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:


 * (±1/2, –$\sqrt{15}$/6, –$\sqrt{15}$/12, –$\sqrt{15}$/20, –$\sqrt{6}$/30),
 * (0, $\sqrt{3}$/3, –$\sqrt{3}$/12, –$\sqrt{3}$/20, –$\sqrt{6}$/30),
 * (0, 0, $\sqrt{10}$/4, –$\sqrt{15}$/20, –$\sqrt{3}$/30),
 * (0, 0, 0, $\sqrt{6}$/5, –$\sqrt{10}$/30),
 * (0, 0, 0, 0, $\sqrt{15}$/6).

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


 * ($\sqrt{6}$/2, 0, 0, 0, 0, 0).

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)