Gursa Tetrahedron

In the Euclidean 3-space there are 3 simple Goursat tetrahedra, which are represented by the symbols [4,3,4], [4,3 ^1,1 ] and [3 ^[4] ]. They can be considered as points on the cube and inside the cube {4,3}.

In geometry, the Goursat tetrahedron is the tetrahedral fundamental domain of the Withoff construction . Each face of the tetrahedron represents a mirror hyperplane on a 3-dimensional surface - a 3-sphere , a Euclidean 3-dimensional space and a hyperbolic 3-dimensional space. Coxeter named the area by the name of Edouard Goursat , who was the first to pay attention to these areas. The Goursat tetrahedron is an extension of the theory of Schwartz triangles for constructing Withhoff on a sphere.

Content

Graphical View

The Goursat tetrahedron can be represented graphically by a tetrahedral graph, which is the dual configuration of the fundamental region in the form of a tetrahedron. In this graph, each node represents the face (mirror) of the Goursat tetrahedron. Each edge is labeled with a rational number corresponding to the order of reflection, which is equal to $\pi$ ${\ displaystyle \ pi}$ dihedral angle .

The 4-vertex Coxeter-Dynkin diagram represents these tetrahedral graphs with hidden second-order edges. If many edges are of order 2, the Coxeter group can be represented by .

For the Goursat tetrahedron to exist, each of the subgraphs with 3 vertices of this graph, (pqr), (pus), (qtu) and (rst), must correspond to the Schwartz triangle .

External Symmetry


The symmetry of the Goursat tetrahedron can be the tetrahedral symmetry of any subgroup of symmetry shown in the tree by the color of the edges.

The extended symmetry of the Goursat tetrahedron is a semidirect product of the Coxeter group of symmetry and the fundamental region of symmetry (Goursat tetrahedron, in this case). supports this symmetry as nested brackets, like [Y [X]], which means the full Coxeter group of symmetry [X] with Y as the symmetry of the Goursat tetrahedron. If Y is pure mirror symmetry, the group will represent another Coxeter group of reflections. If there is only one simple doubling symmetry, Y can be expressed explicitly, like [[X]] with mirror or rotational symmetry, depending on the context.

The expanded symmetry of each Goursat tetrahedron is given below. The highest possible symmetry is in the regular tetrahedron , [3,3], and it is achieved on the prismatic point group [2,2,2], or [2 ^[3,3] ], and on the paracompact hyperbolic group [3 ^[3,3] ].

See tetrahedron symmetries for 7 low-order tetrahedron symmetries.

Total number of decisions

The following sections show all of the complete set of solutions of Goursat tetrahedra for the 3-sphere, Euclidean 3-dimensional space and hyperbolic 3-dimensional space. The expanded symmetry of each tetrahedron is also indicated.

The colored tetrahedral diagrams below are vertex figures of all- polyhedra and honeycombs from each family of symmetries. The edge labels represent the orders of polygonal faces, which are double orders of the branches of the Coxeter graph. The dihedral angle of the edge labeled 2n is $\pi /n$ ${\ displaystyle \ pi / n}$ . The yellow edges marked with the number 4 are obtained from the right angle of the (unconnected) mirrors (nodes) of the Coxeter diagram.

(Final) 3-sphere solutions

Isomorphism of finite Coxeter groups

3-sphere solutions with density 1: ( )

Duoprism and hyperprism :
Coxeter Group and chart	[2,2,2]	[p, 2.2]	[p, 2, q]	[p, 2, p]	[3,3,2]	[4,3,2]	[5,3,2]
Symmetry Group Order	sixteen	8 p	4 pq	4 p ²	48	96	240
Symmetries tetrahedron	[3.3] (order 24)	[2] (order 4)	[2] (order 4)	[2 ⁺ , 4] (order 8)	[] (order 2)	[] ⁺ (order 1)	[] ⁺ (order 1)
Extended symmetries	[(3.3) [2.2.2]] = [4,3,3]	[2 [p, 2.2]] = [2p, 2,4]	[2 [p, 2, q]] = [2p, 2,2q]	[(2 ⁺ , 4) [p, 2, p]] = [2 ⁺ [2p, 2,2p]]	[1 [3,3,2]] = [4,3,2]	[4,3,2]	[5,3,2]
The order of extended symmetry groups	384	32 p	16 pq	32 p ²	96	96	240

Graph type	Linear				Trifoliate
Coxeter Group and chart	Five- cell	Sixteen cell	Twenty four- cell ]]	Six hundred cell	Half-act
Apex figure of all-truncated homogeneous polyhedra
Tetrahedron
Order symmetry groups	120	384	1152	14400	192
Tetrahedral symmetry	[2] ⁺ (order 2)	[] ⁺ (order 1)	[2] ⁺ (order 2)	[] ⁺ (order 1)	[3] (order 6)
Extended symmetry	[2 ⁺ [3,3,3]]	[4,3,3]	[2 ⁺ [3,4,3]]	[5,3,3]	[3 [3 ^1,1,1 ]] = [3,4,3]
Extended symmetry group order	240	384	2304	14400	1152

Solutions in Euclidean 3-dimensional space

Isomorphisms of the Euclidean Coxeter groups

Density Solutions 1:

Graph type	Linear	Trifoliate	Ring	Prismatic			Degenerate
Coxeter Group Coxeter Chart				[4,4,2]	[6,3,2]	[3 ^[3] , 2]	[∞, 2, ∞]
Vertex Figure of a Truncated Honeycomb
Tetrahedron
Tetrahedral symmetry	[2] ⁺ (order 2)	[] (order 2)	[2 ⁺ , 4] (order 8)	[] (order 2)	[] ⁺ (order 1)	[3] (order 6)	[2 ⁺ , 4] (order 8)
Extended symmetry	[(2 ⁺ ) [4,3,4]]	[1 [4.3 ^1.1 ]] = [4,3,4]	[(2 ⁺ , 4) [3 ^[4] ]] = [2 ⁺ [4,3,4]]	[1 [4,4,2]] = [4,4,2]	[6,3,2]	[3 [3 ^[3] , 2]] = [3,6,2]	[(2 ⁺ , 4) [∞, 2, ∞]] = [1 [4.4]]

Solutions for hyperbolic 3-spaces

Density Solutions 1: ( ) ( Compact (Lanner simplex groups) )

Lanner rank 4 simplex groups
Graph type	Linear			Trifoliate
Coxeter Group Coxeter Chart	[3,5,3]	[5,3,4]	[5,3,5]	[5.3 ^1.1 ]
Vertex Shaped Honeycomb Shapes
Tetrahedron
Tetrahedral symmetry	[2] ⁺ (order 2)	[] ⁺ (order 1)	[2] ⁺ (order 2)	[] (order 2)
Extended symmetry	[2 ⁺ [3,5,3]]	[5,3,4]	[2 ⁺ [5,3,5]]	[1 [5.3 ^1.1 ]] = [5,3,4]
Graph type	Ring
Coxeter Group Coxeter Chart	[(4,3,3,3)]	[(4.3) ² ]	[(5,3,3,3)]	[(5,3,4,3)]	[(5.3) ² ]
Vertex Shaped Honeycomb Figures
Tetrahedron
Tetrahedral symmetry	[2] ⁺ (order 2)	[2.2] ⁺ (order 4)	[2] ⁺ (order 2)	[2] ⁺ (order 2)	[2.2] ⁺ (order 4)
Extended symmetry	[2 ⁺ [(4,3,3,3)]]	[(2.2) ⁺ [(4.3) ² ]]	[2 ⁺ [(5,3,3,3)]]	[2 ⁺ [(5,3,4,3)]]	[(2.2) ⁺ [(5.3) ² ]]

Solutions in paracompact hyperbolic 3-spaces

Here, the relationship of the subgroups of a paracompact hyperbolic Goursat tetrahedron is shown. Subgroups of order 2 represent a partition of the Goursat tetrahedron by a plane of mirror symmetry

Density Solutions 1: (See Paracompact (Kozul Simplex Groups) )

Kozul rank 4 simplex groups
Graph type	Line graphs
Coxeter Group Coxeter Chart	[6.3.3]	[3,6,3]	[6,3,4]	[6.3.5]	[6,3,6]	[4,4,3]	[4,4,4]
Tetrahedral symmetry	[] ⁺ (order 1)	[2] ⁺ (order 2)	[] ⁺ (order 1)	[] ⁺ (order 1)	[2] ⁺ (order 2)	[] ⁺ (order 1)	[2] ⁺ (order 2)
Extended symmetry	[6.3.3]	[2 ⁺ [3,6,3]]	[6,3,4]	[6.3.5]	[2 ⁺ [6,3,6]]	[4,4,3]	[2 ⁺ [4,4,4]]
Graph type	Ring graphs
Coxeter Group Coxeter Chart	[3 ^{[] × []} ]	[(4,4,3,3)]	[(4 ³ , 3)]	[4 ^[4] ]	[(6.3 ³ )]	[(6,3,4,3)]	[(6,3,5,3)]	[(6.3) ^[2] ]
Tetrahedral symmetry	[2] (order 4)	[] (order 2)	[2] ⁺ (order 2)	[2 ⁺ , 4] (order 8)	[2] ⁺ (order 2)	[2] ⁺ (order 2)	[2] ⁺ (order 2)	[2.2] ⁺ (order 4)
Extended symmetry	[2 [3 ^{[] × []} ]] = [6.3.4]	[1 [(4,4,3,3)]] = [3.4 ^1.1 ]	[2 ⁺ [(4 ³ , 3)]]	[(2 ⁺ , 4) [4 ^[4] ]] = [2 ⁺ [4,4,4]]	[2 ⁺ [(6.3 ³ )]]	[2 ⁺ [(6,3,4,3)]]	[2 ⁺ [(6,3,5,3)]]	[(2.2) ⁺ [(6.3) ^[2] ]]
Graph type	Trifoliate			Tail ring				Simplex
Coxeter Group Coxeter Chart	[6.3 ^1.1 ]	[3.4 ^1.1 ]	[4 ^1,1,1 ]	[3.3 ^[3] ]	[4.3 ^[3] ]	[5.3 ^[3] ]	[6.3 ^[3] ]	[3 ^[3,3] ]
Tetrahedral symmetry	[] (order 2)	[] (order 2)	[3] (order 6)	[] (order 2)	[] (order 2)	[] (order 2)	[] (order 2)	[3.3] (order 24)
Extended symmetry	[1 [6.3 ^1.1 ]] = [6.3.4]	[1 [3,4 ^1,1 ]] = [3,4,4]	[3 [4 ^1,1,1 ]] = [4,4,3]	[1 [3.3 ^[3] ]] = [3,3,6]	[1 [4,3 ^[3 ]] = [4,3,6]	[1 [5,3 ^[3] ]] = [5,3,6]	[1 [6,3 ^[3 ]] = [6.3.6]	[(3.3) [3 ^[3.3] ]] = [6.3.3]

Rational Solutions

There are hundreds of rational solutions for 3-spheres , including these 6 linear graphs that form the Schlefli – Hess polyhedra , and 11 non-linear:

Line 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]	Counts "ring with a tail": Density 2: Density 3: Density 5: Density 8: Density 9: Density 14: Density 26: Density 30: Density 39: Density 46: Density 115:

Notes

Literature

Coxeter HCM Table 3: Schwarz's Triangles // . - Third edition. - Dover Edition, 1973. - S. 280, Goursat's tetrahedra. - ISBN 0-486-61480-8 .

The Theory of Uniform Polytopes and Honeycombs. - University of Toronto, 1966. - (Ph.D. Dissertation). Johnson proved that the enumeration of Goursat tetrahedrons by Coxeter is complete.
Edouard Goursat. Sur les substitutions orthogonales et les divisions régulières de l'espace // Annales scientifiques de l'École Normale Supérieure. - 1889. - Issue. 6 . - S. 9–102, 80–81 tetrahedra .
Klitzing, Richard. Dynkin Diagrams Goursat tetrahedra
Chapters 11,12,13 // Geometries and Transformations. - 2015.
Johnson NW, Kellerhals R., Ratcliffe JG, Tschantz ST Transformation Groups // The size of a hyperbolic Coxeter simplex . - 1999. - T. 4. - S. 329–353.