# Redshift Academy Wolfram Alpha: Search by keyword: Astronomy

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Celestial Coordinates
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Celestial Navigation
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Distance Units
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Location of North and South Celestial Poles Chemistry

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Avogadro's Number
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Balancing Chemical Equations
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Stochiometry
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The Periodic Table Classical Physics

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Archimedes Principle
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Bernoulli Principle
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Blackbody (Cavity) Radiation and Planck's Hypothesis
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Center of Mass Frame
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Comparison Between Gravitation and Electrostatics
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Compton Effect
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Coriolis Effect
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Cyclotron Resonance
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Dispersion
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Doppler Effect
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Double Slit Experiment
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Elastic and Inelastic Collisions
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Electric Fields
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Error Analysis
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Fick's Law
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Fluid Pressure
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Gauss's Law of Universal Gravity
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Gravity - Force and Acceleration
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Hooke's law
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Ideal and Non-Ideal Gas Laws (van der Waal)
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Impulse Force
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Inclined Plane
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Inertia
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Kepler's Laws
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Kinematics
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Kinetic Theory of Gases
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Nuclear Spin
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One Dimensional Wave Equation
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Planck Radiation Law
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Radioactive Decay
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Specific Heat, Latent Heat and Calorimetry
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The Gas Laws
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Keeling Curve Cosmology

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Cosmic Background Radiation and Decoupling
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Dark Matter
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Geometries of the Universe
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Hubble's Law
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Inflation Theory
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Introduction to Black Holes
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Planck Units
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Stephen Hawking's Last Paper
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The Big Bang Model Finance and Accounting

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Lecture Notes on International Financial Management
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NPV and IRR
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Periodically and Continuously Compounded Interest
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Repurchase versus Dividend Analysis General Relativity

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Accelerated Reference Frames - Rindler Coordinates
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Catalog of Spacetimes
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Curvature and Parallel Transport
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Dirac Equation in Curved Spacetime
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Einstein's Field Equations
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Geodesics
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Vierbein (Frame) Fields
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World Lines Refresher Lagrangian and Hamiltonian Mechanics

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Classical Field Theory
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Euler-Lagrange Equation
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Ex: Newtonian, Lagrangian and Hamiltonian Mechanics
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Hamiltonian Formulation
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Liouville's Theorem
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Symmetry and Conservation Laws - Noether's Theorem Macroeconomics

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Lecture Notes on International Economics
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Lecture Notes on Macroeconomics
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Macroeconomic Policy Mathematics

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Amplitude, Period and Phase
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Asymptotes
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Augmented Matrices and Cramer's Rule
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Basic Group Theory
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Binomial Theorem (Pascal's Triangle)
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Building Groups From Other Groups
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Completing the Square
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Complex Numbers
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Composite Functions
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Conformal Transformations
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Contravariant and Covariant Components of a Vector
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Factoring of a3 +/- b3
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Grassmann and Clifford Algebras
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Heron's Formula
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Introduction to Conformal Field Theory
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Quaternions 1
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Regular Polygons
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The Integers Modulo n Under + and x
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Tic-Tac-Toe Factoring
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Trapezoidal Rule
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Unit Vectors
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Vector Calculus
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Volume Integrals Microeconomics

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Marginal Revenue and Cost Particle Physics

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Feynman Diagrams and Loops
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Field Dimensions
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Helicity and Chirality
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Klein-Gordon and Dirac Equations
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Regularization and Renormalization
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Scattering - Mandelstam Variables
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Spin 1 Eigenvectors
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The Vacuum Catastrophe Probability and Statistics

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Chebyshev's Theorem
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Set Theory - Venn Diagrams
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Hashing
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How this site works ...
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More Programming Topics
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Open Systems Interconnection (OSI) Standard - TCP/IP Protocol
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Public Key Encryption Quantum Field Theory

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Creation and Annihilation Operators
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Field Operators for Bosons and Fermions
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Lagrangians in Quantum Field Theory
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Path Integral Formulation
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Relativistic Quantum Field Theory Quantum Mechanics

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Basic Relationships
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Bell's Theorem
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Bohr Atom
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Clebsch-Gordan Coefficients
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Commutators
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Dyson Series
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Entangled States
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Spin 1/2 Eigenvectors
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The Differential Operator
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The Essential Mathematics of Quantum Mechanics
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The Observer Effect
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The Qubit
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The WKB Approximation
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Time Dependent Perturbation Theory
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Time Evolution and Symmetry Operations
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Time Independent Perturbation Theory
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Wavepackets Semiconductor Reliability

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The Weibull Distribution Solid State Electronics

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Band Theory of Solids
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Fermi-Dirac Statistics
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Intrinsic and Extrinsic Semiconductors
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The MOSFET
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The P-N Junction Special Relativity

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4-vectors
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Electromagnetic 4 - Potential
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Energy and Momentum, E = mc2
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Lorentz Invariance
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Lorentz Transform
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Lorentz Transformation of the EM Field
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Newton versus Einstein
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Spinors - Part 1
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Spinors - Part 2
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Velocity Addition Statistical Mechanics

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Black Body Radiation
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Entropy and the Partition Function
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The Harmonic Oscillator
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The Ideal Gas String Theory

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Bosonic Strings
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Extra Dimensions
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Introduction to String Theory
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Kaluza-Klein Compactification of Closed Strings
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Strings in Curved Spacetime
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Toroidal Compactification Superconductivity

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BCS Theory
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Introduction to Superconductors
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Superconductivity (Lectures 1 - 10)
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Superconductivity (Lectures 11 - 20) Supersymmetry (SUSY) and Grand Unified Theory (GUT)

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Chiral Superfields
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Generators of a Supergroup
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Grassmann Numbers
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Introduction to Supersymmetry
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The Gauge Hierarchy Problem test

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test The Standard Model

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Electroweak Unification (Glashow-Weinberg-Salam)
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Gauge Theories (Yang-Mills)
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Gravitational Force and the Planck Scale
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Introduction to the Standard Model
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Isospin, Hypercharge, Weak Isospin and Weak Hypercharge
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Quantum Flavordynamics and Quantum Chromodynamics
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Special Unitary Groups and the Standard Model - Part 1
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Special Unitary Groups and the Standard Model - Part 2
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Special Unitary Groups and the Standard Model - Part 3
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Standard Model Lagrangian
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The Higgs Mechanism
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The Nature of the Weak Interaction Topology

- Units, Constants and Useful Formulas

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Constants
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Formulas Last modified: January 26, 2018
```Toroidal Compactification
-------------------------

Compactification is the process of reconciling 4-dimensional
spacetime with the extra dimensions required by string theory.
For this purpose it is assumed the extra dimensions are "wrapped"
up on themselves, or "curled" up on manifolds.  In the limit
where these curled up dimensions become very small, one obtains
a theory in which spacetime has effectively a lower number of
dimensions.  To illustrate compactification it is helpful to
consider a cube.  Assume that a particle moving in the interior
of the cube reaches a point on a face and is assumed to go
through it and then appears to come forth from the corresponding
point on the opposite face, in the same direction: A -> A and
B -> B.  This results in compactifying 2 out of 3 dimensions.
Since a torus is nothing more than a curled up cylinder which
in turn is a curled up rectangle, this is also equivalent to a
torus.  Although the torus looks to have a curved geometry, it
can be unfolded, to a Ricci-Flat surface.  This concept can be
extended to 3 dimensions by including the C faces of the cube
which corresponds to a 3-torus. For larger dimensions the manifolds become extremely complex.
A class of manifolds which is of interest to string theorists
are the CALABI-YAU manifolds.  A 2D representation of a 6D
Calabi-Yau manifold that exhibits Ricci flatness is shown
below: Image courtesy of Wikipedia.```
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