Goursat tetrahedron
# Goursat tetrahedron

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.

A 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

* The 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 [Y[X]] 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 $[2^{[3,3]}]$ and the paracompact hyperbolic group $[3^{[3,3]}].$

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

The solutions for the 3-sphere with density 1 solutions are: (Uniform polychora)

Duoprisms and hyperprisms: :

| Coxeter group and diagram | [2,2,2] | [p,2,2] | [p,2,q] | [p,2,p] | [3,3,2] | [4,3,2] | [5,3,2] |
| - | - | - | - | - | - | - | - |
| Group symmetry order | 16 | 8p | 4pq | $4p^2$ | 48 | 96 | 240 |
| Tetrahedron symmetry | [3,3] (order 24) | [2] (order 4) | [2] (order 4) | $[2^+,4]$ (order 8) | [ ] (order 2) | [ $]^+$ (order 1) | [ $]^+$ (order 1) |
| Extended symmetry | [(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] |
| Extended symmetry order | 384 | 32p | 16pq | $32p^2$ | 96 | 96 | 240 |
| Graph type | Linear | Linear | Linear | Linear | Tridental |
| Coxeter group and diagram | Pentachoric | Hexadecachoric | Icositetrachoric | Hexacosichoric | Demitesseractic $[3^{1,1,1}]$ |
| Vertex figure of omnitruncated uniform polychora ||||||
| Tetrahedron | | | | | |
| Group symmetry order | 120 | 384 | 1152 | 14400 | 192 |
| Tetrahedron 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 order | 240 | 384 | 2304 | 14400 | 1152 |

### Euclidean (affine) 3-space solutions

Density 1 solutions: Convex uniform honeycombs:

| Graph type | Linear Orthoscheme | Tri-dental Plagioscheme | Loop Cycloscheme | Prismatic | Prismatic | Prismatic | Degenerate |
| - | - | - | - | - | - | - | - |
| Coxeter group Coxeter diagram | | $[4,3^{1,1}]$ | $[3^{[4]}]$ | [4,4,2] | [6,3,2] | $[3^{[3]},2]$ | [∞,2,∞] |
| Vertex figure of omnitruncated honeycombs ||||||||
| Tetrahedron | | | | | | | |
| Tetrahedron 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]] |

### Compact hyperbolic 3-space solutions

Density 1 solutions: (Convex uniform honeycombs in hyperbolic space) (Coxeter diagram#Compact (Lannér simplex groups))

Rank 4 Lannér simplex groups :

| Graph type | Linear | Linear | Linear | Tri-dental |
| - | - | - | - | - |
| Coxeter group Coxeter diagram | [3,5,3] | [5,3,4] | [5,3,5] | $[5,3^{1,1}]$ | |
| Vertex figures of omnitruncated honeycombs |||||
| Tetrahedron | | | | | |
| Tetrahedron 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 | Loop | Loop | Loop | Loop |
| Coxeter group Coxeter diagram | [(4,3,3,3)] | $[(4,3)^2]$ | [(5,3,3,3)] | [(5,3,4,3)] | $[(5,3)^2]$ |
| Vertex figures of omnitruncated honeycombs |||||
| Tetrahedron | | | | | |
| Tetrahedron Symmetry | $[2]^+$ (order 2) | $[2,2]^+$ (order 4) | $[2]^+$ (order 2) | $[2]^+$ (order 2) | $[2,2]^+$ (order 4) |
| Extended symmetry | $[2^+[(4,3,3,3)]]$ | $[(2,2)^+[(4,3)^2]]$ | $[2^+[(5,3,3,3)]]$ | $[2^+[(5,3,4,3)]]$ | $[(2,2)^+[(5,3)^2]]$ |

### Paracompact hyperbolic 3-space solutions

Density 1 solutions: (See Coxeter diagram#Paracompact (Koszul simplex groups))

Rank 4 Koszul simplex groups :

| Graph type | Linear graphs | Linear graphs | Linear graphs | Linear graphs | Linear graphs | Linear graphs | Linear graphs |
| - | - | - | - | - | - | - | - |
| Coxeter group and diagram | [6,3,3] | [3,6,3] | [6,3,4] | [6,3,5] | [6,3,6] | [4,4,3] | [4,4,4] |
| Tetrahedron 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 | Loop graphs | Loop graphs | Loop graphs | Loop graphs | Loop graphs | Loop graphs | Loop graphs |
| Coxeter group and diagram | $[3^{×}]$ | [(4,4,3,3)] | $[(4^3,3)]$ | $[4^{[4]}]$ | $[(6,3^3)]$ | [(6,3,4,3)] | [(6,3,5,3)] | $[(6,3)^{[2]}]$ |
| Tetrahedron 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 | Tri-dental | Tri-dental | Tri-dental | Loop-n-tail | Loop-n-tail | Loop-n-tail | Loop-n-tail | Simplex |
| Coxeter group and diagram | $[6,3^{1,1}]$ | $[3,4^{1,1}]$ | $[4^{1,1,1}]$ | $[3,3^{[3]}]$ | $[4,3^{[3]}]$ | $[5,3^{[3]}]$ | $[6,3^{[3]}]$ | $[3^{[3,3]}]$ |
| Tetrahedron 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 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.

