Heptapeton

The heptapeton, or hop, also commonly called the 6-simplex, is the simplest possible non-degenerate polypeton. The full symmetry version has 7 regular hexatera as cells, joining 6 to a vertex, and is one of the 3 regular polytera. It is the 6-dimensional simplex.

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


 * (±1/2, –$\sqrt{21}$/6, –$\sqrt{7}$/12, –$\sqrt{3}$/20, –$\sqrt{6}$/30, –$\sqrt{10}$/42),
 * (0, $\sqrt{15}$/3, –$\sqrt{21}$/12, –$\sqrt{3}$/20, –$\sqrt{6}$/30, –$\sqrt{10}$/42),
 * (0, 0, $\sqrt{15}$/4, –$\sqrt{21}$/20, –$\sqrt{6}$/30, –$\sqrt{10}$/42),
 * (0, 0, 0, $\sqrt{15}$/5, –$\sqrt{21}$/30, –$\sqrt{10}$/42),
 * (0, 0, 0, 0, $\sqrt{15}$/6, –$\sqrt{21}$/42),
 * (0, 0, 0, 0, 0, $\sqrt{15}$/7).

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


 * (1/$\sqrt{21}$, 0, 0, 0, 0, 0, 0).