Hexateron

The hexateron, hix or triangular disphenoid, also commonly called the 5-simplex, is the simplest possible non-degenerate polyteron. The full symmetry version has 6 regular pentachora as cells, joining 5 to a vertex, and is one of the 3 regular polytera. It is the 5-dimensional simplex.

It can be viewed as a segmentoteron in three ways: as a pentachoric pyramid, as a dyad atop perpendicular tetrahedron, and as a triangle atop perpendicular triangle. This makes it the triangular member of an infinite family of isogonal polygonal disphenoids.

Vertex coordinates
The vertices of a regular hexateron of edge length 1, centered at the origin, are given by:


 * $$\left(±\frac{1}{2},\,-\frac{\sqrt{3}}{6},\,-\frac{\sqrt{6}}{12},\,-\frac{\sqrt{10}}{20},\,-\frac{\sqrt{15}}{30}\right),$$
 * $$\left(0,\,\frac{\sqrt{3}}{3},\,-\frac{\sqrt{6}}{12},\,-\frac{\sqrt{10}}{20},\,-\frac{\sqrt{15}}{30}\right),$$
 * $$\left(0,\,0,\,\frac{\sqrt{6}}{4},\,-\frac{\sqrt{10}}{20},\,-\frac{\sqrt{15}}{30}\right),$$
 * $$\left(0,\,0,\,0,\,\frac{\sqrt{10}}{5},\,-\frac{\sqrt{15}}{30}\right),$$
 * $$\left(0,\,0,\,0,\,0,\,\frac{\sqrt{15}}{6}\right).$$

Much simpler coordinates can be given in six dimensions, as all permutations of:


 * $$\left(\frac{\sqrt{2}}{2},\,0,\, 0,\, 0,\, 0,\, 0\right).$$

Representations
A regular hexateron has the following Coxeter diagrams:


 * x3o3o3o3o (full symmetry)
 * ox3oo3oo3oo&#x (A4 axial, pentachoric pyramid)
 * xo ox3oo3oo&#x (A3×A1 symmetry, tetrahedral scalene)
 * xo3oo ox3oo&#x (A2×A2 axial, triangular disphenoid)
 * oxo3ooo3ooo&#x (A3 symmetry, tetrahedral pyramidal pyramid)
 * oxo oox3ooo&#x A2×A1 symmetry, triangular scalene pyramid)
 * xoo oxo oox&#x (A1×A1×A1 symmetry, digonal trisphenoid)
 * ooox ooxo&#x (A1×A1 symmetry, disphenoidal pyramidal pyramid)
 * ooox3oooo&#x (A2 symmetry, triangular symmetry only)
 * oooox&#x (A1 symmetrry only)
 * oooooo&#x (no symmetry, fully irregular)

Variations
The regular hexateron has 2 subsymmetrical forms that remain isogonal:


 * Triangular disphenoid - triangle atop an orthogonal triangle, facets and vertex figures are triangular scalenes
 * Digonal trisphenoid - Cells and vertex figures are disphenoidal pyramids