Pentadecatradakon

The pentadecatradakon, also commonly called the 14-simplex, is the simplest possible non-degenerate polytradakon. The full symmetry version has 15 regular tetradecadoka as facets, joining 3 to a hendon and 14 to a vertex, and is one of the 3 regular polytradaka. It is the 14-dimensional simplex.

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


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

Much simpler coordinates can be given in 15 dimensions, as all permutations of:


 * ($\sqrt{105}$/2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0).