Hexadecatedakon

Hexadecatedakon
Rank	15
Type	Regular
Space	Spherical
Notation
Coxeter diagram	x3o3o3o3o3o3o3o3o3o3o3o3o3o3o ()
Schläfli symbol	{3,3,3,3,3,3,3,3,3,3,3,3,3,3}
Elements
Tedaka	16 pentadecatradaka
Tradaka	120 tetradecadoka
Doka	560 tridecahenda
Henda	1820 dodecadaka
Daka	4368 hendecaxenna
Xenna	8008 decayotta
Yotta	11440 enneazetta
Zetta	12870 octaexa
Exa	11440 heptapeta
Peta	8008 hexatera
Tera	4368 pentachora
Cells	1820 tetrahedra
Faces	560 triangles
Edges	120
Vertices	16
Vertex figure	Pentadecatradakon, edge length 1
Measures (edge length 1)
Circumradius	$\frac{\sqrt{30}}{8} \approx 0.68465$
Inradius	$\frac{\sqrt{30}}{120} \approx 0.045644$
Hypervolume	$\frac{\sqrt2}{83691159552000} \approx 1.6898×10^{-14}$
Dihedral angle	$\arccos\left(\frac{1}{15}\right) \approx 86.17745°$
Height	$\frac{2\sqrt{30}}{15} \approx 0.73030$
Central density	1
Number of external pieces	16
Level of complexity	1
Related polytopes
Army	Hexadecatedakon
Regiment	Hexadecatedakon
Dual	Hexadecatedakon
Conjugate	None
Abstract & topological properties
Flag count	20922789888000
Euler characteristic	2
Orientable	Yes
Properties
Symmetry	A15, order 20922789888000
Convex	Yes
Nature	Tame

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√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:

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

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:

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