Goursat tetrahedron

Wythoffian polychora can be classified by their Goursat tetrahedron, which is the 3D extension of the Schwarz triangle. The equivalent for polychora are the 3-dimensional Goursat tetrahedra.

To quote Coxeter: Goursat [ref] proposed a problem analogous to that of Schwarz [ref]: to find all spherical tetrahedra which lead, by repeated reflection in their faces, to a finite set of congruent tetrahedra, i.e., to a honeycomb covering the hyperspace a finite number of times. Clearly, the reflections generate a group, viz. &hellip;,

[m] &times; [n] or [3, 3] &times; [1] or [3, 4] &times; [1] or [3, 5] &times; [1] or [3, 3, 3] or [3, 3, 4] or [3, 3, 5] or [3, 4, 3] or [31, 1, 1].

Hence the faces and their transforms dissect such a tetrahedron into a set of congruent tetrahedra &hellip;

When we compare this with the corresponding statement for Schwarz&rsquo;s triangles, we are not surprised to find Goursat&rsquo;s tetrahedra running into hundreds. Their complete enumeration will (perhaps !) be published elsewhere. The essential tool for that formidable work is the following process of deriving them from one another.

To be fair to Coxeter, when Coxeter penned those words, computers were not so prevalent as they are today, and the easiest way to generate the tetrahedra along with the densities (which cannot be calculated by simply looking at the vertex volumes, as with Schwarz triangles) is to consider the great spheres, taking four at a time, and looking at the sixteen 4-volumes (two sets of eight) that these divide the polychoron into.

Full list
The full list of tetrahedra (including [m] &times; [n], where {m, n} &#8834; {3, 4, 5}.