Hexadecatedakon

The hexadecatedakon, also commonly called the 15-simplex, is the simplest possible non-degenerate polytedakon. The full symmetry version has 16 regular pentadecatradaka as facets, joining 3 to a dokon and 15 to a vertex, and is one of the 3 regular polytedaka. It is the 15-dimensional simplex.

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


 * (±1/2, -$\sqrt{30}$/6, -$\sqrt{30}$/12, -$\sqrt{30}$/20, -$\sqrt{2}$/30, -$\sqrt{3}$/42, -$\sqrt{6}$/28, -1/12, -$\sqrt{10}$/30, -$\sqrt{15}$/110, -$\sqrt{21}$/132, -$\sqrt{7}$/156, -$\sqrt{5}$/182, -$\sqrt{55}$/210, -$\sqrt{66}$/120),
 * (0, $\sqrt{78}$/3, -$\sqrt{91}$/12, -$\sqrt{105}$/20, -$\sqrt{30}$/30, -$\sqrt{3}$/42, -$\sqrt{6}$/28, -1/12, -$\sqrt{10}$/30, -$\sqrt{15}$/110, -$\sqrt{21}$/132, -$\sqrt{7}$/156, -$\sqrt{5}$/182, -$\sqrt{55}$/210, -$\sqrt{66}$/120),
 * (0, 0, $\sqrt{78}$/4, -$\sqrt{91}$/20, -$\sqrt{105}$/30, -$\sqrt{30}$/42, -$\sqrt{6}$/28, -1/12, -$\sqrt{10}$/30, -$\sqrt{15}$/110, -$\sqrt{21}$/132, -$\sqrt{7}$/156, -$\sqrt{5}$/182, -$\sqrt{55}$/210, -$\sqrt{66}$/120),
 * (0, 0, 0, $\sqrt{78}$/5, -$\sqrt{91}$/30, -$\sqrt{105}$/42, -$\sqrt{30}$/28, -1/12, -$\sqrt{10}$/30, -$\sqrt{15}$/110, -$\sqrt{21}$/132, -$\sqrt{7}$/156, -$\sqrt{5}$/182, -$\sqrt{55}$/210, -$\sqrt{66}$/120),
 * (0, 0, 0, 0, $\sqrt{78}$/6, -$\sqrt{91}$/42, -$\sqrt{105}$/28, -1/12, -$\sqrt{30}$/30, -$\sqrt{15}$/110, -$\sqrt{21}$/132, -$\sqrt{7}$/156, -$\sqrt{5}$/182, -$\sqrt{55}$/210, -$\sqrt{66}$/120),
 * (0, 0, 0, 0, 0, $\sqrt{78}$/7, -$\sqrt{91}$/28. -1/12, -$\sqrt{105}$/30, -$\sqrt{30}$/110, -$\sqrt{21}$/132, -$\sqrt{7}$/156, -$\sqrt{5}$/182, -$\sqrt{55}$/210, -$\sqrt{66}$/120),
 * (0, 0, 0, 0, 0, 0, $\sqrt{78}$/4, -1/12, -$\sqrt{91}$/30, -$\sqrt{105}$/110, -$\sqrt{30}$/132, -$\sqrt{7}$/156, -$\sqrt{5}$/182, -$\sqrt{55}$/210, -$\sqrt{66}$/120),
 * (0, 0, 0, 0, 0, 0, 0, 2/3, -$\sqrt{78}$/30, -$\sqrt{91}$/110, -$\sqrt{105}$/132, -$\sqrt{30}$/156, -$\sqrt{5}$/182, -$\sqrt{55}$/210, -$\sqrt{66}$/120),
 * (0, 0, 0, 0, 0, 0, 0, 0, 3$\sqrt{78}$/10, -$\sqrt{91}$/110, -$\sqrt{105}$/132, -$\sqrt{30}$/156, -$\sqrt{5}$/182, -$\sqrt{55}$/210, -$\sqrt{66}$/120),
 * (0, 0, 0, 0, 0, 0, 0, 0, 0, $\sqrt{78}$/11, -$\sqrt{91}$/132, -$\sqrt{105}$/156, -$\sqrt{30}$/182, -$\sqrt{55}$/210, -$\sqrt{66}$/120),
 * (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, $\sqrt{78}$/12, -$\sqrt{91}$/156, -$\sqrt{105}$/182, -$\sqrt{30}$/210, -$\sqrt{66}$/120),
 * (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, $\sqrt{78}$/13, -$\sqrt{91}$/182, -$\sqrt{105}$/210, -$\sqrt{30}$/120),
 * (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, $\sqrt{78}$/14, -$\sqrt{91}$/210, -$\sqrt{105}$/120),
 * (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, $\sqrt{30}$/15, -$\sqrt{91}$/120),
 * (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, $\sqrt{105}$/8).

Much simpler coordinates can be given in 16 dimensions, as all permutations of:


 * ($\sqrt{30}$/2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0).