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.

The following sections show all of the whole number Goursat tetrahedral solutions on the 3-sphere, Euclidean 3-space, and Hyperbolic 3-space.

The colored tetrahedal diagrams below are vertex figures for omnitruncated polytopes and honeycombs from each symmetry family.

Yellow edges labeled 4 come from right angle (unconnected) mirror nodes in the Coxeter diagram.

For Euclidean 3-space, there are 3 simple and related Goursat tetrahedra, represented by [4,3,4], [4,3 1,1 ], and [3 [4] ]. They can be seen inside as points on and within a cube, {4,3}.
Finite Coxeter groups isomorphisms
Euclidean Coxeter group isomorphisms
This show subgroup relations of paracompact hyperbolic Goursat tetrahedra. Order 2 subgroups represent bisecting a Goursat tetrahedron with a plane of mirror symmetry