Hendecaxennon

The hendecaxennon, or ux, also commonly called the 10-simplex, is the simplest possible non-degenerate polyxennon. The full symmetry version has 11 regular decayotta as facets, joining 3 to an octaexon peak and 10 to a vertex, and is one of the 3 regular polyxenna. It is the 10-dimensional simplex.

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


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

Much simpler coordinates can be given in 11 dimensions, as all permutations of:


 * ($\sqrt{55}$/2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0).