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.

A regular hexadecatedakon of edge length 2$\sqrt{2}$ can be inscribed in the unit pentadekeract.

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


 * $$\left(±\frac{1}{2},\,-\frac{\sqrt{3}}{6},\,-\frac{\sqrt{6}}{12},\,-\frac{\sqrt{10}}{20},\,-\frac{\sqrt{15}}{30},\,-\frac{\sqrt{21}}{42},\,-\frac{\sqrt7}{28},\,-\frac{1}{12},\,-\frac{\sqrt5}{30},\,-\frac{\sqrt{55}}{110},\,-\frac{\sqrt{66}}{132},\,-\frac{\sqrt{78}}{156},\,-\frac{\sqrt{91}}{182},\,-\frac{\sqrt{105}}{210},\,-\frac{\sqrt{30}}{120}\right),$$
 * $$\left(0,\,\frac{\sqrt{3}}{3},\,-\frac{\sqrt{6}}{12},\,-\frac{\sqrt{10}}{20},\,-\frac{\sqrt{15}}{30},\,-\frac{\sqrt{21}}{42},\,-\frac{\sqrt7}{28},\,-\frac{1}{12},\,-\frac{\sqrt5}{30},\,-\frac{\sqrt{55}}{110},\,-\frac{\sqrt{66}}{132},\,-\frac{\sqrt{78}}{156},\,-\frac{\sqrt{91}}{182},\,-\frac{\sqrt{105}}{210},\,-\frac{\sqrt{30}}{120}\right),$$
 * $$\left(0,\,0,\,\frac{\sqrt{6}}{4},\,-\frac{\sqrt{10}}{20},\,-\frac{\sqrt{15}}{30},\,-\frac{\sqrt{21}}{42},\,-\frac{\sqrt7}{28},\,-\frac{1}{12},\,-\frac{\sqrt5}{30},\,-\frac{\sqrt{55}}{110},\,-\frac{\sqrt{66}}{132},\,-\frac{\sqrt{78}}{156},\,-\frac{\sqrt{91}}{182},\,-\frac{\sqrt{105}}{210},\,-\frac{\sqrt{30}}{120}\right),$$
 * $$\left(0,\,0,\,0,\,\frac{\sqrt{10}}{5},\,-\frac{\sqrt{15}}{30},\,-\frac{\sqrt{21}}{42},\,-\frac{\sqrt7}{28},\,-\frac{1}{12},\,-\frac{\sqrt5}{30},\,-\frac{\sqrt{55}}{110},\,-\frac{\sqrt{66}}{132},\,-\frac{\sqrt{78}}{156},\,-\frac{\sqrt{91}}{182},\,-\frac{\sqrt{105}}{210},\,-\frac{\sqrt{30}}{120}\right),$$
 * $$\left(0,\,0,\,0,\,0,\,\frac{\sqrt{15}}{6},\,-\frac{\sqrt{21}}{42},\,-\frac{\sqrt7}{28},\,-\frac{1}{12},\,-\frac{\sqrt5}{30},\,-\frac{\sqrt{55}}{110},\,-\frac{\sqrt{66}}{132},\,-\frac{\sqrt{78}}{156},\,-\frac{\sqrt{91}}{182},\,-\frac{\sqrt{105}}{210},\,-\frac{\sqrt{30}}{120}\right),$$
 * $$\left(0,\,0,\,0,\,0,\,0,\,\frac{\sqrt{21}}{7},\,-\frac{\sqrt7}{28},\,-\frac{1}{12},\,-\frac{\sqrt5}{30},\,-\frac{\sqrt{55}}{110},\,-\frac{\sqrt{66}}{132},\,-\frac{\sqrt{78}}{156},\,-\frac{\sqrt{91}}{182},\,-\frac{\sqrt{105}}{210},\,-\frac{\sqrt{30}}{120}\right),$$
 * $$\left(0,\,0,\,0,\,0,\,0,\,0,\,\frac{\sqrt7}{4},\,-\frac{1}{12},\,-\frac{\sqrt5}{30},\,-\frac{\sqrt{55}}{110},\,-\frac{\sqrt{66}}{132},\,-\frac{\sqrt{78}}{156},\,-\frac{\sqrt{91}}{182},\,-\frac{\sqrt{105}}{210},\,-\frac{\sqrt{30}}{120}\right),$$
 * $$\left(0,\,0,\,0,\,0,\,0,\,0,\,0,\,\frac23,\,-\frac{\sqrt5}{30},\,-\frac{\sqrt{55}}{110},\,-\frac{\sqrt{66}}{132},\,-\frac{\sqrt{78}}{156},\,-\frac{\sqrt{91}}{182},\,-\frac{\sqrt{105}}{210},\,-\frac{\sqrt{30}}{120}\right),$$
 * $$\left(0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,\frac{3\sqrt5}{10},\,-\frac{\sqrt{55}}{110},\,-\frac{\sqrt{66}}{132},\,-\frac{\sqrt{78}}{156},\,-\frac{\sqrt{91}}{182},\,-\frac{\sqrt{105}}{210},\,-\frac{\sqrt{30}}{120}\right),$$
 * $$\left(0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,\frac{\sqrt{55}}{11},\,-\frac{\sqrt{66}}{132},\,-\frac{\sqrt{78}}{156},\,-\frac{\sqrt{91}}{182},\,-\frac{\sqrt{105}}{210},\,-\frac{\sqrt{30}}{120}\right),$$
 * $$\left(0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,\frac{\sqrt{66}}{12},\,-\frac{\sqrt{78}}{156},\,-\frac{\sqrt{91}}{182},\,-\frac{\sqrt{105}}{210},\,-\frac{\sqrt{30}}{120}\right),$$
 * $$\left(0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,\frac{\sqrt{78}}{13},\,-\frac{\sqrt{91}}{182},\,-\frac{\sqrt{105}}{210},\,-\frac{\sqrt{30}}{120}\right),$$
 * $$\left(0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,\frac{\sqrt{91}}{14},\,-\frac{\sqrt{105}}{210},\,-\frac{\sqrt{30}}{120}\right),$$
 * $$\left(0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,\frac{\sqrt{105}}{15},\,-\frac{\sqrt{30}}{120}\right),$$
 * $$\left(0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,\frac{\sqrt{30}}{8}\right).$$

Simpler sets of coordinates can be found by inscribing the hexadecatedakon into the pentadekeract. One such set is given by:


 * $$\left(\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,\frac{\sqrt2}{8}\right),$$
 * $$\left(\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8}\right),$$
 * $$\left(\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8}\right),$$
 * $$\left(\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,\frac{\sqrt2}{8}\right),$$
 * $$\left(\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8}\right),$$
 * $$\left(\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,\frac{\sqrt2}{8}\right),$$
 * $$\left(\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,\frac{\sqrt2}{8}\right),$$
 * $$\left(\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8}\right),$$
 * $$\left(-\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8}\right),$$
 * $$\left(-\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,\frac{\sqrt2}{8}\right),$$
 * $$\left(-\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,\frac{\sqrt2}{8}\right),$$
 * $$\left(-\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8}\right),$$
 * $$\left(-\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,\frac{\sqrt2}{8}\right),$$
 * $$\left(-\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8}\right),$$
 * $$\left(-\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8}\right),$$
 * $$\left(-\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,-\frac{\sqrt2}{8},\,\frac{\sqrt2}{8}\right).$$

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


 * $$\left(\frac{\sqrt2}{2},\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0\right).$$