Goursat tetrahedron
Introduction File:Coxeter-Dynkin 3-space groups.png|480px|thumb|For Euclidean 3-space, there are 3 simple and related Goursat tetrahedra, represented by 4,3,4, 4,31,1, and 34.They can be seen inside as points on and within a cube, {4,3}.In geometry, a '''Goursat tetrahedron''' is a tetrahedral fundamental domain of a Wythoff construction. Each tetrahedral face represents a reflection hyperplane on 3-dimensional surfaces: the 3-sphere, Euclidean 3-space, and hyperbolic 3-space. Coxeter named them after Édouard Goursat who first looked into these domains. It is an extension of the theory of Schwarz triangles for Wythoff constructions on the sphere.
Graphical representation A '''Goursat tetrahedron''' can be represented graphically by a tetrahedral graph, which is in a dual configuration of the fundamental domain tetrahedron. In the graph, each node represents a face (mirror) of the Goursat tetrahedron. Each edge is labeled by a rational value corresponding to the reflection order, being π/dihedral angle. :100pxA 4-node Coxeter-Dynkin diagram represents this tetrahedral graph with order-2 edges hidden. If many edges are order 2, the Coxeter group can be represented by a bracket notation.Existence requires each of the 3-node subgraphs of this graph, (p q r), (p u s), (q t u), and (r s t), must correspond to a Schwarz triangle.
Extended symmetry 360px240pxThe symmetry of a Goursat tetrahedron can be tetrahedral symmetry of any subgroup symmetry shown in this tree, with subgroups below with subgroup indices labeled in the colored edges.An extended symmetry of the Goursat tetrahedron is a semidirect product of the Coxeter group symmetry and the fundamental domain symmetry (the Goursat tetrahedron in these cases). Coxeter notation supports this symmetry as double-brackets like YX means full Coxeter group symmetry X, with '''Y''' as a symmetry of the Goursat tetrahedron. If '''Y''' is a pure reflective symmetry, the group will represent another Coxeter group of mirrors. If there is only one simple doubling symmetry, '''Y''' can be implicit like X with either reflectional or rotational symmetry depending on the context.The extended symmetry of each Goursat tetrahedron is also given below. The highest possible symmetry is that of the regular tetrahedron as 3,3, and this occurs in the prismatic point group 2,2,2 or 23,3 and the paracompact hyperbolic group 33,3.See Tetrahedron#Isometries of irregular tetrahedra for 7 lower symmetry isometries of the tetrahedron.
Whole number solutions The following sections show all of the whole number Goursat tetrahedral solutions on the 3-sphere, Euclidean 3-space, and Hyperbolic 3-space. The extended symmetry of each tetrahedron is also given.The colored tetrahedal diagrams below are vertex figures for omnitruncated polytopes and honeycombs from each symmetry family. The edge labels represent polygonal face orders, which is double the Coxeter graph branch order. The dihedral angle of an edge labeled ''2n'' is π/''n''. Yellow edges labeled 4 come from right angle (unconnected) mirror nodes in the Coxeter diagram.===3-sphere (finite) solutions===Finite Coxeter groups isomorphismsThe solutions for the 3-sphere with density 1 solutions are: (Uniform polychora)+ Duoprisms and hyperprisms:Coxeter groupand diagram2,2,2p,2,2p,2,qp,2,p3,3,24,3,25,3,2Group symmetry order168''p''4''pq''4''p''24896240Tetrahedronsymmetry3,3(order 24)40px2(order 4)40px2(order 4)40px2+,4(order 8)40px (order 2)40px +(order 1)40px +(order 1)40pxExtended symmetry(3,3)2,2,2=4,3,32p,2,2=2p,2,42p,2,q=2p,2,2q(2+,4)p,2,p=2+2p,2,2p13,3,2=4,3,24,3,25,3,2Extended symmetry order38432''p''16''pq''32''p''29696240Graph typeLinearTridentalCoxeter groupand diagramPentachoricOmnitruncated 5-cell|3,3,3HexadecachoricOmnitruncated 16-cell|4,3,3IcositetrachoricOmnitruncated 24-cell|3,4,3HexacosichoricOmnitruncated 120-cell|5,3,3DemitesseracticTruncated_24-cell#Truncated_24-cell|31,1,1Vertex figure of omnitruncated uniform polychoraTetrahedron80px80px80px80px80pxGroup symmetry order120384115214400192Tetrahedronsymmetry2+(order 2)40px +(order 1)40px2+(order 2)40px +(order 1)40px3(order 6)40pxExtended symmetry2+3,3,34,3,32+3,4,35,3,3331,1,1=3,4,3Extended symmetry order2403842304144001152===Euclidean (affine) 3-space solutions===Euclidean Coxeter group isomorphismsDensity 1 solutions: Convex uniform honeycombs:Graph typeLinearOrthoschemeTri-dentalPlagioschemeLoopCycloschemePrismaticDegenerateCoxeter groupCoxeter diagramOmnitruncated cubic honeycomb|4,3,4Cantitruncated cubic honeycomb|4,31,1Bitruncated cubic honeycomb|344,4,26,3,233,2∞,2,∞Vertex figure of omnitruncated honeycombsTetrahedron80px80px80pxTetrahedronSymmetry2+(order 2)40px (order 2)40px2+,4(order 8)40px (order 2)40px +(order 1)40px3(order 6)40px2+,4(order 8)40pxExtended symmetry(2+)4,3,414,31,1=4,3,4(2+,4)34=2+4,3,414,4,2=4,4,26,3,2333,2=3,6,2(2+,4)∞,2,∞=14,4===Compact hyperbolic 3-space solutions===Density 1 solutions: (Convex uniform honeycombs in hyperbolic space) (Coxeter diagram#Compact (Lannér simplex groups))+ Rank 4 Lannér simplex groupsGraph typeLinearTri-dentalCoxeter groupCoxeter diagram3,5,35,3,45,3,55,31,1 Vertex figures of omnitruncated honeycombsTetrahedron80px80px80px80pxTetrahedronSymmetry2+(order 2)40px +(order 1)40px2+(order 2)40px (order 2)40pxExtended symmetry2+3,5,35,3,42+5,3,515,31,1=5,3,4Graph typeLoopCoxeter groupCoxeter diagram(4,3,3,3)(4,3)2(5,3,3,3)(5,3,4,3)(5,3)2 Vertex figures of omnitruncated honeycombsTetrahedron80px80px80px80px80pxTetrahedronSymmetry2+(order 2)40px2,2+(order 4)40px2+(order 2)40px2+(order 2)40px2,2+(order 4)40pxExtended symmetry2+(4,3,3,3)(2,2)+(4,3)22+(5,3,3,3)2+(5,3,4,3)(2,2)+(5,3)2===Paracompact hyperbolic 3-space solutions===This show subgroup relations of paracompact hyperbolic Goursat tetrahedra. Order 2 subgroups represent bisecting a Goursat tetrahedron with a plane of mirror symmetrythumbDensity 1 solutions: (See Coxeter diagram#Paracompact (Koszul simplex groups))+ Rank 4 Koszul simplex groupsGraph typeLinear graphsCoxeter groupand diagram6,3,33,6,36,3,46,3,56,3,64,4,34,4,4Tetrahedronsymmetry +(order 1)40px2+(order 2)40px +(order 1)40px +(order 1)40px2+(order 2)40px +(order 1)40px2+(order 2)40pxExtended symmetry6,3,32+3,6,36,3,46,3,52+6,3,64,4,32+4,4,4Graph typeLoop graphsCoxeter groupand diagram3 × (4,4,3,3)(43,3)44(6,33)(6,3,4,3)(6,3,5,3)(6,3)2Tetrahedronsymmetry2(order 4)40px (order 2)40px2+(order 2)40px2+,4(order 8)40px2+(order 2)40px2+(order 2)40px2+(order 2)40px2,2+(order 4)40pxExtended symmetry23 × =6,3,41(4,4,3,3)=3,41,12+(43,3)(2+,4)44=2+4,4,42+(6,33)2+(6,3,4,3)2+(6,3,5,3)(2,2)+(6,3)2Graph typeTri-dentalLoop-n-tailSimplexCoxeter groupand diagram6,31,13,41,141,1,13,334,335,336,3333,3Tetrahedronsymmetry (order 2)40px (order 2)40px3(order 6)40px (order 2)40px (order 2)40px (order 2)40px (order 2)40px3,3(order 24)40pxExtended symmetry16,31,1=6,3,413,41,1=3,4,4341,1,1=4,4,313,33=3,3,614,33=4,3,615,33=5,3,616,33=6,3,6(3,3)33,3=6,3,3
Rational solutions There are hundreds of rational solutions for the 3-sphere, including these 6 linear graphs which generate the Schläfli-Hess polychora, and 11 nonlinear ones from Coxeter: Linear graphs# Density 4: 3,5,5/2 # Density 6: 5,5/2,5 # Density 20: 5,3,5/2 # Density 66: 5/2,5,5/2 # Density 76: 5,5/2,3 # Density 191: 3,3,5/2 Loop-n-tail graphs:# Density 2: # Density 3: # Density 5: # Density 8: # Density 9: # Density 14: # Density 26: # Density 30: # Density 39: # Density 46: # Density 115: In all, there are 59 sporadic tetrahedra with rational angles, and 2 infinite families.
See also * Point group for ''n''-simplex solutions on (''n''-1)-sphere.
References * ''Regular Polytopes'', (3rd edition, 1973), Dover edition,  (page 280, Goursat's tetrahedra) * Norman Johnson ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. (1966) He proved the enumeration of the Goursat tetrahedra by Coxeter is complete* Goursat, Edouard, ''Sur les substitutions orthogonales et les divisions régulières de l'espace'', Annales Scientifiques de l'École Normale Supérieure, Sér. 3, 6 (1889), (pp. 9–102, pp. 80–81 tetrahedra)* *Norman Johnson, ''Geometries and Transformations'' (2018), Chapters 11,12,13*N. W. Johnson, R. Kellerhals, J. G. Ratcliffe, S. T. Tschantz, ''The size of a hyperbolic Coxeter simplex'', Transformation Groups 1999, Volume 4, Issue 4, pp 329–353