15-simplex

15-simplex
Rank	15
Type	Regular
Notation
Coxeter diagram	x3o3o3o3o3o3o3o3o3o3o3o3o3o3o ()
Schläfli symbol	{3,3,3,3,3,3,3,3,3,3,3,3,3,3}
Elements
Tedaka	16 14-simplices
Tradaka	120 13-simplices
Doka	560 12-simplices
Henda	1820 11-simplices
Daka	4368 10-simplices
Xenna	8008 9-simplices
Yotta	11440 8-simplices
Zetta	12870 7-simplices
Exa	11440 6-simplices
Peta	8008 hexatera
Tera	4368 pentachora
Cells	1820 tetrahedra
Faces	560 triangles
Edges	120
Vertices	16
Vertex figure	14-simplex, edge length 1
Measures (edge length 1)
Circumradius	${\frac {\sqrt {30}}{8}}\approx 0.68465$
Inradius	${\frac {\sqrt {30}}{120}}\approx 0.045644$
Hypervolume	${\frac {\sqrt {2}}{83691159552000}}\approx 1.6898\times 10^{-14}$
Dihedral angle	$\arccos \left({\frac {1}{15}}\right)\approx 86.17745^{\circ }$
Height	${\frac {2{\sqrt {30}}}{15}}\approx 0.73030$
Central density	1
Number of external pieces	16
Level of complexity	1
Related polytopes
Army	15-simplex
Regiment	15-simplex
Dual	15-simplex
Conjugate	None
Abstract & topological properties
Flag count	20922789888000
Euler characteristic	2
Orientable	Yes
Properties
Symmetry	A15, order 20922789888000
Convex	Yes
Nature	Tame

The 15-simplex (also called the hexadecatedakon) is the simplest possible non-degenerate 15-polytope. The full symmetry version has 16 regular 14-simplices as facets, joining 3 to a facet and 15 to a vertex, and is regular.

A regular hexadecatedakon of edge length 2√2 can be inscribed in the unit 15-cube.

Vertex coordinates

The vertices of a regular 15-simplex of edge length 1, centered at the origin, are given by:

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

Simpler sets of coordinates can be found by inscribing the 15-simplex into the 15-cube. One such set is given by:

• $\left({\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}}\right)$,
• $\left({\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}}\right)$,
• $\left({\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}}\right)$,
• $\left({\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}}\right)$,
• $\left({\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}}\right)$,
• $\left({\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}}\right)$,
• $\left({\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}}\right)$,
• $\left({\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}}\right)$,
• $\left(-{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}}\right)$,
• $\left(-{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}}\right)$,
• $\left(-{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}}\right)$,
• $\left(-{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}}\right)$,
• $\left(-{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}}\right)$,
• $\left(-{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}}\right)$,
• $\left(-{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}}\right)$,
• $\left(-{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,-{\frac {\sqrt {2}}{8}},\,{\frac {\sqrt {2}}{8}}\right)$.

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

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