# 15-simplex

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.

15-simplex
Rank15
TypeRegular
Notation
Coxeter diagramx3o3o3o3o3o3o3o3o3o3o3o3o3o3o ()
Schläfli symbol{3,3,3,3,3,3,3,3,3,3,3,3,3,3}
Elements
Tedaka16 14-simplices
Doka560 12-simplices
Henda1820 11-simplices
Daka4368 10-simplices
Xenna8008 9-simplices
Yotta11440 8-simplices
Zetta12870 7-simplices
Exa11440 6-simplices
Peta8008 hexatera
Tera4368 pentachora
Cells1820 tetrahedra
Faces560 triangles
Edges120
Vertices16
Vertex figure14-simplex, edge length 1
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {30}}{8}}\approx 0.68465}$
Inradius${\displaystyle {\frac {\sqrt {30}}{120}}\approx 0.045644}$
Hypervolume${\displaystyle {\frac {\sqrt {2}}{83691159552000}}\approx 1.6898\times 10^{-14}}$
Dihedral angle${\displaystyle \arccos \left({\frac {1}{15}}\right)\approx 86.17745^{\circ }}$
Height${\displaystyle {\frac {2{\sqrt {30}}}{15}}\approx 0.73030}$
Central density1
Number of external pieces16
Level of complexity1
Related polytopes
Army15-simplex
Regiment15-simplex
Dual15-simplex
ConjugateNone
Abstract & topological properties
Flag count20922789888000
Euler characteristic2
OrientableYes
Properties
SymmetryA15, order 20922789888000
ConvexYes
NatureTame

A regular hexadecatedakon of edge length 22 can be inscribed in the unit 15-cube.[1]

## Vertex coordinates

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

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

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

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

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

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

## References

1. Sloane, N. J. A. (ed.). "Sequence A019442". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.