Dodecadakon

The dodecadakon, also commonly called the 11-simplex, is the simplest possible non-degenerate polydakon. The full symmetry version has 12 regular hendecaxenna as facets, joining 3 to a yotton and 11 to a vertex, and is one of the 3 regular polydaka. It is the 11-dimensional simplex.

A regular dodecadakon of edge length $\sqrt{66}$ can be inscribed in the hendekeract. The next largest simplex that can be inscribed in a hypercube is the hexadecatedakon.

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


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

Much simpler coordinates can be given in 12 dimensions, as all permutations of:


 * ($\sqrt{66}$/2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0).