For Euclidean 3-space, there are 3 simple and related Goursat tetrahedra, represented by [4,3,4], [4,31,1], and [3[4]]. They can be seen inside as points on and within a cube, {4,3}.
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.
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]].
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.
This show subgroup relations of paracompact hyperbolic Goursat tetrahedra. Order 2 subgroups represent bisecting a Goursat tetrahedron with a plane of mirror symmetry
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.[1]
See also
Point group for n-simplex solutions on (n-1)-sphere.
References
^https://arxiv.org/abs/2011.14232 Space vectors forming rational angles, Kiran S. Kedlaya, Alexander Kolpakov, Bjorn Poonen, Michael Rubinstein, 2020
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)
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]
