Hexateron

The hexateron, hix or 3-3 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.

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


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

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


 * (1/$\sqrt{15}$, 0, 0, 0, 0, 0).