's philosophy of education might adequately be summarized in his statement that "knowledge does not keep any better than fish."[79] In other words, bits of disconnected knowledge are meaningless; all knowledge must find some imaginative application to the students' own lives, or else it becomes so much useless trivia, and the students themselves become good at parroting facts but not thinking for themselves.
Philosophy and metaphysics
Whitehead did not begin his career as a philosopher.[26] In fact, he never had any formal training in philosophy beyond his undergraduate education. Early in his life, he showed great interest in and respect for philosophy and metaphysics, but it is evident that he considered himself a rank amateur. In one letter to his friend and former student Bertrand Russell, after discussing whether science aimed to be explanatory or merely descriptive, he wrote: "This further question lands us in the ocean of metaphysic, onto which my profound ignorance of that science forbids me to enter."[81] Ironically, in later life, Whitehead would become one of the 20th century's foremost metaphysicians.
However, interest in metaphysics – the philosophical investigation of the nature of the universe and existence – had become unfashionable by the time Whitehead began writing in earnest about it in the 1920s. The ever-more impressive accomplishments of empirical science had led to a general consensus in academia that the development of comprehensive metaphysical systems was a waste of time because they were not subject to empirical testing.[82]
Whitehead was unimpressed by this objection. In the notes of one of his students for a 1927 class, Whitehead was quoted as saying: "Every scientific man in order to preserve his reputation has to say he dislikes metaphysics. What he means is he dislikes having his metaphysics criticized."[83] In Whitehead's view, scientists and philosophers make metaphysical assumptions about how the universe works all the time, but such assumptions are not easily seen precisely because they remain unexamined and unquestioned. While Whitehead acknowledged that "philosophers can never hope finally to formulate these metaphysical first principles,"[84] he argued that people need to continually reimagine their basic assumptions about how the universe works if philosophy and science are to make any real progress, even if that progress remains permanently asymptotic. For this reason, Whitehead regarded metaphysical investigations as essential to both good science and good philosophy.[85]
Perhaps foremost among what Whitehead considered faulty metaphysical assumptions was the Cartesian idea that reality is fundamentally constructed of bits of matter that exist totally independently of one another, which he rejected in favour of an event-based or "process" ontology in which events are primary and are fundamentally interrelated and dependent on one another.[86] He also argued that the most basic elements of reality can all be regarded as experiential, indeed that everything is constituted by its experience. He used the term "experience" very broadly so that even inanimate processes such as electron collisions are said to manifest some degree of experience. In this, he went against Descartes' separation of two different kinds of real existence, either exclusively material or else exclusively mental.[87] Whitehead referred to his metaphysical system as "philosophy of organism," but it would become known more widely as "process philosophy."[87]
Whitehead's philosophy was highly original, and soon garnered interest in philosophical circles. After publishing The Concept of Nature in 1920, he served as president of the Aristotelian Society from 1922 to 1923, and Henri Bergson was quoted as saying that Whitehead was "the best philosopher writing in English."[88] So impressive and different was Whitehead's philosophy that in 1924 he was invited to join the faculty at Harvard University as a professor of philosophy at 63 years of age.[43]
This is not to say that Whitehead's thought was widely accepted or even well understood. His philosophical work is generally considered to be among the most difficult to understand in all of the Western canon.[24] Even professional philosophers struggled to follow Whitehead's writings. One famous story illustrating the level of difficulty of Whitehead's philosophy centres around the delivery of Whitehead's Gifford lectures in 1927–28 – following Arthur Eddington's lectures of the year previous – which Whitehead would later publish as Process and Reality:
Eddington was a marvellous popular lecturer who had enthralled an audience of 600 for his entire course. The same audience turned up to Whitehead's first lecture but it was completely unintelligible, not merely to the world at large but to the elect. My father remarked to me afterwards that if he had not known Whitehead well he would have suspected that it was an imposter making it up as he went along... The audience at subsequent lectures was only about half a dozen in all.[90]
It may not be inappropriate to speculate that some fair portion of the respect generally shown to Whitehead by his philosophical peers at the time arose from their sheer bafflement. The Chicago theologian Shailer Mathews once remarked of Whitehead's 1926 book Religion in the Making: "It is infuriating, and I must say embarrassing as well, to read page after page of relatively familiar words without understanding a single sentence."[91]
However, Mathews' frustration with Whitehead's books did not negatively affect his interest. In fact, there were numerous philosophers and theologians at Chicago's Divinity School that perceived the importance of what Whitehead was doing without fully grasping all of the details and implications. In 1927, they invited one of America's only Whitehead experts, Henry Nelson Wieman, to Chicago to give a lecture explaining Whitehead's thoughts.[91] Wieman's lecture was so brilliant that he was promptly hired to the faculty and taught there for twenty years, and for at least thirty years afterwards Chicago's Divinity School was closely associated with Whitehead's thought.[89]
Shortly after Whitehead's book Process and Reality appeared in 1929, Wieman famously wrote in his 1930 review:
Not many people will read Whitehead's recent book in this generation; not many will read it in any generation. But its influence will radiate through concentric circles of popularization until the common man will think and work in the light of it, not knowing whence the light came. After a few decades of discussion and analysis, one will be able to understand it more readily than can now be done.[92]
Wieman's words proved prophetic. Though Process and Reality has been called "arguably the most impressive single metaphysical text of the twentieth century,"[93] it has been little-read and little-understood, partly because it demands – as Isabelle Stengers puts it – "that its readers accept the adventure of the questions that will separate them from every consensus."[94] Whitehead questioned Western philosophy's most dearly held assumptions about how the universe works — but in doing so, he managed to anticipate a number of 21st century scientific and philosophical problems and provide novel solutions.[95]
Whitehead's conception of reality
Whitehead was convinced that the scientific notion of matter was misleading as a way of describing the ultimate nature of things. In his 1925 book Science and the Modern World, he wrote that:
There persists ... [a] fixed scientific cosmology which presupposes the ultimate fact of an irreducible brute matter, or material, spread through space in a flux of configurations. In itself, such a material is senseless, valueless, purposeless. It just does what it does do, following a fixed routine imposed by external relations which do not spring from the nature of its being. It is this assumption that I call "scientific materialism." Also, it is an assumption which I shall challenge as being entirely unsuited to the scientific situation at which we have now arrived.[86]
In Whitehead's view, there are a number of problems with this notion of "irreducible brute matter." First, it obscures and minimizes the importance of change. By thinking of any material thing (like a rock, or a person) as being fundamentally the same thing throughout time, with any changes to it being secondary to its "nature," scientific materialism hides the fact that nothing ever stays the same. For Whitehead, change is fundamental and inescapable; he emphasizes that "all things flow."[96]
In Whitehead's view, then, concepts such as "quality," "matter," and "form" are problematic. These "classical" concepts fail to adequately account for change, and overlook the active and experiential nature of the most basic elements of the world. They are useful abstractions but are not the world's basic building blocks.[97] What is ordinarily conceived of as a single person, for instance, is philosophically described as a continuum of overlapping events.[98] After all, people change all the time, if only because they have aged by another second and had some further experience. These occasions of experience are logically distinct but are progressively connected in what Whitehead calls a "society" of events.[99] By assuming that enduring objects are the most real and fundamental things in the universe, materialists have mistaken the abstract for the concrete (what Whitehead calls the "fallacy of misplaced concreteness").[87][100]
To put it another way, a thing or person is often seen as having a "defining essence" or a "core identity" that is unchanging, and describes what the thing or person really is. In this way of thinking, things and people are seen as fundamentally the same through time, with any changes being qualitative and secondary to their core identity (e.g., "Mark's hair has turned grey as he has gotten older, but he is still the same person"). But in Whitehead's cosmology, the only fundamentally existent things are discrete "occasions of experience" that overlap one another in time and space, and jointly make up the enduring person or thing. On the other hand, what ordinary thinking often regards as "the essence of a thing" or "the identity/core of a person" is an abstract generalization of what is regarded as that person or thing's most important or salient features across time. Identities do not define people; people define identities. Everything changes from moment to moment and to think of anything as having an "enduring essence" misses the fact that "all things flow," though it is often a useful way of speaking.
Whitehead pointed to the limitations of language as one of the main culprits in maintaining a materialistic way of thinking and acknowledged that it may be difficult to ever wholly move past such ideas in everyday speech.[101] After all, every moment of each person's life can hardly be given a different proper name, and it is easy and convenient to think of people and objects as remaining fundamentally the same things, rather than constantly keeping in mind that each thing is a different thing from what it was a moment ago. Yet the limitations of everyday living and everyday speech should not prevent people from realizing that "material substances" or "essences" are a convenient generalized description of a continuum of particular, concrete processes. No one questions that a ten-year-old person is quite different by the time he or she turns thirty years old, and in many ways is not the same person at all; Whitehead points out that it is not philosophically or ontologically sound to think that a person is the same from one second to the next.
A second problem with materialism is that it obscures the importance of relations. It sees every object as distinct and discrete from all other objects. Each object is simply an inert clump of matter that is only externally related to other things. The idea of matter as primary makes people think of objects as being fundamentally separate in time and space, and not necessarily related to anything. But in Whitehead's view, relations take a primary role, perhaps even more important than the relata themselves.[102] A student taking notes in one of Whitehead's fall 1924 classes wrote that, "Reality applies to connections, and only relatively to the things connected. (A) is real for (B), and (B) is real for (A), but [they are] not absolutely real independent of each other."[103] In fact, Whitehead describes any entity as in some sense nothing more and nothing less than the sum of its relations to other entities – its synthesis of and reaction to the world around it.[104] A real thing is just that which forces the rest of the universe to in some way conform to it; that is to say, if theoretically, a thing made strictly no difference to any other entity (i.e., it was not related to any other entity), it could not be said to really exist.[105] Relations are not secondary to what a thing is; they are what the thing is.
It must be emphasized, however, that an entity is not merely a sum of its relations, but also a valuation of them and reaction to them.[106] For Whitehead, creativity is the absolute principle of existence, and every entity (whether it is a human being, a tree, or an electron) has some degree of novelty in how it responds to other entities and is not fully determined by causal or mechanistic laws.[107] Most entities do not have consciousness.[108] As a human being's actions cannot always be predicted, the same can be said of where a tree's roots will grow, or how an electron will move, or whether it will rain tomorrow. Moreover, the inability to predict an electron's movement (for instance) is not due to faulty understanding or inadequate technology; rather, the fundamental creativity/freedom of all entities means that there will always remain phenomena that are unpredictable.[109]
The other side of creativity/freedom as the absolute principle is that every entity is constrained by the social structure of existence (i.e., its relations); each actual entity must conform to the settled conditions of the world around it.[105] Freedom always exists within limits. But an entity's uniqueness and individuality arise from its own self-determination as to just how it will take account of the world within the limits that have been set for it.[110]
In summary, Whitehead rejects the idea of separate and unchanging bits of matter as the most basic building blocks of reality, in favour of the idea of reality as interrelated events in the process. He conceives of reality as composed of processes of dynamic "becoming" rather than static "being," emphasizing that all physical things change and evolve and that changeless "essences" such as matter are mere abstractions from the interrelated events that are the final real things that make up the world.[87]
Theory of perception
Since Whitehead's metaphysics described a universe in which all entities experience, he needed a new way of describing perception that was not limited to living, self-conscious beings. The term he coined was "prehension," which comes from the Latin prehensio, meaning "to seize".[111] The term is meant to indicate a kind of perception that can be conscious or unconscious, applying to people as well as electrons. It is also intended to make clear Whitehead's rejection of the theory of representative perception, in which the mind only has private ideas about other entities.[111] For Whitehead, the term "prehension" indicates that the perceiver actually incorporates aspects of the perceived thing into itself.[111] In this way, entities are constituted by their perceptions and relations, rather than being independent of them. Further, Whitehead regards perception as occurring in two modes, causal efficacy (or "physical prehension") and presentational immediacy (or "conceptual prehension").[108]
Whitehead describes causal efficacy as "the experience dominating the primitive living organisms, which have a sense for the fate from which they have emerged, and the fate towards which they go."[112] It is, in other words, the sense of causal relations between entities, a feeling of being influenced and affected by the surrounding environment, unmediated by the senses. Presentational immediacy, on the other hand, is what is usually referred to as "pure sense perception," unmediated by any causal or symbolic interpretation, even unconscious interpretation. In other words, it is pure appearance, which may or may not be delusive (e.g., mistaking an image in a mirror for "the real thing").[113]
In higher organisms (like people), these two modes of perception combine into what Whitehead terms "symbolic reference," which links appearance with causation in a process that is so automatic that both people and animals have difficulty refraining from it. By way of illustration, Whitehead uses the example of a person's encounter with a chair. An ordinary person looks up, sees a coloured shape, and immediately infers that it is a chair. However, an artist, Whitehead supposes, "might not have jumped to the notion of a chair," but instead "might have stopped at the mere contemplation of a beautiful colour and a beautiful shape."[114] This is not the normal human reaction; most people place objects in categories by habit and instinct, without even thinking about it. Moreover, animals do the same thing. Using the same example, Whitehead points out that a dog "would have acted immediately on the hypothesis of a chair and would have jumped onto it by way of using it as such."[115] In this way, symbolic reference is a fusion of pure sense perceptions on the one hand and causal relations on the other, and that it is in fact the causal relationships that dominate the more basic mentality (as the dog illustrates), while it is the sense perceptions which indicate a higher grade mentality (as the artist illustrates).[116]
Evolution and value
Whitehead believed that when asking questions about the basic facts of existence, questions about value and purpose can never be fully escaped. This is borne out in his thoughts on abiogenesis, or the hypothetical natural process by which life arises from simple organic compounds.
Whitehead makes the startling observation that "life is comparatively deficient in survival value."[117] If humans can only exist for about a hundred years, and rocks for eight hundred million, then one is forced to ask why complex organisms ever evolved in the first place; as Whitehead humorously notes, "they certainly did not appear because they were better at that game than the rocks around them."[118] He then observes that the mark of higher forms of life is that they are actively engaged in modifying their environment, an activity which he theorizes is directed toward the three-fold goal of living, living well, and living better.[119] In other words, Whitehead sees life as directed toward the purpose of increasing its own satisfaction. Without such a goal, he sees the rise of life as totally unintelligible.
For Whitehead, there is no such thing as wholly inert matter. Instead, all things have some measure of freedom or creativity, however small, which allows them to be at least partly self-directed. The process philosopher David Ray Griffin coined the term "panexperientialism" (the idea that all entities experience) to describe Whitehead's view, and to distinguish it from panpsychism (the idea that all matter has consciousness).[120]
God
Henri Bergson
William James
John Dewey
"I am also greatly indebted to Bergson, William James, and John Dewey. One of my preoccupations has been to rescue their type of thought from the charge of anti-intellectualism, which rightly or wrongly has been associated with it." – Alfred North Whitehead, Process and Reality, preface.[3]
Whitehead's idea of God differs from traditional monotheistic notions.[121] Perhaps his most famous and pointed criticism of the Christian conception of God is that "the Church gave unto God the attributes which belonged exclusively to Caesar."[122] Here, Whitehead is criticizing Christianity for defining God as primarily a divine king who imposes his will on the world, and whose most important attribute is power. As opposed to the most widely accepted forms of Christianity, Whitehead emphasized an idea of God that he called "the brief Galilean vision of humility":
It does not emphasize the ruling Caesar, or the ruthless moralist, or the unmoved mover. It dwells upon the tender elements in the world, which slowly and in quietness operates by love; and it finds purpose in the present immediacy of a kingdom not of this world. Love neither rules, nor is it unmoved; also it is a little oblivious as to morals. It does not look to the future; for it finds its own reward in the immediate present.[123]
For Whitehead, God is not necessarily tied to religion.[124] Rather than springing primarily from religious faith, Whitehead saw God as necessary for his metaphysical system.[124] His system required that an order exist among possibilities, an order that allowed for novelty in the world and provided an aim to all entities. Whitehead posited that these ordered potentials exist in what he called the primordial nature of God. However, Whitehead was also interested in religious experience. This led him to reflect more intensively on what he saw as the second nature of God, the consequent nature. Whitehead's conception of God as a "dipolar"[125] entity has called for fresh theological thinking.
The primordial nature he described as "the unlimited conceptual realization of the absolute wealth of potentiality"[123] — i.e., the unlimited possibility of the universe. This primordial nature is eternal and unchanging, providing entities in the universe with possibilities for realization. Whitehead also calls this primordial aspect "the lure for feeling, the eternal urge of desire,"[126] pulling the entities in the universe toward as-yet unrealized possibilities.
God's consequent nature, on the other hand, is anything but unchanging; it is God's reception of the world's activity. As Whitehead puts it, "[God] saves the world as it passes into the immediacy of his own life. It is the judgment of a tenderness which loses nothing that can be saved."[127] In other words, God saves and cherishes all experiences forever, and those experiences go on to change the way God interacts with the world. In this way, God is really changed by what happens in the world and the wider universe, lending the actions of finite creatures an eternal significance.
Whitehead thus sees God and the world as fulfilling one another. He sees entities in the world as fluent and changing things that yearn for a permanence which only God can provide by taking them into God's self, thereafter changing God and affecting the rest of the universe throughout time. On the other hand, he sees God as permanent but as deficient in actuality and change: alone, God is merely eternally unrealized possibilities and requires the world to actualize them. God gives creatures permanence, while the creatures give God actuality and change. Here it is worthwhile to quote Whitehead at length:
"In this way God is completed by the individual, fluent satisfactions of finite fact, and the temporal occasions are completed by their everlasting union with their transformed selves, purged into conformation with the eternal order which is the final absolute 'wisdom.' The final summary can only be expressed in terms of a group of antitheses, whose apparent self-contradictions depend on neglect of the diverse categories of existence. In each antithesis there is a shift of meaning which converts the opposition into a contrast.
"It is as true to say that God is permanent and the World fluent, as that the World is permanent and God is fluent.
"It is as true to say that God is one and the World many, as that the World is one and God many.
"It is as true to say that, in comparison with the World, God is actual eminently, as that, in comparison with God, the World is actual eminently.
"It is as true to say that the World is immanent in God, as that God is immanent in the World.
"It is as true to say that God transcends the World, as that the World transcends God.
"It is as true to say that God creates the World, as that the World creates God...
"What is done in the world is transformed into a reality in heaven, and the reality in heaven passes back into the world... In this sense, God is the great companion – the fellow-sufferer who understands."[128]
The above is some of Whitehead's most evocative writing about God, and was powerful enough to inspire the movement known as process theology, a vibrant theological school of thought that continues to thrive today.[129][130]
Religion
For Whitehead, the core of religion was individual. While he acknowledged that individuals cannot ever be fully separated from their society, he argued that life is an internal fact for its own sake before it is an external fact relating to others.[131] His most famous remark on religion is that "religion is what the individual does with his own solitariness ... and if you are never solitary, you are never religious."[132] Whitehead saw religion as a system of general truths that transformed a person's character.[133] He took special care to note that while religion is often a good influence, it is not necessarily good – an idea which he called a "dangerous delusion" (e.g., a religion might encourage the violent extermination of a rival religion's adherents).[134]
However, while Whitehead saw religion as beginning in solitariness, he also saw religion as necessarily expanding beyond the individual. In keeping with his process metaphysics in which relations are primary, he wrote that religion necessitates the realization of "the value of the objective world which is a community derivative from the interrelations of its component individuals."[135] In other words, the universe is a community which makes itself whole through the relatedness of each individual entity to all the others; meaning and value do not exist for the individual alone, but only in the context of the universal community. Whitehead writes further that each entity "can find no such value till it has merged its individual claim with that of the objective universe. Religion is world loyalty. The spirit at once surrenders itself to this universal claim and appropriates it for itself."[136] In this way, the individual and universal/social aspects of religion are mutually dependent. A connection between the works of William DeWitt Hyde and Whitehead further elucidates this necessary duality of social and individual roles in religious experience.[137] Whitehead also described religion more technically as "an ultimate craving to infuse into the insistent particularity of emotion that non-temporal generality which primarily belongs to conceptual thought alone."[138] In other words, religion takes deeply felt emotions and contextualizes them within a system of general truths about the world, helping people to identify their wider meaning and significance. For Whitehead, religion served as a kind of bridge between philosophy and the emotions and purposes of a particular society.[139] It is the task of religion to make philosophy applicable to the everyday lives of ordinary people.
Influence
Isabelle Stengers wrote that "Whiteheadians are recruited among both philosophers and theologians, and the palette has been enriched by practitioners from the most diverse horizons, from ecology to feminism, practices that unite political struggle and spirituality with the sciences of education."[94] In recent decades, attention to Whitehead's work has become more widespread, with interest extending to intellectuals in Europe and China, and coming from such diverse fields as ecology, physics, biology, education, economics, and psychology. One of the first theologians to attempt to interact with Whitehead's thought was the future Archbishop of Canterbury, William Temple. In Temple's Gifford Lectures of 1932-1934 (subsequently published as "Nature, Man and God"), Whitehead is one of a number of philosophers of the emergent evolution approach with which Temple interacts.[140] However, it was not until the 1970s and 1980s that Whitehead's thought drew much attention outside of a small group of philosophers and theologians, primarily Americans, and even today he is not considered especially influential outside of relatively specialized circles.
Early followers of Whitehead were found primarily at the University of Chicago Divinity School, where Henry Nelson Wieman initiated an interest in Whitehead's work that would last for about thirty years.[89] Professors such as Wieman, Charles Hartshorne, Bernard Loomer, Bernard Meland, and Daniel Day Williams made Whitehead's philosophy arguably the most important intellectual thread running through the divinity school.[141] They taught generations of Whitehead scholars, the most notable of whom is John B. Cobb.
Although interest in Whitehead has since faded at Chicago's divinity school, Cobb effectively grabbed the torch and planted it firmly in Claremont, California, where he began teaching at Claremont School of Theology in 1958 and founded the Center for Process Studies with David Ray Griffin in 1973.[142] Largely due to Cobb's influence, today Claremont remains strongly identified with Whitehead's process thought.[143][144]
But while Claremont remains the most concentrated hub of Whiteheadian activity, the place where Whitehead's thought currently seems to be growing the most quickly is in China. In order to address the challenges of modernization and industrialization, China has begun to blend traditions of Taoism, Buddhism, and Confucianism with Whitehead's "constructive post-modern" philosophy in order to create an "ecological civilization".[76] To date, the Chinese government has encouraged the building of twenty-three university-based centres for the study of Whitehead's philosophy,[76][145] and books by process philosophers John Cobb and David Ray Griffin are becoming required reading for Chinese graduate students.[76] Cobb has attributed China's interest in process philosophy partly to Whitehead's stress on the mutual interdependence of humanity and nature, as well as his emphasis on an educational system that includes the teaching of values rather than simply bare facts.[76]
Overall, however, Whitehead's influence is very difficult to characterize. In English-speaking countries, his primary works are little-studied outside of Claremont and a select number of liberal graduate-level theology and philosophy programs. Outside of these circles, his influence is relatively small and diffuse and has tended to come chiefly through the work of his students and admirers rather than Whitehead himself.[146] For instance, Whitehead was a teacher and long-time friend and collaborator of Bertrand Russell, and he also taught and supervised the dissertation of Willard Van Orman Quine,[147] both of whom are important figures in analytic philosophy – the dominant strain of philosophy in English-speaking countries in the 20th century.[148] Whitehead has also had high-profile admirers in the continental tradition, such as French post-structuralist philosopher Gilles Deleuze, who once dryly remarked of Whitehead that "he stands provisionally as the last great Anglo-American philosopher before Wittgenstein's disciples spread their misty confusion, sufficiency, and terror."[149] French sociologist and anthropologist Bruno Latour even went so far as to call Whitehead "the greatest philosopher of the 20th century."[150]
Deleuze's and Latour's opinions, however, are minority ones, as Whitehead has not been recognized as particularly influential within the most dominant philosophical schools.[151] It is impossible to say exactly why Whitehead's influence has not been more widespread, but it may be partly due to his metaphysical ideas seeming somewhat counterintuitive (such as his assertion that matter is an abstraction), or his inclusion of theistic elements in his philosophy,[152] or the perception of metaphysics itself as passé, or simply the sheer difficulty and density of his prose.[24]
Process philosophy and theology
Historically, Whitehead's work has been most influential in the field of American progressive theology.[129][144] The most important early proponent of Whitehead's thought in a theological context was Charles Hartshorne, who spent a semester at Harvard as Whitehead's teaching assistant in 1925, and is widely credited with developing Whitehead's process philosophy into a full-blown process theology.[153] Other notable process theologians include John B. Cobb, David Ray Griffin, Marjorie Hewitt Suchocki, C. Robert Mesle, Roland Faber, and Catherine Keller.
Process theology typically stresses God's relational nature. Rather than seeing God as impassive or emotionless, process theologians view God as "the fellow sufferer who understands," and as the being who is supremely affected by temporal events.[154] Hartshorne points out that people would not praise a human ruler who was unaffected by either the joys or sorrows of his followers – so why would this be a praiseworthy quality in God?[155] Instead, as the being who is most affected by the world, God is the being who can most appropriately respond to the world. However, process theology has been formulated in a wide variety of ways. C. Robert Mesle, for instance, advocates a "process naturalism" — i.e., a process theology without God.[156]
In fact, process theology is difficult to define because process theologians are so diverse and transdisciplinary in their views and interests. John B. Cobb is a process theologian who has also written books on biology and economics. Roland Faber and Catherine Keller integrate Whitehead with poststructuralist, postcolonialist, and feminist theory. Charles Birch was both a theologian and a geneticist. Franklin I. Gamwell writes on theology and political theory. In Syntheism - Creating God in The Internet Age, futurologists Alexander Bard and Jan Söderqvist repeatedly credit Whitehead for the process theology they see rising out of the participatory culture expected to dominate the digital era.
Process philosophy is even more difficult to pin down than process theology. In practice, the two fields cannot be neatly separated. The 32-volume State University of New York series in constructive postmodern thought edited by process philosopher and theologian David Ray Griffin displays the range of areas in which different process philosophers work, including physics, ecology, medicine, public policy, nonviolence, politics, and psychology.[157]
One philosophical school which has historically had a close relationship with process philosophy is American pragmatism. Whitehead himself thought highly of William James and John Dewey, and acknowledged his indebtedness to them in the preface to Process and Reality.[3] Charles Hartshorne (along with Paul Weiss) edited the collected papers of Charles Sanders Peirce, one of the founders of pragmatism. Noted neopragmatist Richard Rorty was in turn a student of Hartshorne.[158] Today, Nicholas Rescher is one example of a philosopher who advocates both process philosophy and pragmatism.
In addition, while they might not properly be called process philosophers, Whitehead has been influential in the philosophy of Gilles Deleuze, Milič Čapek, Isabelle Stengers, Bruno Latour, Susanne Langer, and Maurice Merleau-Ponty.
Science
Scientists of the early 20th century for whom Whitehead's work has been influential include physical chemist Ilya Prigogine, biologist Conrad Hal Waddington, and geneticists Charles Birch and Sewall Wright.[18] Henry Murray dedicated his "Explorations in Personality" to Whitehead, a contemporary at Harvard.
In physics, Whitehead's theory of gravitation articulated a view that might perhaps be regarded as dual to Albert Einstein's general relativity. It has been severely criticized.[160][161] Yutaka Tanaka suggested that the gravitational constant disagrees with experimental findings, and proposed that Einstein's work does not actually refute Whitehead's formulation.[162] Whitehead's view has now been rendered obsolete, with the discovery of gravitational waves, phenomena observed locally that largely violate the kind of local flatness of space that Whitehead assumes. Consequently, Whitehead's cosmology must be regarded as a local approximation, and his assumption of a uniform spatio-temporal geometry, Minkowskian in particular, as an often-locally-adequate approximation. An exact replacement of Whitehead's cosmology would need to admit a Riemannian geometry. Also, although Whitehead himself gave only secondary consideration to quantum theory, his metaphysics of processes has proved attractive to some physicists in that field. Henry Stapp and David Bohm are among those whose work has been influenced by Whitehead.[159]
In the 21st century, Whiteheadian thought is still a stimulating influence: Timothy E. Eastman and Hank Keeton's Physics and Whitehead (2004)[163] and Michael Epperson's Quantum Mechanics and the Philosophy of Alfred North Whitehead (2004)[164] and Foundations of Relational Realism: A Topological Approach to Quantum Mechanics and the Philosophy of Nature (2013),[165] aim to offer Whiteheadian approaches to physics. Brian G. Henning, Adam Scarfe, and Dorion Sagan's Beyond Mechanism (2013) and Rupert Sheldrake's Science Set Free (2012) are examples of Whiteheadian approaches to biology.
Ecology, economy, and sustainability
One of the most promising applications of Whitehead's thought in recent years has been in the area of ecological civilization, sustainability, and environmental ethics.
"Because Whitehead's holistic metaphysics of value lends itself so readily to an ecological point of view, many see his work as a promising alternative to the traditional mechanistic worldview, providing a detailed metaphysical picture of a world constituted by a web of interdependent relations."[24]
This work has been pioneered by John B. Cobb, whose book Is It Too Late? A Theology of Ecology (1971) was the first single-authored book in environmental ethics.[170] Cobb also co-authored a book with leading ecological economist and steady-state theorist Herman Daly entitled For the Common Good: Redirecting the Economy toward Community, the Environment, and a Sustainable Future (1989), which applied Whitehead's thought to economics, and received the Grawemeyer Award for Ideas Improving World Order. Cobb followed this with a second book, Sustaining the Common Good: A Christian Perspective on the Global Economy (1994), which aimed to challenge "economists' zealous faith in the great god of growth."[171]
Education
Whitehead is widely known for his influence in education theory. His philosophy inspired the formation of the Association for Process Philosophy of Education (APPE), which published eleven volumes of a journal titled Process Papers on process philosophy and education from 1996 to 2008.[172] Whitehead's theories on education also led to the formation of new modes of learning and new models of teaching.
One such model is the ANISA model developed by Daniel C. Jordan, which sought to address a lack of understanding of the nature of people in current education systems. As Jordan and Raymond P. Shepard put it: "Because it has not defined the nature of man, education is in the untenable position of having to devote its energies to the development of curricula without any coherent ideas about the nature of the creature for whom they are intended."[173]
Another model is the FEELS model developed by Xie Bangxiu and deployed successfully in China. "FEELS" stands for five things in curriculum and education: Flexible-goals, Engaged-learner, Embodied-knowledge, Learning-through-interactions, and Supportive-teacher.[174] It is used for understanding and evaluating educational curriculum under the assumption that the purpose of education is to "help a person become whole." This work is in part the product of cooperation between Chinese government organizations and the Institute for the Postmodern Development of China.[76]
Whitehead's philosophy of education has also found institutional support in Canada, where the University of Saskatchewan created a Process Philosophy Research Unit and sponsored several conferences on process philosophy and education.[175] Howard Woodhouse at the University of Saskatchewan remains a strong proponent of Whiteheadian education.[176]
Three recent books which further develop Whitehead's philosophy of education include: Modes of Learning: Whitehead's Metaphysics and the Stages of Education (2012) by George Allan; The Adventure of Education: Process Philosophers on Learning, Teaching, and Research (2009) by Adam Scarfe; and "Educating for an Ecological Civilization: Interdisciplinary, Experiential, and Relational Learning" (2017) edited by Marcus Ford and Stephen Rowe. "Beyond the Modern University: Toward a Constructive Postmodern University," (2002) is another text that explores the importance of Whitehead's metaphysics for thinking about higher education.
Business administration
Whitehead has had some influence on philosophy of business administration and organizational theory. This has led in part to a focus on identifying and investigating the effect of temporal events (as opposed to static things) within organizations through an "organization studies" discourse that accommodates a variety of 'weak' and 'strong' process perspectives from a number of philosophers.[177] One of the leading figures having an explicitly Whiteheadian and panexperientialist stance towards management is Mark Dibben,[178] who works in what he calls "applied process thought" to articulate a philosophy of management and business administration as part of a wider examination of the social sciences through the lens of process metaphysics. For Dibben, this allows "a comprehensive exploration of life as perpetually active experiencing, as opposed to occasional – and thoroughly passive – happening."[179] Dibben has published two books on applied process thought, Applied Process Thought I: Initial Explorations in Theory and Research (2008), and Applied Process Thought II: Following a Trail Ablaze (2009), as well as other papers in this vein in the fields of philosophy of management and business ethics.[180]
Margaret Stout and Carrie M. Staton have also written recently on the mutual influence of Whitehead and Mary Parker Follett, a pioneer in the fields of organizational theory and organizational behavior. Stout and Staton see both Whitehead and Follett as sharing an ontology that "understands becoming as a relational process; difference as being related, yet unique; and the purpose of becoming as harmonizing difference."[181] This connection is further analyzed by Stout and Jeannine M. Love in Integrative Process: Follettian Thinking from Ontology to Administration[182]
Political views
Whitehead's political views sometimes appear to be libertarian without the label. He wrote:
Now the intercourse between individuals and between social groups takes one of two forms, force or persuasion. Commerce is the great example of intercourse by way of persuasion. War, slavery, and governmental compulsion exemplify the reign of force.[183]
On the other hand, many Whitehead scholars read his work as providing a philosophical foundation for the social liberalism of the New Liberal movement that was prominent throughout Whitehead's adult life. Morris wrote that "... there is good reason for claiming that Whitehead shared the social and political ideals of the new liberals."[184]
Primary works
Books written by Whitehead, listed by date of publication.
• A Treatise on Universal Algebra. Cambridge: Cambridge University Press, 1898. ISBN 1-4297-0032-7. Available online at http://projecteuclid.org/euclid.chmm/1263316509 Archived 3 September 2014 at the Wayback Machine.
• The Axioms of Descriptive Geometry. Cambridge: Cambridge University Press, 1907.[185] Available online at http://quod.lib.umich.edu/u/umhistmath/ABN2643.0001.001.
• with Bertrand Russell. Principia Mathematica, Volume I. Cambridge: Cambridge University Press, 1910. Available online at http://www.hti.umich.edu/cgi/b/bib/bibperm?q1=AAT3201.0001.001. Vol. 1 to *56 is available as a CUP paperback.[186][187][188]
• An Introduction to Mathematics. Cambridge: Cambridge University Press, 1911. Available online at http://quod.lib.umich.edu/u/umhistmath/AAW5995.0001.001. Vol. 56 of the Great Books of the Western World series.
• with Bertrand Russell. Principia Mathematica, Volume II. Cambridge: Cambridge University Press, 1912. Available online at http://www.hti.umich.edu/cgi/b/bib/bibperm?q1=AAT3201.0002.001.
• with Bertrand Russell. Principia Mathematica, Volume III. Cambridge: Cambridge University Press, 1913. Available online at http://www.hti.umich.edu/cgi/b/bib/bibperm?q1=AAT3201.0003.001.
• The Organization of Thought Educational and Scientific. London: Williams & Norgate, 1917. Available online at https://archive.org/details/organisationofth00whit.
• An Enquiry Concerning the Principles of Natural Knowledge. Cambridge: Cambridge University Press, 1919. Available online at https://archive.org/details/enquiryconcernpr00whitrich.
• The Concept of Nature. Cambridge: Cambridge University Press, 1920. Based on the November 1919 Tarner Lectures delivered at Trinity College. Available online at https://archive.org/details/cu31924012068593.
• The Principle of Relativity with Applications to Physical Science. Cambridge: Cambridge University Press, 1922. Available online at https://archive.org/details/theprincipleofre00whituoft.
• Science and the Modern World. New York: Macmillan Company, 1925. Vol. 55 of the Great Books of the Western World series.
• Religion in the Making. New York: Macmillan Company, 1926. Based on the 1926 Lowell Lectures.
• Symbolism, Its Meaning and Effect. New York: Macmillan Co., 1927. Based on the 1927 Barbour-Page Lectures delivered at the University of Virginia.
• Process and Reality: An Essay in Cosmology. New York: Macmillan Company, 1929. Based on the 1927–28 Gifford Lectures delivered at the University of Edinburgh. The 1978 Free Press "corrected edition" edited by David Ray Griffin and Donald W. Sherburne corrects many errors in both the British and American editions, and also provides a comprehensive index.
• The Aims of Education and Other Essays. New York: Macmillan Company, 1929.
• The Function of Reason. Princeton: Princeton University Press, 1929. Based on the March 1929 Louis Clark Vanuxem Foundation Lectures delivered at Princeton University.
• Adventures of Ideas. New York: Macmillan Company, 1933. Also published by Cambridge: Cambridge University Press, 1933.
• Nature and Life. Chicago: University of Chicago Press, 1934.
• Modes of Thought. New York: MacMillan Company, 1938.
• "Mathematics and the Good." In The Philosophy of Alfred North Whitehead, edited by Paul Arthur Schilpp, 666–681. Evanston and Chicago: Northwestern University Press, 1941.
• "Immortality." In The Philosophy of Alfred North Whitehead, edited by Paul Arthur Schilpp, 682–700. Evanston and Chicago: Northwestern University Press, 1941.
• Essays in Science and Philosophy. London: Philosophical Library, 1947.
• with Allison Heartz Johnson, ed. The Wit and Wisdom of Whitehead. Boston: Beacon Press, 1948.
In addition, the Whitehead Research Project of the Center for Process Studies is currently working on a critical edition of Whitehead's writings, which is set to include notes taken by Whitehead's students during his Harvard classes, correspondence, and corrected editions of his books.[53]
• Paul A. Bogaard and Jason Bell, eds. The Harvard Lectures of Alfred North Whitehead, 1924–1925: Philosophical Presuppositions of Science. Cambridge: Cambridge University Press, 2017.
See also
• Great refusal
• Pancreativism
• Relationalism
• Speculative realism
• Whitehead's point-free geometry
• A.N. Whitehead at Sherborne School
References
1. Alfred North Whitehead at the Mathematics Genealogy Project
2. Alfred North Whitehead, Process and Reality (New York: The Free Press, 1978), 39.
3. Alfred North Whitehead, Process and Reality (New York: The Free Press, 1978), xii.
4. Alfred North Whitehead, Process and Reality (New York: The Free Press, 1978), xiii.
5. Alfred North Whitehead, Process and Reality (New York: The Free Press, 1978), xi.
6. Michel Weber and Will Desmond, eds., Handbook of Whiteheadian Process Thought, Volume 1 (Frankfurt: Ontos Verlag, 2008), 17.
7. John B. Cobb Jr. and David Ray Griffin, Process Theology: An Introductory Exposition (Philadelphia: Westminster Press, 1976), 174.
8. Michel Weber and Will Desmond, eds., Handbook of Whiteheadian Process Thought, Volume 1 (Frankfurt: Ontos Verlag, 2008), 26.
9. An Interview with Donald Davidson.
10. Gilles Deleuze and Claire Parnet, Dialogues II, Columbia University Press, 2007, p. vii.
11. John B. Cobb Jr. and David Ray Griffin, Process Theology: An Introductory Exposition (Philadelphia: Westminster Press, 1976), 164-165.
12. John B. Cobb Jr. and David Ray Griffin, Process Theology: An Introductory Exposition (Philadelphia: Westminster Press, 1976), 175.
13. Thomas J. Fararo, "On the Foundations of the Theory of Action in Whitehead and Parsons", in Explorations in General Theory in Social Science, ed. Jan J. Loubser et al. (New York: The Free Press, 1976), chapter 5.
14. Michel Weber and Will Desmond, eds., Handbook of Whiteheadian Process Thought, Volume 1 (Frankfurt: Ontos Verlag, 2008), 25.
15. "Alfred North Whitehead - Biography". European Graduate School. Archived from the original on 3 September 2013. Retrieved 12 December 2013.
16. Wolfgang Smith, Cosmos and Transcendence: Breaking Through the Barrier of Scientistic Belief (Peru, Illinois: Sherwood Sugden and Company, 1984), 3.
17. Michel Weber and Will Desmond, eds., Handbook of Whiteheadian Process Thought, Volume 1 (Frankfurt: Ontos Verlag, 2008), 13.
18. Charles Birch, "Why Aren't We Zombies? Neo-Darwinism and Process Thought", in Back to Darwin: A Richer Account of Evolution, ed. John B. Cobb Jr. (Grand Rapids: William B. Eerdmans Publishing Company, 2008), 252.
19. "Young Voegelin in America". 6 March 2011.
20. "Integrating Whitehead".
21. Griffin, David Ray (2001). Reenchantment without Supernaturalism: A Process Philosophy of Religion. Ithaca: Cornell University Press, vii.
22. "The Modern Library's Top 100 Nonfiction Books of the Century". 30 April 1999. The New York Times. Accessed 21 November 2013.
23. C. Robert Mesle, Process-Relational Philosophy: An Introduction to Alfred North Whitehead (West Conshohocken: Templeton Foundation Press, 2009), 9.
24. Philip Rose, On Whitehead (Belmont: Wadsworth, 2002), preface.
25. Cobb, John B., Jr.; Schwartz, Wm. Andrew (2018). Putting Philosophy to Work: Toward an Ecological Civilization. Process Century Press. ISBN 978-1-940447-33-9.{{cite book}}: CS1 maint: multiple names: authors list (link)
26. Victor Lowe, Alfred North Whitehead: The Man and his Work, Vol I (Baltimore: The Johns Hopkins Press, 1985), 2.
27. Lowe, Victor (1985). Alfred North Whitehead: The Man and his Work, Vol I Baltimore: The Johns Hopkins Press, 13.
28. "Olympedia – Walter Buckmaster".
29. Griffin Ed., Nicholas (1992). The Selected Letters of Bertrand Russell, Volume 1, pp.215-217. New York: Houghton Miffin. ISBN 0-395-56269-4.
30. Britannica, The Editors of Encyclopaedia. "Bloomsbury group". Encyclopedia Britannica, 20 Feb. 2021, https://www.britannica.com/topic/Bloomsbury-group. Accessed 29 May 2022.
31. "Alfred North Whitehead (1861–1947)". The Old Shirburnian Society. 10 October 2020. Archived from the original on 7 November 2021. Retrieved 8 November 2021.
32. Victor Lowe, Alfred North Whitehead: The Man and his Work, Vol I (Baltimore: The Johns Hopkins Press, 1985), 54–60.
33. Lowe, Victor (1985). Alfred North Whitehead: The Man and his Work, Vol I. Baltimore: The Johns Hopkins Press, 63.
34. Victor Lowe, Alfred North Whitehead: The Man and his Work, Vol I (Baltimore: The Johns Hopkins Press, 1985), 72.
35. Victor Lowe, Alfred North Whitehead: The Man and his Work, Vol I (Baltimore: The Johns Hopkins Press, 1985), 103-109.
36. On Whitehead the mathematician and logician, see Ivor Grattan-Guinness, The Search for Mathematical Roots 1870–1940: Logics, Set Theories, and the Foundations of Mathematics from Cantor through Russell to Gödel (Princeton: Princeton University Press, 2000), and Quine's chapter in Paul Schilpp, The Philosophy of Alfred North Whitehead (New York: Tudor Publishing Company, 1941), 125–163.
37. Victor Lowe, Alfred North Whitehead: The Man and his Work, Vol I (Baltimore: The Johns Hopkins Press, 1985), 112.
38. Victor Lowe, Alfred North Whitehead: The Man and his Work, Vol II (Baltimore: The Johns Hopkins Press, 1990), 2.
39. Victor Lowe, Alfred North Whitehead: The Man and his Work, Vol II (Baltimore: The Johns Hopkins Press, 1990), 6-8.
40. Victor Lowe, Alfred North Whitehead: The Man and his Work, Vol II (Baltimore: The Johns Hopkins Press, 1990), 26-27.
41. Victor Lowe, Alfred North Whitehead: The Man and his Work, Vol II (Baltimore: The Johns Hopkins Press, 1990), 72-74.
42. Victor Lowe, Alfred North Whitehead: The Man and his Work, Vol II (Baltimore: The Johns Hopkins Press, 1990), 127.
43. Victor Lowe, Alfred North Whitehead: The Man and his Work, Vol II (Baltimore: The Johns Hopkins Press, 1990), 132.
44. Victor Lowe, Alfred North Whitehead: The Man and his Work, Vol I (Baltimore: The Johns Hopkins Press, 1985), 3–4.
45. "Alfred North Whitehead". American Academy of Arts & Sciences. 9 February 2023. Retrieved 11 August 2023.
46. "APS Member History". search.amphilsoc.org. Retrieved 11 August 2023.
47. Victor Lowe, Alfred North Whitehead: The Man and his Work, Vol II (Baltimore: The Johns Hopkins Press, 1990), 34.
48. "Valley Heritage booklet". Fyfield and West Overton Parish Council. 1987. Retrieved 29 November 2020.
49. Lowe, Victor (31 March 1974). "Whitehead's 1911 Criticism of The Problems of Philosophy". Journal of Bertrand Russell Studies. 13: 1–28.
50. Victor Lowe, Alfred North Whitehead: The Man and his Work, Vol II (Baltimore: The Johns Hopkins Press, 1990), 262.
51. Victor Lowe, Alfred North Whitehead: The Man and his Work, Vols I & II (Baltimore: The Johns Hopkins Press, 1985 & 1990).
52. Victor Lowe, Alfred North Whitehead: The Man and his Work, Vol I (Baltimore: The Johns Hopkins Press, 1985), 7.
53. "Critical Edition of Whitehead", last modified 16 July 2013, Whitehead Research Project, accessed 21 November 2013, http://whiteheadresearch.org/research/cew/press-release.shtml Archived 9 December 2013 at the Wayback Machine.
54. "The Edinburgh Critical Edition of the Complete Works of Alfred North Whitehead". Edinburgh University Press Books. Retrieved 22 May 2018.
55. Christoph Wassermann, "The Relevance of An Introduction to Mathematics to Whitehead's Philosophy", Process Studies 17 (1988): 181. Available online at "The Relevance of An Introduction to Mathematics to Whitehead?s Philosophy". Archived from the original on 3 December 2013. Retrieved 21 November 2013.
56. "Whitehead, Alfred North", last modified 8 May 2007, Gary L. Herstein, Internet Encyclopedia of Philosophy, accessed 21 November 2013, http://www.iep.utm.edu/whitehed/.
57. George Grätzer, Universal Algebra (Princeton: Van Nostrand Co., Inc., 1968), v.
58. Cf. Michel Weber and Will Desmond (eds.). Handbook of Whiteheadian Process Thought (Frankfurt / Lancaster, Ontos Verlag, Process Thought X1 & X2, 2008) and Ronny Desmet & Michel Weber (edited by), Whitehead. The Algebra of Metaphysics. Applied Process Metaphysics Summer Institute Memorandum, Louvain-la-Neuve, Les Éditions Chromatika, 2010.
59. Alexander Macfarlane, "Review of A Treatise on Universal Algebra", Science 9 (1899): 325.
60. G. B. Mathews (1898) A Treatise on Universal Algebra from Nature 58:385 to 7 (#1504)
61. Victor Lowe, Alfred North Whitehead: The Man and his Work, Vol I (Baltimore: The Johns Hopkins Press, 1985), 190–191.
62. Alfred North Whitehead, A Treatise on Universal Algebra (Cambridge: Cambridge University Press, 1898), v. Available online at http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?handle=euclid.chmm/1263316510&view=body&content-type=pdf_1
63. Barron Brainerd, "Review of Universal Algebra by P. M. Cohn", American Mathematical Monthly, 74 (1967): 878–880.
64. Alfred North Whitehead, Principia Mathematica Volume 2, Second Edition (Cambridge: Cambridge University Press, 1950), 83.
65. Hal Hellman, Great Feuds in Mathematics: Ten of the Liveliest Disputes Ever (Hoboken: John Wiley & Sons, 2006). Available online at https://books.google.com/books?id=ft8bEGf_OOcC&pg=PT12
66. "Principia Mathematica", last modified 3 December 2013, Andrew David Irvine, ed. Edward N. Zalta, The Stanford Encyclopedia of Philosophy (Winter 2013 Edition), accessed 5 December 2013, http://plato.stanford.edu/entries/principia-mathematica/#HOPM.
67. Stephen Cole Kleene, Mathematical Logic (New York: Wiley, 1967), 250.
68. "'Principia Mathematica' Celebrates 100 Years", last modified 22 December 2010, NPR, accessed 21 November 2013, https://www.npr.org/2010/12/22/132265870/Principia-Mathematica-Celebrates-100-Years
69. "Principia Mathematica", last modified 3 December 2013, Andrew David Irvine, ed. Edward N. Zalta, The Stanford Encyclopedia of Philosophy (Winter 2013 Edition), accessed 5 December 2013, http://plato.stanford.edu/entries/principia-mathematica/#SOPM.
70. Alfred North Whitehead, An Introduction to Mathematics, (New York: Henry Holt and Company, 1911), 8.
71. Christoph Wassermann, "The Relevance of An Introduction to Mathematics to Whitehead's Philosophy", Process Studies 17 (1988): 181–182. Available online at "The Relevance of An Introduction to Mathematics to Whitehead?s Philosophy". Archived from the original on 3 December 2013. Retrieved 21 November 2013.
72. Christoph Wassermann, "The Relevance of An Introduction to Mathematics to Whitehead's Philosophy", Process Studies 17 (1988): 182. Available online at "The Relevance of An Introduction to Mathematics to Whitehead?s Philosophy". Archived from the original on 3 December 2013. Retrieved 21 November 2013.
73. Committee To Inquire into the Position of Classics in the Educational System of the United Kingdom, Report of the Committee Appointed by the Prime Minister to Inquire into the Position of Classics in the Educational System of the United Kingdom, (London: His Majesty's Stationery Office, 1921), 1, 282. Available online at https://archive.org/details/reportofcommitt00grea.
74. Alfred North Whitehead, The Aims of Education and Other Essays (New York: The Free Press, 1967), 1–2.
75. Alfred North Whitehead, The Aims of Education and Other Essays (New York: The Free Press, 1967), 2.
76. "China embraces Alfred North Whitehead", last modified 10 December 2008, Douglas Todd, The Vancouver Sun, accessed 5 December 2013, http://blogs.vancouversun.com/2008/12/10/china-embraces-alfred-north-whitehead/ Archived 10 March 2016 at the Wayback Machine.
77. Alfred North Whitehead, The Aims of Education and Other Essays (New York: The Free Press, 1967), 13.
78. Alfred North Whitehead, The Aims of Education and Other Essays (New York: The Free Press, 1967), 93.
79. Alfred North Whitehead, The Aims of Education and Other Essays (New York: The Free Press, 1967), 98.
80. "An Iconic College View: Harvard University, circa 1900. Richard Rummell (1848–1924)", last modified 6 July 2011, Graham Arader, accessed 5 December 2013, http://grahamarader.blogspot.com/2011/07/iconic-college-view-harvard-university.html.
81. Alfred North Whitehead to Bertrand Russell, 13 February 1895, Bertrand Russell Archives, Archives and Research Collections, McMaster Library, McMaster University, Hamilton, Ontario, Canada.
82. A. J. Ayer, Language, Truth and Logic, (New York: Penguin, 1971), 22.
83. George P. Conger, "Whitehead lecture notes: Seminary in Logic: Logical and Metaphysical Problems", 1927, Manuscripts and Archives, Yale University Library, Yale University, New Haven, Connecticut.
84. Alfred North Whitehead, Process and Reality (New York: The Free Press, 1978), 4.
85. Alfred North Whitehead, Process and Reality (New York: The Free Press, 1978), 11.
86. Alfred North Whitehead, Science and the Modern World (New York: The Free Press, 1967), 17.
87. Alfred North Whitehead, Process and Reality (New York: The Free Press, 1978), 18.
88. Victor Lowe, Alfred North Whitehead: The Man and his Work, Vol II (Baltimore: The Johns Hopkins Press, 1990), 127, 133.
89. Gary Dorrien, The Making of American Liberal Theology: Crisis, Irony, and Postmodernity, 1950–2005 (Louisville: Westminster John Knox Press, 2006), 123–124.
90. Victor Lowe, Alfred North Whitehead: The Man and his Work, Vol II (Baltimore: The Johns Hopkins Press, 1990), 250.
91. Gary Dorrien, "The Lure and Necessity of Process Theology", CrossCurrents 58 (2008): 320.
92. Henry Nelson Wieman, "A Philosophy of Religion", The Journal of Religion 10 (1930): 137.
93. Peter Simons, "Metaphysical systematics: A lesson from Whitehead", Erkenntnis 48 (1998), 378.
94. Isabelle Stengers, Thinking with Whitehead: A Free and Wild Creation of Concepts, trans. Michael Chase (Cambridge, Massachusetts: Harvard University Press, 2011), 6.
95. David Ray Griffin, Whitehead's Radically Different Postmodern Philosophy: An Argument for Its Contemporary Relevance (Albany: State University of New York Press, 2007), viii–ix.
96. Alfred North Whitehead, Process and Reality (New York: The Free Press, 1978), 208.
97. Alfred North Whitehead, Science and the Modern World (New York: The Free Press, 1967), 52–55.
98. Alfred North Whitehead, Process and Reality (New York: The Free Press, 1978), 34–35.
99. Alfred North Whitehead, Process and Reality (New York: The Free Press, 1978), 34.
100. Alfred North Whitehead, Science and the Modern World (New York: The Free Press, 1967), 54–55.
101. Alfred North Whitehead, Process and Reality (New York: The Free Press, 1978), 183.
102. Alfred North Whitehead, Symbolism: Its Meaning and Effect (New York: Fordham University Press, 1985), 38–39.
103. Louise R. Heath, "Notes on Whitehead's Philosophy 3b: Philosophical Presuppositions of Science", 27 September 1924, Whitehead Research Project, Center for Process Studies, Claremont, California.
104. Alfred North Whitehead, Symbolism: Its Meaning and Effect (New York: Fordham University Press, 1985), 26.
105. Alfred North Whitehead, Symbolism: Its Meaning and Effect (New York: Fordham University Press, 1985), 39.
106. Alfred North Whitehead, Process and Reality (New York: The Free Press, 1978), 19.
107. Alfred North Whitehead, Process and Reality (New York: The Free Press, 1978), 21.
108. Alfred North Whitehead, Process and Reality (New York: The Free Press, 1978), 23.
109. Charles Hartshorne, "Freedom Requires Indeterminism and Universal Causality", The Journal of Philosophy 55 (1958): 794.
110. John B. Cobb, A Christian Natural Theology (Louisville: Westminster John Knox Press, 1978), 52.
111. David Ray Griffin, Reenchantment Without Supernaturalism: A Process Philosophy of Religion (Ithaca: Cornell University Press, 2001), 79.
112. Alfred North Whitehead, Symbolism: Its Meaning and Effect (New York: Fordham University Press, 1985), 44.
113. Alfred North Whitehead, Symbolism: Its Meaning and Effect (New York: Fordham University Press, 1985), 24.
114. Alfred North Whitehead, Symbolism: Its Meaning and Effect (New York: Fordham University Press, 1985), 3.
115. Alfred North Whitehead, Symbolism: Its Meaning and Effect (New York: Fordham University Press, 1985), 4.
116. Alfred North Whitehead, Symbolism: Its Meaning and Effect (New York: Fordham University Press, 1985), 49.
117. Alfred North Whitehead, The Function of Reason (Boston: Beacon Press, 1958), 4.
118. Alfred North Whitehead, The Function of Reason (Boston: Beacon Press, 1958), 4–5.
119. Alfred North Whitehead, The Function of Reason (Boston: Beacon Press, 1958), 8.
120. David Ray Griffin, Reenchantment Without Supernaturalism: A Process Philosophy of Religion (Ithaca: Cornell University Press, 2001), 97.
121. Roland Faber, God as Poet of the World: Exploring Process Theologies (Louisville: Westminster John Knox Press, 2008), chapters 4–5.
122. Alfred North Whitehead, Process and Reality (New York: The Free Press, 1978), 342.
123. Alfred North Whitehead, Process and Reality (New York: The Free Press, 1978), 343.
124. Alfred North Whitehead, Process and Reality (New York: The Free Press, 1978), 207.
125. Alfred North Whitehead, Process and Reality (New York: The Free Press, 1978), 345.
126. Alfred North Whitehead, Process and Reality (New York: The Free Press, 1978), 344.
127. Alfred North Whitehead, Process and Reality (New York: The Free Press, 1978), 346.
128. Alfred North Whitehead, Process and Reality (New York: The Free Press, 1978), 347–348, 351.
129. Bruce G. Epperly, Process Theology: A Guide for the Perplexed (New York: T&T Clark, 2011), 12.
130. Roland Faber, God as Poet of the World: Exploring Process Theologies (Louisville: Westminster John Knox Press, 2008), chapter 1.
131. Alfred North Whitehead, Religion in the Making (New York: Fordham University Press, 1996), 15–16.
132. Alfred North Whitehead, Religion in the Making (New York: Fordham University Press, 1996), 16–17.
133. Alfred North Whitehead, Religion in the Making (New York: Fordham University Press, 1996), 15.
134. Alfred North Whitehead, Religion in the Making (New York: Fordham University Press, 1996), 18.
135. Alfred North Whitehead, Religion in the Making (New York: Fordham University Press, 1996), 59.
136. Alfred North Whitehead, Religion in the Making (New York: Fordham University Press, 1996), 60.
137. Unraveling the Seven Riddles of the Universe. Hamilton Books. 2022. ISBN 9780761872900.
138. Alfred North Whitehead, Process and Reality (New York: The Free Press, 1978), 16.
139. Alfred North Whitehead, Process and Reality (New York: The Free Press, 1978), 15.
140. George Garin, "Theistic Evolution in a Sacramental Universe: The Theology of William Temple Against the Background of Process Thinkers (Whitehead, Alexander, Etc.)," (Protestant University Press, Kinshasa, The Congo, 1991).
141. Gary Dorrien, "The Lure and Necessity of Process Theology", CrossCurrents 58 (2008): 321–322.
142. David Ray Griffin, "John B. Cobb, Jr.: A Theological Biography", in Theology and the University: Essays in Honor of John B. Cobb, Jr., ed. David Ray Griffin and Joseph C. Hough, Jr. (Albany: State University of New York Press, 1991), 229.
143. Gary Dorrien, "The Lure and Necessity of Process Theology", CrossCurrents 58 (2008): 334.
144. Victor Lowe, Alfred North Whitehead: The Man and his Work, Vol I (Baltimore: The Johns Hopkins Press, 1985), 5.
145. "About Us". www.postmodernchina.org. The Institute for the Postmodern Development of China. Archived from the original on 2 December 2013. Retrieved 21 November 2013.
146. "Whitehead, Alfred North", last modified 8 May 2007, Gary L. Herstein, Internet Encyclopedia of Philosophy, accessed 20 July 2015, http://www.iep.utm.edu/whitehed/.
147. "Quine Biography", last modified October 2003, John J. O'Connor and Edmund F. Robertson, MacTutor History of Mathematics archive, University of St Andrews, accessed 5 December 2013, http://www-history.mcs.st-andrews.ac.uk/Biographies/Quine.html.
148. John Searle, "Contemporary Philosophy in the United States", in N. Bunnin and E.P. Tsui-James, eds., The Blackwell Companion to Philosophy, 2nd ed., (Oxford: Blackwell, 2003), 1.
149. Gilles Deleuze, The Fold: Leibniz and the Baroque, trans. Tom Conley (Minneapolis: University of Minnesota Press, 1993), 76.
150. Bruno Latour, preface to Thinking with Whitehead: A Free and Wild Creation of Concepts, by Isabelle Stengers, trans. Michael Chase (Cambridge, Massachusetts: Harvard University Press, 2011), x.
151. "Alfred North Whitehead", last modified 10 March 2015, Andrew David Irvine, ed. Edward N. Zalta, The Stanford Encyclopedia of Philosophy (Spring 2015 Edition), accessed 20 July 2015, http://plato.stanford.edu/entries/whitehead/#WI
152. "Alfred North Whitehead", last modified 1 October 2013, Andrew David Irvine, ed. Edward N. Zalta, The Stanford Encyclopedia of Philosophy (Winter 2013 Edition), accessed 21 November 2013, http://plato.stanford.edu/entries/whitehead/#WI
153. Charles Hartshorne, A Christian Natural Theology, 2nd edition (Louisville, Westminster John Knox Press, 2007), 112.
154. Alfred North Whitehead, Process and Reality (New York: The Free Press, 1978), 351.
155. Charles Hartshorne, The Divine Relativity: A Social Conception of God (New Haven: Yale University Press, 1964), 42–43.
156. See part IV of Mesle's Process Theology: A Basic Introduction (St. Louis: Chalice Press, 1993).
157. "Search Results For: SUNY series in Constructive Postmodern Thought", Sunypress.edu, accessed 5 December 2013, http://www.sunypress.edu/Searchadv.aspx?IsSubmit=true&CategoryID=6899 Archived 19 November 2013 at the Wayback Machine.
158. "Richard Rorty", last modified 16 June 2007, Bjørn Ramberg, ed. Edward N. Zalta, The Stanford Encyclopedia of Philosophy (Spring 2009 Edition), accessed 5 December 2013, http://plato.stanford.edu/archives/spr2009/entries/rorty/.
159. See David Ray Griffin, Physics and the Ultimate Significance of Time (Albany: State University of New York Press, 1986).
160. Chandrasekhar, S. (1979). Einstein and general relativity, Am. J. Phys. 47: 212–217.
161. Will, C.M. (1981/1993). Theory and Experiment in Gravitational Physics, revised edition, Cambridge University Press, Cambridge UK, ISBN 978-0-521-43973-2, p. 139.
162. Yutaka Tanaka, "The Comparison between Whitehead's and Einstein's Theories of Relativity", Historia Scientiarum 32 (1987).
163. Timothy E. Eastman and Hank Keeton, eds., Physics and Whitehead: Quantum, Process, and Experience (Albany: State University of New York Press, 2004).
164. Michael Epperson, Quantum Mechanics and the Philosophy of Alfred North Whitehead (New York: Fordham University Press, 2004).
165. Michael Epperson & Elias Zafiris, Foundations of Relational Realism: A Topological Approach to Quantum Mechanics and the Philosophy of Nature (New York: Rowman & Littlefield, 2013).
166. Roland Faber, God as Poet of the World: Exploring Process Theologies (Louisville: Westminster John Knox Press, 2008), 35.
167. C. Robert Mesle, Process Theology (St. Louis: Chalice Press, 1993), 126.
168. Gary Dorrien, "The Lure and Necessity of Process Theology", CrossCurrents 58 (2008): 316.
169. Monica Coleman, Nancy R. Howell, and Helene Tallon Russell, Creating Women's Theology: A Movement Engaging Process Thought (Wipf and Stock, 2011), 13.
170. "History of Environmental Ethics for the Novice", last modified 15 March 2011, The Center for Environmental Philosophy, accessed 21 November 2013, http://www.cep.unt.edu/novice.html.
171. John B. Cobb Jr., Sustaining the Common Good: A Christian Perspective on the Global Economy (Cleveland: The Pilgrim Press, 1994), back cover.
172. See Process Papers, a publication of the Association for Process Philosophy of Education. Volume 1 published in 1996, Volume 11 (final volume) published in 2008.
173. Daniel C. Jordan and Raymond P. Shepard, "The Philosophy of the ANISA Model", Process Papers 6, 38–39.
174. "FEELS: A Constructive Postmodern Approach To Curriculum and Education", Xie Bangxiu, JesusJazzBuddhism.org, accessed 5 December 2013, http://www.jesusjazzbuddhism.org/feels.html Archived 2 November 2013 at the Wayback Machine.
175. "International Conferences – University of Saskatchewan", University of Saskatchewan, accessed 5 December 2013, https://www.usask.ca/usppru/international-conferences.php Archived 7 May 2016 at the Wayback Machine.
176. "Dr. Howard Woodhouse" Archived 7 May 2016 at the Wayback Machine, University of Saskatchewan, accessed 5 December 2013
177. Tor Hernes, A Process Theory of Organization (Oxford University Press, 2014)
178. Mark R. Dibben and John B. Cobb Jr., "Special Focus: Process Thought and Organization Studies," in Process Studies 32 (2003).
179. "Mark Dibben – School of Management – University of Tasmania, Australia", last modified 16 July 2013, University of Tasmania, accessed 21 November 2013, http://www.utas.edu.au/business-and-economics/people/profiles/accounting/Mark-Dibben Archived 13 December 2013 at the Wayback Machine.
180. Mark Dibben, "Exploring the Processual Nature of Trust and Cooperation in Organisations: A Whiteheadian Analysis," in Philosophy of Management 4 (2004): 25-39; Mark Dibben, "Organisations and Organising: Understanding and Applying Whitehead's Processual Account," in Philosophy of Management 7 (2009); Cristina Neesham and Mark Dibben, "The Social Value of Business: Lessons from Political Economy and Process Philosophy," in Applied Ethics: Remembering Patrick Primeaux (Research in Ethical Issues in Organizations, Volume 8), ed. Michael Schwartz and Howard Harris (Emerald Group Publishing Limited, 2012): 63-83.
181. Margaret Stout & Carrie M. Staton, "The Ontology of Process Philosophy in Follett's Administrative Theory" Administrative Theory & Praxis 33 (2011): 268.
182. Margaret Stout & Jeannine M. Love, Integrative Process: Follettian Thinking from Ontology to Administration, (Anoka, MN: Process Century Press 2015).
183. Adventures of Ideas p. 105, 1933 edition; p. 83, 1967 ed.
184. Morris, Randall C., Journal of the History of Ideas 51: 75-92. p. 92.
185. F.W. Owens, "Review: The Axioms of Descriptive Geometry by A. N. Whitehead", Bulletin of the American Mathematical Society 15 (1909): 465–466. Available online at http://www.ams.org/journals/bull/1909-15-09/S0002-9904-1909-01815-4/S0002-9904-1909-01815-4.pdf.
186. James Byrnie Shaw, "Review: Principia Mathematica by A. N. Whitehead and B. Russell, Vol. I, 1910", Bulletin of the American Mathematical Society 18 (1912): 386–411. Available online at http://www.ams.org/journals/bull/1912-18-08/S0002-9904-1912-02233-4/S0002-9904-1912-02233-4.pdf.
187. Benjamin Abram Bernstein, "Review: Principia Mathematica by A. N. Whitehead and B. Russell, Vol. I, Second Edition, 1925", Bulletin of the American Mathematical Society 32 (1926): 711–713. Available online at http://www.ams.org/journals/bull/1926-32-06/S0002-9904-1926-04306-8/S0002-9904-1926-04306-8.pdf.
188. Alonzo Church, "Review: Principia Mathematica by A. N. Whitehead and B. Russell, Volumes II and III, Second Edition, 1927", Bulletin of the American Mathematical Society 34 (1928): 237–240. Available online at http://www.ams.org/journals/bull/1928-34-02/S0002-9904-1928-04525-1/S0002-9904-1928-04525-1.pdf.
Further reading
For the most comprehensive list of resources related to Whitehead, see the thematic bibliography of the Center for Process Studies.
• Casati, Roberto, and Achille C. Varzi. Parts and Places: The Structures of Spatial Representation. Cambridge, Massachusetts: The MIT Press, 1999.
• Ford, Lewis. Emergence of Whitehead's Metaphysics, 1925–1929. Albany: State University of New York Press, 1985.
• Hartshorne, Charles. Whitehead's Philosophy: Selected Essays, 1935–1970. Lincoln and London: University of Nebraska Press, 1972.
• Henning, Brian G. The Ethics of Creativity: Beauty, Morality, and Nature in a Processive Cosmos. Pittsburgh: University of Pittsburgh Press, 2005.
• Holtz, Harald and Ernest Wolf-Gazo, eds. Whitehead und der Prozeßbegriff / Whitehead and The Idea of Process. Proceedings of the First International Whitehead-Symposion. Verlag Karl Alber, Freiburg i. B. / München, 1984. ISBN 3-495-47517-6
• Jones, Judith A. Intensity: An Essay in Whiteheadian Ontology. Nashville: Vanderbilt University Press, 1998.
• Kraus, Elizabeth M. The Metaphysics of Experience. New York: Fordham University Press, 1979.
• Malik, Charles H. The Systems of Whitehead's Metaphysics. Zouq Mosbeh, Lebanon: Notre Dame Louaize, 2016. 436 pp.
• McDaniel, Jay. What is Process Thought?: Seven Answers to Seven Questions. Claremont: P&F Press, 2008.
• McHenry, Leemon. The Event Universe: The Revisionary Metaphysics of Alfred North Whitehead. Edinburgh: Edinburgh University Press, 2015.
• Nobo, Jorge L. Whitehead's Metaphysics of Extension and Solidarity. Albany: State University of New York Press, 1986.
• Price, Lucien. Dialogues of Alfred North Whitehead. New York: Mentor Books, 1956.
• Quine, Willard Van Orman. "Whitehead and the rise of modern logic." In The Philosophy of Alfred North Whitehead, edited by Paul Arthur Schilpp, 125–163. Evanston and Chicago: Northwestern University Press, 1941.
• Rapp, Friedrich and Reiner Wiehl, eds. Whiteheads Metaphysik der Kreativität. Internationales Whitehead-Symposium Bad Homburg 1983. Verlag Karl Alber, Freiburg i. B. / München, 1986. ISBN 3-495-47612-1
• Rescher, Nicholas. Process Metaphysics. Albany: State University of New York Press, 1995.
• Rescher, Nicholas. Process Philosophy: A Survey of Basic Issues. Pittsburgh: University of Pittsburgh Press, 2001.
• Roelker, Nancy Lyman. An Application Of Whitehead’s Concepts Of Conformity and Novelty to the Philosophy of History. Unpublished dissertation, 1940, Harvard University. Held in John Hay Library's Special Collections at Brown University. [1]
• Schilpp, Paul Arthur, ed. The Philosophy of Alfred North Whitehead. Evanston and Chicago: Northwestern University Press, 1941. Part of the Library of Living Philosophers series.
• Siebers, Johan. The Method of Speculative Philosophy: An Essay on the Foundations of Whitehead's Metaphysics. Kassel: Kassel University Press GmbH, 2002. ISBN 3-933146-79-8
• Smith, Olav Bryant. Myths of the Self: Narrative Identity and Postmodern Metaphysics. Lanham: Lexington Books, 2004. ISBN 0-7391-0843-3
– Contains a section called "Alfred North Whitehead: Toward a More Fundamental Ontology" that is an overview of Whitehead's metaphysics.
• Weber, Michel. Whitehead's Pancreativism — The Basics. Frankfurt: Ontos Verlag, 2006.
• Weber, Michel. Whitehead's Pancreativism — Jamesian Applications, Frankfurt / Paris: Ontos Verlag, 2011.
• Weber, Michel and Will Desmond (eds.). Handbook of Whiteheadian Process Thought, Frankfurt / Lancaster: Ontos Verlag, 2008.
• Alan Van Wyk and Michel Weber (eds.). Creativity and Its Discontents. The Response to Whitehead's Process and Reality, Frankfurt / Lancaster: Ontos Verlag, 2009.
• Will, Clifford. Theory and Experiment in Gravitational Physics. Cambridge: Cambridge University Press, 1993.
External links
Wikiquote has quotations related to Alfred North Whitehead.
Wikisource has original works by or about:
Alfred North Whitehead
Wikimedia Commons has media related to Alfred North Whitehead.
• The Philosophy of Organism in Philosophy Now magazine. An accessible summary of Alfred North Whitehead's philosophy.
• Center for Process Studies in Claremont, California. A faculty research center of Claremont School of Theology, in association with Claremont Graduate University. The Center organizes conferences and events and publishes materials pertaining to Whitehead and process thought. It also maintains extensive Whitehead-related bibliographies.
• Summary of Whitehead's Philosophy A Brief Introduction to Whitehead's Metaphysics
• Society for the Study of Process Philosophies, a scholarly society that holds periodic meetings in conjunction with each of the divisional meetings of the American Philosophical Association, as well as at the annual meeting of the Society for the Advancement of American Philosophy.
• "Alfred North Whitehead" in the MacTutor History of Mathematics archive, by John J. O'Connor and Edmund F. Robertson.
• "Alfred North Whitehead: New World Philosopher" at the Harvard Square Library.
• Jesus, Jazz, and Buddhism: Process Thinking for a More Hospitable World
• "What is Process Thought?" an introductory video series to process thought by Jay McDaniel.
• Centre de philosophie pratique « Chromatiques whiteheadiennes »
• "Whitehead's Principle of Relativity" by John Lighton Synge on arXiv.org
• Whitehead at Monoskop.org, with extensive bibliography.
• Works by Alfred North Whitehead at Project Gutenberg
• Works by or about Alfred North Whitehead at Internet Archive
• Works by Alfred North Whitehead at LibriVox (public domain audiobooks)
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1. The Nancy Lyman Roelker papers, Brown University, John Hay Library, Special Collections, Box A, Series 1, Box 2. List of contents at this link accessed 15 August 2023, https://www.riamco.org/render?eadid=US-RPB-ms2012.006&view=all
| Wikipedia |
A. V. Balakrishnan
Alampallam Venkatachalaiyer Balakrishnan (1922-2015) was an American applied mathematician and professor at the University of California, Los Angeles.
A. V. Balakrishnan
BornDecember 4, 1922[1]
India
DiedMarch 17, 2015(2015-03-17) (aged 92)[2]
Los Angeles
NationalityAmerican
Alma materUniversity of Madras
University of Southern California
AwardsRichard E. Bellman Control Heritage Award (2001)
Scientific career
FieldsControl theory
InstitutionsThe Henry Samueli School of Engineering and Applied Sciences, University of California, Los Angeles
Doctoral advisorRalph S. Phillips
Education and career
Balakrishnan grew up in Chennai, India, and entered the University of Madras in the early 1940s. While there he earned a scholarship from the Indian government to study in the United States and learn to produce documentaries. Upon arriving at the University of Southern California, known for its film school, he initially wanted to become a sound engineer on Hollywood films. At the time, he was unable to get a position because he was not a member of any of the guilds, which controlled who was able to get a jobs. Therefore, after earning his first master's degree in cinema in 1949, he switched to electrical engineering. Balakrishnan received his M.S. in electrical engineering and his Ph.D. in mathematics from the University of Southern California in 1950 and 1954, respectively.
He has been professor of engineering and professor of mathematics since 1965 at The University of California, Los Angeles. He was chair of the Department of Systems Science in the (then) School of Engineering from 1969 to 1975, and director of the NASA-UCLA Flight Systems Research Center since 1985.
Recognition
He was a recipient of the Richard E. Bellman Control Heritage Award, which is the highest recognition of professional achievement for US control systems engineers and scientists in 2001 for "pioneering contributions to stochastic and distributed systems theory, optimization, control, and aerospace flight systems research".[3]
He has received honors and awards from the International Federation of Information Processing Society (1977), NASA (1978, 1992,1995, and 1996), and, in 1980, the Guillemin Prize in recognition of the major impact that his original contributions have had in setting the research direction of communications and control.
In 2016, USC Viterbi Professor Petros Ioannou became the inaugural A.V. “Bal” Balakrishnan Chair of the Ming Hsieh Department of Electrical Engineering.
References
1. "In Memory of Alampallam V. Balakrishnan".
2. "In Memoriam: A.V. Balakrishnan". UCLA. 24 March 2015.
3. "Richard E. Bellman Control Heritage Award". American Automatic Control Council. Archived from the original on 2018-10-01. Retrieved February 10, 2013.
External links
• UCLA faculty page
• USC Viterbi School of Engineering biography
• AACC profile
• A. V. Balakrishnan at the Mathematics Genealogy Project
• USC A.V.(Bal) Balakrishnan Chair of the Ming Hsieh Department of Electrical Engineering
AACC Richard E. Bellman Control Heritage Award
1979–2000
• Hendrik Wade Bode (1979)
• Nathaniel B. Nichols (1980)
• Charles Stark Draper (1981)
• Irving Lefkowitz (1982)
• John V. Breakwell (1983)
• Richard E. Bellman (1984)
• Harold Chestnut (1985)
• John Zaborszky (1986)
• John C. Lozier (1987)
• Walter R. Evans (1988)
• Roger W. Brockett (1989)
• Arthur E. Bryson (1990)
• John G. Truxal (1991)
• Rutherford Aris (1992)
• Eliahu I. Jury (1993)
• Jose B. Cruz Jr. (1994)
• Michael Athans (1995)
• Elmer G. Gilbert (1996)
• Rudolf E. Kalman (1997)
• Lotfi A. Zadeh (1998)
• Yu-Chi Ho (1999)
• W. Harmon Ray (2000)
2001–present
• A. V. Balakrishnan (2001)
• Petar V. Kokotovic (2002)
• Kumpati S. Narendra (2003)
• Harold J. Kushner (2004)
• Gene F. Franklin (2005)
• Tamer Başar (2006)
• Sanjoy K. Mitter (2007)
• Pravin Varaiya (2008)
• George Leitmann (2009)
• Dragoslav D. Šiljak (2010)
• Manfred Morari (2011)
• Arthur J. Krener (2012)
• A. Stephen Morse (2013)
• Dimitri Bertsekas (2014)
• Thomas F. Edgar (2015)
• Jason L. Speyer (2016)
• John S. Baras (2017)
• Masayoshi Tomizuka (2018)
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• Stephen P. Boyd (2023)
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| Wikipedia |
5-simplex honeycomb
In five-dimensional Euclidean geometry, the 5-simplex honeycomb or hexateric honeycomb is a space-filling tessellation (or honeycomb or pentacomb). Each vertex is shared by 12 5-simplexes, 30 rectified 5-simplexes, and 20 birectified 5-simplexes. These facet types occur in proportions of 2:2:1 respectively in the whole honeycomb.
5-simplex honeycomb
(No image)
TypeUniform 5-honeycomb
FamilySimplectic honeycomb
Schläfli symbol{3[6]}
Coxeter diagram
5-face types{34} , t1{34}
t2{34}
4-face types{33} , t1{33}
Cell types{3,3} , t1{3,3}
Face types{3}
Vertex figuret0,4{34}
Coxeter groups${\tilde {A}}_{5}$×2, <[3[6]]>
Propertiesvertex-transitive
A5 lattice
This vertex arrangement is called the A5 lattice or 5-simplex lattice. The 30 vertices of the stericated 5-simplex vertex figure represent the 30 roots of the ${\tilde {A}}_{5}$ Coxeter group.[1] It is the 5-dimensional case of a simplectic honeycomb.
The A2
5
lattice is the union of two A5 lattices:
∪
The A3
5
is the union of three A5 lattices:
∪ ∪ .
The A*
5
lattice (also called A6
5
) is the union of six A5 lattices, and is the dual vertex arrangement to the omnitruncated 5-simplex honeycomb, and therefore the Voronoi cell of this lattice is an omnitruncated 5-simplex.
∪ ∪ ∪ ∪ ∪ = dual of
Related polytopes and honeycombs
This honeycomb is one of 12 unique uniform honeycombs[2] constructed by the ${\tilde {A}}_{5}$ Coxeter group. The extended symmetry of the hexagonal diagram of the ${\tilde {A}}_{5}$ Coxeter group allows for automorphisms that map diagram nodes (mirrors) on to each other. So the various 12 honeycombs represent higher symmetries based on the ring arrangement symmetry in the diagrams:
A5 honeycombs
Hexagon
symmetry
Extended
symmetry
Extended
diagram
Extended
group
Honeycomb diagrams
a1 [3[6]] ${\tilde {A}}_{5}$
d2 <[3[6]]> ${\tilde {A}}_{5}$×21 1, , , ,
p2 [[3[6]]] ${\tilde {A}}_{5}$×22 2,
i4 [<[3[6]]>] ${\tilde {A}}_{5}$×21×22 ,
d6 <3[3[6]]> ${\tilde {A}}_{5}$×61
r12 [6[3[6]]] ${\tilde {A}}_{5}$×12 3
Projection by folding
The 5-simplex honeycomb can be projected into the 3-dimensional cubic honeycomb by a geometric folding operation that maps two pairs of mirrors into each other, sharing the same vertex arrangement:
${\tilde {A}}_{5}$
${\tilde {C}}_{3}$
See also
Regular and uniform honeycombs in 5-space:
• 5-cubic honeycomb
• 5-demicube honeycomb
• Truncated 5-simplex honeycomb
• Omnitruncated 5-simplex honeycomb
Notes
1. "The Lattice A5".
2. mathworld: Necklace, OEIS sequence A000029 13-1 cases, skipping one with zero marks
References
• Norman Johnson Uniform Polytopes, Manuscript (1991)
• Kaleidoscopes: Selected Writings of H. S. M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6
• (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380–407, MR 2,10] (1.9 Uniform space-fillings)
• (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
Fundamental convex regular and uniform honeycombs in dimensions 2–9
Space Family ${\tilde {A}}_{n-1}$ ${\tilde {C}}_{n-1}$ ${\tilde {B}}_{n-1}$ ${\tilde {D}}_{n-1}$ ${\tilde {G}}_{2}$ / ${\tilde {F}}_{4}$ / ${\tilde {E}}_{n-1}$
E2 Uniform tiling {3[3]} δ3 hδ3 qδ3 Hexagonal
E3 Uniform convex honeycomb {3[4]} δ4 hδ4 qδ4
E4 Uniform 4-honeycomb {3[5]} δ5 hδ5 qδ5 24-cell honeycomb
E5 Uniform 5-honeycomb {3[6]} δ6 hδ6 qδ6
E6 Uniform 6-honeycomb {3[7]} δ7 hδ7 qδ7 222
E7 Uniform 7-honeycomb {3[8]} δ8 hδ8 qδ8 133 • 331
E8 Uniform 8-honeycomb {3[9]} δ9 hδ9 qδ9 152 • 251 • 521
E9 Uniform 9-honeycomb {3[10]} δ10 hδ10 qδ10
E10 Uniform 10-honeycomb {3[11]} δ11 hδ11 qδ11
En-1 Uniform (n-1)-honeycomb {3[n]} δn hδn qδn 1k2 • 2k1 • k21
| Wikipedia |
6-simplex honeycomb
In six-dimensional Euclidean geometry, the 6-simplex honeycomb is a space-filling tessellation (or honeycomb). The tessellation fills space by 6-simplex, rectified 6-simplex, and birectified 6-simplex facets. These facet types occur in proportions of 1:1:1 respectively in the whole honeycomb.
6-simplex honeycomb
(No image)
TypeUniform 6-honeycomb
FamilySimplectic honeycomb
Schläfli symbol{3[7]}
Coxeter diagram
6-face types{35} , t1{35}
t2{35}
5-face types{34} , t1{34}
t2{34}
4-face types{33} , t1{33}
Cell types{3,3} , t1{3,3}
Face types{3}
Vertex figuret0,5{35}
Symmetry${\tilde {A}}_{6}$×2, [[3[7]]]
Propertiesvertex-transitive
A6 lattice
This vertex arrangement is called the A6 lattice or 6-simplex lattice. The 42 vertices of the expanded 6-simplex vertex figure represent the 42 roots of the ${\tilde {A}}_{6}$ Coxeter group.[1] It is the 6-dimensional case of a simplectic honeycomb. Around each vertex figure are 126 facets: 7+7 6-simplex, 21+21 rectified 6-simplex, 35+35 birectified 6-simplex, with the count distribution from the 8th row of Pascal's triangle.
The A*
6
lattice (also called A7
6
) is the union of seven A6 lattices, and has the vertex arrangement of the dual to the omnitruncated 6-simplex honeycomb, and therefore the Voronoi cell of this lattice is the omnitruncated 6-simplex.
∪ ∪ ∪ ∪ ∪ ∪ = dual of
Related polytopes and honeycombs
This honeycomb is one of 17 unique uniform honeycombs[2] constructed by the ${\tilde {A}}_{6}$ Coxeter group, grouped by their extended symmetry of the Coxeter–Dynkin diagrams:
A6 honeycombs
Heptagon
symmetry
Extended
symmetry
Extended
diagram
Extended
group
Honeycombs
a1 [3[7]] ${\tilde {A}}_{6}$
i2 [[3[7]]] ${\tilde {A}}_{6}$×2
1
2
r14 [7[3[7]]] ${\tilde {A}}_{6}$×14
3
Projection by folding
The 6-simplex honeycomb can be projected into the 3-dimensional cubic honeycomb by a geometric folding operation that maps two pairs of mirrors into each other, sharing the same vertex arrangement:
${\tilde {A}}_{6}$
${\tilde {C}}_{3}$
See also
Regular and uniform honeycombs in 6-space:
• 6-cubic honeycomb
• 6-demicubic honeycomb
• Truncated 6-simplex honeycomb
• Omnitruncated 6-simplex honeycomb
• 222 honeycomb
Notes
1. "The Lattice A6". Archived from the original on 2012-01-19. Retrieved 2011-05-11.
• Weisstein, Eric W. "Necklace". MathWorld., OEIS sequence A000029 18-1 cases, skipping one with zero marks
References
• Norman Johnson Uniform Polytopes, Manuscript (1991)
• Kaleidoscopes: Selected Writings of H. S. M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6
• (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380–407, MR 2,10] (1.9 Uniform space-fillings)
• (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
Fundamental convex regular and uniform honeycombs in dimensions 2–9
Space Family ${\tilde {A}}_{n-1}$ ${\tilde {C}}_{n-1}$ ${\tilde {B}}_{n-1}$ ${\tilde {D}}_{n-1}$ ${\tilde {G}}_{2}$ / ${\tilde {F}}_{4}$ / ${\tilde {E}}_{n-1}$
E2 Uniform tiling {3[3]} δ3 hδ3 qδ3 Hexagonal
E3 Uniform convex honeycomb {3[4]} δ4 hδ4 qδ4
E4 Uniform 4-honeycomb {3[5]} δ5 hδ5 qδ5 24-cell honeycomb
E5 Uniform 5-honeycomb {3[6]} δ6 hδ6 qδ6
E6 Uniform 6-honeycomb {3[7]} δ7 hδ7 qδ7 222
E7 Uniform 7-honeycomb {3[8]} δ8 hδ8 qδ8 133 • 331
E8 Uniform 8-honeycomb {3[9]} δ9 hδ9 qδ9 152 • 251 • 521
E9 Uniform 9-honeycomb {3[10]} δ10 hδ10 qδ10
E10 Uniform 10-honeycomb {3[11]} δ11 hδ11 qδ11
En-1 Uniform (n-1)-honeycomb {3[n]} δn hδn qδn 1k2 • 2k1 • k21
| Wikipedia |
7-simplex honeycomb
In seven-dimensional Euclidean geometry, the 7-simplex honeycomb is a space-filling tessellation (or honeycomb). The tessellation fills space by 7-simplex, rectified 7-simplex, birectified 7-simplex, and trirectified 7-simplex facets. These facet types occur in proportions of 2:2:2:1 respectively in the whole honeycomb.
7-simplex honeycomb
(No image)
TypeUniform 7-honeycomb
FamilySimplectic honeycomb
Schläfli symbol{3[8]}
Coxeter diagram
6-face types{36} , t1{36}
t2{36} , t3{36}
6-face types{35} , t1{35}
t2{35}
5-face types{34} , t1{34}
t2{34}
4-face types{33} , t1{33}
Cell types{3,3} , t1{3,3}
Face types{3}
Vertex figuret0,6{36}
Symmetry${\tilde {A}}_{7}$×21, <[3[8]]>
Propertiesvertex-transitive
A7 lattice
This vertex arrangement is called the A7 lattice or 7-simplex lattice. The 56 vertices of the expanded 7-simplex vertex figure represent the 56 roots of the ${\tilde {A}}_{7}$ Coxeter group.[1] It is the 7-dimensional case of a simplectic honeycomb. Around each vertex figure are 254 facets: 8+8 7-simplex, 28+28 rectified 7-simplex, 56+56 birectified 7-simplex, 70 trirectified 7-simplex, with the count distribution from the 9th row of Pascal's triangle.
${\tilde {E}}_{7}$ contains ${\tilde {A}}_{7}$ as a subgroup of index 144.[2] Both ${\tilde {E}}_{7}$ and ${\tilde {A}}_{7}$ can be seen as affine extensions from $A_{7}$ from different nodes:
The A2
7
lattice can be constructed as the union of two A7 lattices, and is identical to the E7 lattice.
∪ = .
The A4
7
lattice is the union of four A7 lattices, which is identical to the E7* lattice (or E2
7
).
∪ ∪ ∪ = + = dual of .
The A*
7
lattice (also called A8
7
) is the union of eight A7 lattices, and has the vertex arrangement to the dual honeycomb of the omnitruncated 7-simplex honeycomb, and therefore the Voronoi cell of this lattice is an omnitruncated 7-simplex.
∪ ∪ ∪ ∪ ∪ ∪ ∪ = dual of .
Related polytopes and honeycombs
This honeycomb is one of 29 unique uniform honeycombs[3] constructed by the ${\tilde {A}}_{7}$ Coxeter group, grouped by their extended symmetry of rings within the regular octagon diagram:
A7 honeycombs
Octagon
symmetry
Extended
symmetry
Extended
diagram
Extended
group
Honeycombs
a1 [3[8]] ${\tilde {A}}_{7}$
d2 <[3[8]]> ${\tilde {A}}_{7}$×21
1
p2 [[3[8]]] ${\tilde {A}}_{7}$×22
2
d4 <2[3[8]]> ${\tilde {A}}_{7}$×41
p4 [2[3[8]]] ${\tilde {A}}_{7}$×42
d8 [4[3[8]]] ${\tilde {A}}_{7}$×8
r16 [8[3[8]]] ${\tilde {A}}_{7}$×16 3
Projection by folding
The 7-simplex honeycomb can be projected into the 4-dimensional tesseractic honeycomb by a geometric folding operation that maps two pairs of mirrors into each other, sharing the same vertex arrangement:
${\tilde {A}}_{7}$
${\tilde {C}}_{4}$
See also
Regular and uniform honeycombs in 7-space:
• 7-cubic honeycomb
• 7-demicubic honeycomb
• Truncated 7-simplex honeycomb
• Omnitruncated 7-simplex honeycomb
• E7 honeycomb
Notes
1. "The Lattice A7".
2. N.W. Johnson: Geometries and Transformations, (2018) 12.4: Euclidean Coxeter groups, p.294
3. Weisstein, Eric W. "Necklace". MathWorld., OEIS sequence A000029 30-1 cases, skipping one with zero marks
References
• Norman Johnson Uniform Polytopes, Manuscript (1991)
• Kaleidoscopes: Selected Writings of H. S. M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley–Interscience Publication, 1995, ISBN 978-0-471-01003-6
• (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380–407, MR 2,10] (1.9 Uniform space-fillings)
• (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3–45]
Fundamental convex regular and uniform honeycombs in dimensions 2–9
Space Family ${\tilde {A}}_{n-1}$ ${\tilde {C}}_{n-1}$ ${\tilde {B}}_{n-1}$ ${\tilde {D}}_{n-1}$ ${\tilde {G}}_{2}$ / ${\tilde {F}}_{4}$ / ${\tilde {E}}_{n-1}$
E2 Uniform tiling {3[3]} δ3 hδ3 qδ3 Hexagonal
E3 Uniform convex honeycomb {3[4]} δ4 hδ4 qδ4
E4 Uniform 4-honeycomb {3[5]} δ5 hδ5 qδ5 24-cell honeycomb
E5 Uniform 5-honeycomb {3[6]} δ6 hδ6 qδ6
E6 Uniform 6-honeycomb {3[7]} δ7 hδ7 qδ7 222
E7 Uniform 7-honeycomb {3[8]} δ8 hδ8 qδ8 133 • 331
E8 Uniform 8-honeycomb {3[9]} δ9 hδ9 qδ9 152 • 251 • 521
E9 Uniform 9-honeycomb {3[10]} δ10 hδ10 qδ10
E10 Uniform 10-honeycomb {3[11]} δ11 hδ11 qδ11
En-1 Uniform (n-1)-honeycomb {3[n]} δn hδn qδn 1k2 • 2k1 • k21
| Wikipedia |
8-simplex honeycomb
In eighth-dimensional Euclidean geometry, the 8-simplex honeycomb is a space-filling tessellation (or honeycomb). The tessellation fills space by 8-simplex, rectified 8-simplex, birectified 8-simplex, and trirectified 8-simplex facets. These facet types occur in proportions of 1:1:1:1 respectively in the whole honeycomb.
8-simplex honeycomb
(No image)
TypeUniform 8-honeycomb
FamilySimplectic honeycomb
Schläfli symbol{3[9]}
Coxeter diagram
6-face types{37} , t1{37}
t2{37} , t3{37}
6-face types{36} , t1{36}
t2{36} , t3{36}
6-face types{35} , t1{35}
t2{35}
5-face types{34} , t1{34}
t2{34}
4-face types{33} , t1{33}
Cell types{3,3} , t1{3,3}
Face types{3}
Vertex figuret0,7{37}
Symmetry${\tilde {A}}_{8}$×2, [[3[9]]]
Propertiesvertex-transitive
A8 lattice
This vertex arrangement is called the A8 lattice or 8-simplex lattice. The 72 vertices of the expanded 8-simplex vertex figure represent the 72 roots of the ${\tilde {A}}_{8}$ Coxeter group.[1] It is the 8-dimensional case of a simplectic honeycomb. Around each vertex figure are 510 facets: 9+9 8-simplex, 36+36 rectified 8-simplex, 84+84 birectified 8-simplex, 126+126 trirectified 8-simplex, with the count distribution from the 10th row of Pascal's triangle.
${\tilde {E}}_{8}$ contains ${\tilde {A}}_{8}$ as a subgroup of index 5760.[2] Both ${\tilde {E}}_{8}$ and ${\tilde {A}}_{8}$ can be seen as affine extensions of $A_{8}$ from different nodes:
The A3
8
lattice is the union of three A8 lattices, and also identical to the E8 lattice.[3]
∪ ∪ = .
The A*
8
lattice (also called A9
8
) is the union of nine A8 lattices, and has the vertex arrangement of the dual honeycomb to the omnitruncated 8-simplex honeycomb, and therefore the Voronoi cell of this lattice is an omnitruncated 8-simplex
∪ ∪ ∪ ∪ ∪ ∪ ∪ ∪ = dual of .
Related polytopes and honeycombs
This honeycomb is one of 45 unique uniform honeycombs[4] constructed by the ${\tilde {A}}_{8}$ Coxeter group. The symmetry can be multiplied by the ring symmetry of the Coxeter diagrams:
A8 honeycombs
Enneagon
symmetry
Symmetry Extended
diagram
Extended
group
Honeycombs
a1 [3[9]] ${\tilde {A}}_{8}$
i2 [[3[9]]] ${\tilde {A}}_{8}$×2
1 2
i6 [3[3[9]]] ${\tilde {A}}_{8}$×6
r18 [9[3[9]]] ${\tilde {A}}_{8}$×18 3
Projection by folding
The 8-simplex honeycomb can be projected into the 4-dimensional tesseractic honeycomb by a geometric folding operation that maps two pairs of mirrors into each other, sharing the same vertex arrangement:
${\tilde {A}}_{8}$
${\tilde {C}}_{4}$
See also
• Regular and uniform honeycombs in 8-space:
• 8-cubic honeycomb
• 8-demicubic honeycomb
• Truncated 8-simplex honeycomb
• 521 honeycomb
• 251 honeycomb
• 152 honeycomb
Notes
1. "The Lattice A8".
2. N.W. Johnson: Geometries and Transformations, (2018) Chapter 12: Euclidean symmetry groups, p.294
3. Kaleidoscopes: Selected Writings of H. S. M. Coxeter, Paper 18, "Extreme forms" (1950)
• Weisstein, Eric W. "Necklace". MathWorld., OEIS sequence A000029 46-1 cases, skipping one with zero marks
References
• Norman Johnson Uniform Polytopes, Manuscript (1991)
• Kaleidoscopes: Selected Writings of H. S. M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6
• (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380–407, MR 2,10] (1.9 Uniform space-fillings)
• (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3–45]
Fundamental convex regular and uniform honeycombs in dimensions 2–9
Space Family ${\tilde {A}}_{n-1}$ ${\tilde {C}}_{n-1}$ ${\tilde {B}}_{n-1}$ ${\tilde {D}}_{n-1}$ ${\tilde {G}}_{2}$ / ${\tilde {F}}_{4}$ / ${\tilde {E}}_{n-1}$
E2 Uniform tiling {3[3]} δ3 hδ3 qδ3 Hexagonal
E3 Uniform convex honeycomb {3[4]} δ4 hδ4 qδ4
E4 Uniform 4-honeycomb {3[5]} δ5 hδ5 qδ5 24-cell honeycomb
E5 Uniform 5-honeycomb {3[6]} δ6 hδ6 qδ6
E6 Uniform 6-honeycomb {3[7]} δ7 hδ7 qδ7 222
E7 Uniform 7-honeycomb {3[8]} δ8 hδ8 qδ8 133 • 331
E8 Uniform 8-honeycomb {3[9]} δ9 hδ9 qδ9 152 • 251 • 521
E9 Uniform 9-honeycomb {3[10]} δ10 hδ10 qδ10
E10 Uniform 10-honeycomb {3[11]} δ11 hδ11 qδ11
En-1 Uniform (n-1)-honeycomb {3[n]} δn hδn qδn 1k2 • 2k1 • k21
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AA postulate
In Euclidean geometry, the AA postulate states that two triangles are similar if they have two corresponding angles congruent.
The AA postulate follows from the fact that the sum of the interior angles of a triangle is always equal to 180°. By knowing two angles, such as 32° and 64° degrees, we know that the next angle is 84°, because 180-(32+64)=84. (This is sometimes referred to as the AAA Postulate—which is true in all respects, but two angles are entirely sufficient.)
The postulate can be better understood by working in reverse order. The two triangles on grids A and B are similar, by a 1.5 dilation from A to B. If they are aligned, as in grid C, it is apparent that the angle on the origin is congruent with the other (D). We also know that the pair of sides opposite the origin are parallel. We know this because the pairs of sides around them are similar, stem from the same point, and line up with each other. We can then look at the sides around the parallels as transversals, and therefore the corresponding angles are congruent. Using this reasoning we can tell that similar triangles have congruent angles.
Now, because this article is practically over, you might want to know what AA postulate can be used for. It is used proving the Angle Bisector Theorem. AA postulate is one of the many similarity ways for determining similarity in a triangle.
References
• http://hanlonmath.com/pdfFiles/464Chapter7Sim.Poly.pdf (Unused Source)
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AB5 category
In mathematics, Alexander Grothendieck (1957) in his "Tôhoku paper" introduced a sequence of axioms of various kinds of categories enriched over the symmetric monoidal category of abelian groups. Abelian categories are sometimes called AB2 categories, according to the axiom (AB2). AB3 categories are abelian categories possessing arbitrary coproducts (hence, by the existence of quotients in abelian categories, also all colimits). AB5 categories are the AB3 categories in which filtered colimits of exact sequences are exact. Grothendieck categories are the AB5 categories with a generator.
References
• Grothendieck, Alexander (1957), "Sur quelques points d'algèbre homologique", Tohoku Mathematical Journal, Second Series, 9: 119–221, doi:10.2748/tmj/1178244839, ISSN 0040-8735, MR 0102537
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ABACABA pattern
The ABACABA pattern is a recursive fractal pattern that shows up in many places in the real world (such as in geometry, art, music, poetry, number systems, literature and higher dimensions).[1][2][3][4] Patterns often show a DABACABA type subset. AA, ABBA, and ABAABA type forms are also considered.[5]
Generating the pattern
In order to generate the next sequence, first take the previous pattern, add the next letter from the alphabet, and then repeat the previous pattern. The first few steps are listed here.[4] A generator can be found here
StepPatternLetters
1A21 − 1 = 1
2ABA3
3ABACABA7
4ABACABADABACABA15
5ABACABADABACABAEABACABADABACABA31
6ABACABADABACABAEABACABADABACABAFABACABADABACABAEABACABADABACABA63
ABACABA is a "quickly growing word", often described as chiastic or "symmetrically organized around a central axis" (see: Chiastic structure and Χ).[4] The number of members in each iteration is a(n) = 2n − 1, the Mersenne numbers (OEIS: A000225).
Gallery
• Sierpinski triangle[1][2]:
ABACABA
• Ruler,[1][2] excluding 1 and 2:
ABACABADABACABA
excluding 2:
EABACABADABACABA
• Cantor set:
ABACABADABACABA
• Binary tree[1][2]/upside down family tree:
ABACABADABACABA
• Koch curve:[1] $n=1$ is ABA, $n=2$ is ABACABA, and $n=3$: ABACABADABACABA
• Metric hierarchy:
ABACABADABACABA[lower-alpha 1]
• Metric levels:[1]
EABACABADABACABA
• When counting in binary (here 4-bit), the final 0s form an ABACABA pattern[1]
• A staircase built with the largest possible squares/cubes while allowing equally sized steps:
ABACABADABACABA[1]
• A "circle fractal"[1] superimposed with a 2 × 2 box fractal:
ABACABADABACABA
• The Tower of Hanoi[1] with four disks:
ABACABADABACABA
• Binary tree array:
to O
• Binary-reflected Gray code (BRGC):
to G
• Rotary encoder:
to I
• 3-bit Gray code visualized as a traversal of vertices of a cube (0,1,3,2,6,7,5,4):[1]
ABACABA
• Double harmonic scale (Play ) with steps of H-3H-H-W-H-3H-H:
ABACABA
• Château de Chambord:
ABACABA[6]
• Gray code along the number line[1] (OEIS: A003188):
ABACABADABACABAEABACABADABACABA
• Devil's needle:[1]
ABACABADABACABA
See also
• Arch form
• Farey sequence
• Rondo
• Sesquipower
Notes
1. The strength, emphasis, or importance of the beginning of each duration $1/8$ the length of a single measure in 4
4
(eighth-notes) is, divisively ($2/2^{1}=1$, $4/2^{2}=1$, $8/2^{3}=1$), determined by each eighth-note's position in a DABACABA structure, while the eighth notes of two measures grouped, additively ($8\times 2=16$), are determined by an EABACABADABACABA structure.[3]
References
1. Naylor, Mike (February 2013). "ABACABA Amazing Pattern, Amazing Connections". Math Horizons. Retrieved June 13, 2019.
2. SheriOZ (2016-04-21). "Exploring Fractals with ABACABA". Chicago Geek Guy. Archived from the original on 22 January 2021. Retrieved January 22, 2021.
3. Naylor, Mike (2011). "Abacaba! – Using a mathematical pattern to connect art, music, poetry and literature" (PDF). Bridges. Retrieved October 6, 2017.
4. Conley, Craig (2008-10-01). Magic Words: A Dictionary. Weiser Books. p. 53. ISBN 9781609250508.
5. Halter-Koch, Franz and Tichy, Robert F.; eds. (2000). Algebraic Number Theory and Diophantine Analysis, p.478. W. de Gruyter. ISBN 9783110163049.
6. Wright, Craig (2016). Listening to Western Music, p.48. Cengage Learning. ISBN 9781305887039.
External links
• Naylor, Mike: abacaba.org
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Subsets and Splits