Goursat tetrahedron

Wythoffian polychora can be classified by their Goursat tetrahedron, which is the 3D extension of the Schwarz triangle.

Schwarz triangles
Consider a regular convex polyhedron projected onto a sphere, and consider the three sets of lines, viz, the edges, lines connecting the face centres to the vertices of the said face, and lines connecting the face centres to the middles of the surrounding edges. These lines form a number of great circles on the sphere, and separate the spherical polyhedron into a number of triangles. These fundamental triangles are called M&ouml;bius triangles.

If, instead of restricting oneself to fundamental triangles, larger triangles can be created. These are called Schwarz triangles.

Each Schwarz triangle generates up to eight different polyhedra (seven simple and one snub form).

The density of a polyhedron is generally equal to the number of fundamental triangles enclosed. This number is easy to compute, by adding up the three angles for the Schwarz triangle.

Goursat tetrahedra
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 tetrahedram 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}.