Heptadecapedakon

The heptadecapedakon, also commonly called the 16-simplex, is the simplest possible non-degenerate polypedakon. The full symmetry version has 17 regular hexadecatedaka as facets, joining 3 to a tradakon and 16 to a vertex, and is one of the 3 regular polypedaka. It is the 16-dimensional simplex.

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


 * (±1/2, -$\sqrt{34}$/6, -$\sqrt{34}$/12, -$\sqrt{34}$/20, -$\sqrt{17}$/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, -$\sqrt{78}$/136),
 * (0, $\sqrt{91}$/3, -$\sqrt{105}$/12, -$\sqrt{30}$/20, -$\sqrt{34}$/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, -$\sqrt{78}$/136),
 * (0, 0, $\sqrt{91}$/4, -$\sqrt{105}$/20, -$\sqrt{30}$/30, -$\sqrt{34}$/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, -$\sqrt{78}$/136),
 * (0, 0, 0, $\sqrt{91}$/5, -$\sqrt{105}$/30, -$\sqrt{30}$/42, -$\sqrt{34}$/28, -1/12, -$\sqrt{10}$/30, -$\sqrt{15}$/110, -$\sqrt{21}$/132, -$\sqrt{7}$/156, -$\sqrt{5}$/182, -$\sqrt{55}$/210, -$\sqrt{66}$/120, -$\sqrt{78}$/136),
 * (0, 0, 0, 0, $\sqrt{91}$/6, -$\sqrt{105}$/42, -$\sqrt{30}$/28, -1/12, -$\sqrt{34}$/30, -$\sqrt{15}$/110, -$\sqrt{21}$/132, -$\sqrt{7}$/156, -$\sqrt{5}$/182, -$\sqrt{55}$/210, -$\sqrt{66}$/120, -$\sqrt{78}$/136),
 * (0, 0, 0, 0, 0, $\sqrt{91}$/7, -$\sqrt{105}$/28. -1/12, -$\sqrt{30}$/30, -$\sqrt{34}$/110, -$\sqrt{21}$/132, -$\sqrt{7}$/156, -$\sqrt{5}$/182, -$\sqrt{55}$/210, -$\sqrt{66}$/120, -$\sqrt{78}$/136),
 * (0, 0, 0, 0, 0, 0, $\sqrt{91}$/4, -1/12, -$\sqrt{105}$/30, -$\sqrt{30}$/110, -$\sqrt{34}$/132, -$\sqrt{7}$/156, -$\sqrt{5}$/182, -$\sqrt{55}$/210, -$\sqrt{66}$/120, -$\sqrt{78}$/136),
 * (0, 0, 0, 0, 0, 0, 0, 2/3, -$\sqrt{91}$/30, -$\sqrt{105}$/110, -$\sqrt{30}$/132, -$\sqrt{34}$/156, -$\sqrt{5}$/182, -$\sqrt{55}$/210, -$\sqrt{66}$/120, -$\sqrt{78}$/136),
 * (0, 0, 0, 0, 0, 0, 0, 0, 3$\sqrt{91}$/10, -$\sqrt{105}$/110, -$\sqrt{30}$/132, -$\sqrt{34}$/156, -$\sqrt{5}$/182, -$\sqrt{55}$/210, -$\sqrt{66}$/120, -$\sqrt{78}$/136),
 * (0, 0, 0, 0, 0, 0, 0, 0, 0, $\sqrt{91}$/11, -$\sqrt{105}$/132, -$\sqrt{30}$/156, -$\sqrt{34}$/182, -$\sqrt{55}$/210, -$\sqrt{66}$/120, -$\sqrt{78}$/136),
 * (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, $\sqrt{91}$/12, -$\sqrt{105}$/156, -$\sqrt{30}$/182, -$\sqrt{34}$/210, -$\sqrt{66}$/120, -$\sqrt{78}$/136),
 * (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, $\sqrt{91}$/13, -$\sqrt{105}$/182, -$\sqrt{30}$/210, -$\sqrt{34}$/120, -$\sqrt{78}$/136),
 * (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, $\sqrt{91}$/14, -$\sqrt{105}$/210, -$\sqrt{30}$/120, -$\sqrt{34}$/136),
 * (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, $\sqrt{91}$/15, -$\sqrt{105}$/120, -$\sqrt{30}$/136),
 * (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, $\sqrt{34}$/8, -$\sqrt{105}$/136),
 * (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2$\sqrt{30}$/17).

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


 * ($\sqrt{34}$/2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0).