Tridecahendon

The tridecahendakon, also commonly called the 12-simplex, is the simplest possible non-degenerate polyhendakon. The full symmetry version has 13 regular dodecadaka as facets, joining 3 to a xennon and 11 to a vertex, and is one of the 3 regular polyhendaka. It is the 12-dimensional simplex.

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


 * (±1/2, -$\sqrt{78}$/6, -$\sqrt{78}$/12, -$\sqrt{78}$/20, -$\sqrt{13}$/30, -$\sqrt{3}$/42, -$\sqrt{6}$/28, -1/12, -$\sqrt{10}$/30, -$\sqrt{15}$/110, -$\sqrt{21}$/132, -$\sqrt{7}$/156),
 * (0, $\sqrt{5}$/3, -$\sqrt{55}$/12, -$\sqrt{66}$/20, -$\sqrt{78}$/30, -$\sqrt{3}$/42, -$\sqrt{6}$/28, -1/12, -$\sqrt{10}$/30, -$\sqrt{15}$/110, -$\sqrt{21}$/132, -$\sqrt{7}$/156),
 * (0, 0, $\sqrt{5}$/4, -$\sqrt{55}$/20, -$\sqrt{66}$/30, -$\sqrt{78}$/42, -$\sqrt{6}$/28, -1/12, -$\sqrt{10}$/30, -$\sqrt{15}$/110, -$\sqrt{21}$/132, -$\sqrt{7}$/156),
 * (0, 0, 0, $\sqrt{5}$/5, -$\sqrt{55}$/30, -$\sqrt{66}$/42, -$\sqrt{78}$/28, -1/12, -$\sqrt{10}$/30, -$\sqrt{15}$/110, -$\sqrt{21}$/132, -$\sqrt{7}$/156),
 * (0, 0, 0, 0, $\sqrt{5}$/6, -$\sqrt{55}$/42, -$\sqrt{66}$/28, -1/12, -$\sqrt{78}$/30, -$\sqrt{15}$/110, -$\sqrt{21}$/132, -$\sqrt{7}$/156),
 * (0, 0, 0, 0, 0, $\sqrt{5}$/7, -$\sqrt{55}$/28. -1/12, -$\sqrt{66}$/30, -$\sqrt{78}$/110, -$\sqrt{21}$/132, -$\sqrt{7}$/156),
 * (0, 0, 0, 0, 0, 0, $\sqrt{5}$/4, -1/12, -$\sqrt{55}$/30, -$\sqrt{66}$/110, -$\sqrt{78}$/132, -$\sqrt{7}$/156),
 * (0, 0, 0, 0, 0, 0, 0, 2/3, -$\sqrt{5}$/30, -$\sqrt{55}$/110, -$\sqrt{66}$/132, -$\sqrt{78}$/156),
 * (0, 0, 0, 0, 0, 0, 0, 0, 3$\sqrt{5}$/10, -$\sqrt{55}$/110, -$\sqrt{66}$/132, -$\sqrt{78}$/156),
 * (0, 0, 0, 0, 0, 0, 0, 0, 0, $\sqrt{5}$/11, -$\sqrt{55}$/132, -$\sqrt{66}$/156),
 * (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, $\sqrt{78}$/12, -$\sqrt{55}$/156),
 * (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, $\sqrt{66}$/13).

Much simpler coordinates can be given in 13 dimensions, as all permutations of:


 * ($\sqrt{78}$/2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0).