Ingeometry, aGoursat tetrahedron is atetrahedralfundamental domain of aWythoff construction. Each tetrahedral face represents a reflection hyperplane on 3-dimensional surfaces: the3-sphere, Euclidean 3-space, and hyperbolic 3-space.Coxeter named them afterÉdouard Goursat who first looked into these domains. It is an extension of the theory ofSchwarz triangles for Wythoff constructions on the sphere.

AGoursat 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-nodeCoxeter-Dynkin diagram represents this tetrahedral graph with order-2 edges hidden. If many edges are order 2, theCoxeter group can be represented by abracket 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 aSchwarz triangle.

The symmetry of a Goursat tetrahedron can betetrahedral 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 asemidirect product of theCoxeter group symmetry and thefundamental 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], withY as a symmetry of the Goursat tetrahedron. IfY 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 regulartetrahedron 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]].

SeeTetrahedron#Isometries of irregular tetrahedra for 7 lower symmetry isometries of the tetrahedron.

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 arevertex figures foromnitruncated polytopes and honeycombs from each symmetry family. The edge labels represent polygonal face orders, which is double the Coxeter graph branch order. Thedihedral angle of an edge labeled2n is π/n. Yellow edges labeled 4 come from right angle (unconnected) mirror nodes in the Coxeter diagram.

The solutions for the3-sphere with density 1 solutions are: (Uniform polychora)

Duoprisms andhyperprisms:

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	4p2	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	32p2	96	96	240

Graph type	Linear				Tridental
Coxeter group and diagram	Pentachoric [3,3,3]	Hexadecachoric [4,3,3]	Icositetrachoric [3,4,3]	Hexacosichoric [5,3,3]	Demitesseractic [31,1,1]
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[31,1,1]] =[3,4,3]
Extended symmetry order	240	384	2304	14400	1152

Density 1 solutions:Convex uniform honeycombs:

Graph type	Linear Orthoscheme	Tri-dental Plagioscheme	Loop Cycloscheme	Prismatic			Degenerate
Coxeter group Coxeter diagram	[4,3,4]	[4,31,1]	[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[4,31,1]] =[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]]

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

Rank 4 Lannér simplex groups

Graph type	Linear			Tri-dental
Coxeter group Coxeter diagram	[3,5,3]	[5,3,4]	[5,3,5]	[5,31,1]
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[5,31,1]] =[5,3,4]
Graph type	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]]

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

Rank 4 Koszul simplex groups

Graph type	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
Coxeter group and diagram	[3[ ]×[ ]]	[(4,4,3,3)]	[(43,3)]	[4[4]]	[(6,33)]	[(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)]] =[3,41,1]	[2+[(43,3)]]	[(2+,4)[4[4]]] =[2+[4,4,4]]	[2+[(6,33)]]	[2+[(6,3,4,3)]]	[2+[(6,3,5,3)]]	[(2,2)+[(6,3)[2]]]
Graph type	Tri-dental			Loop-n-tail				Simplex
Coxeter group and diagram	[6,31,1]	[3,41,1]	[41,1,1]	[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[6,31,1]] =[6,3,4]	[1[3,41,1]] =[3,4,4]	[3[41,1,1]] =[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]

There are hundreds of rational solutions for the3-sphere, including these 6 linear graphs which generate theSchläfli-Hess polychora, and 11 nonlinear ones from Coxeter:

In all, there are 59 sporadic tetrahedra with rational angles, and 2 infinite families.^[1]

• Point group forn-simplex solutions on (n−1)-sphere.

• Regular Polytopes, (3rd edition, 1973), Dover edition,ISBN 0-486-61480-8 (page 280, Goursat's tetrahedra)[1]
• Norman JohnsonThe 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)
• Klitzing, Richard."Dynkin Diagrams Goursat 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[2]

• Tetrahedra