# 16-simplex

16-simplex
Rank16
TypeRegular
Notation
Coxeter diagramx3o3o3o3o3o3o3o3o3o3o3o3o3o3o3o ()
Schläfli symbol{3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
Elements
Pedaka17 15-simplices
Tedaka136 14-simplices
Doka2380 12-simplices
Henda6188 11-simplices
Daka12376 10-simplices
Xenna19448 9-simplices
Yotta24310 8-simplices
Zetta24310 7-simplices
Exa19448 6-simplices
Peta12376 hexatera
Tera6188 pentachora
Cells2380 tetrahedra
Faces680 triangles
Edges136
Vertices17
Vertex figure15-simplex, edge length 1
Measures (edge length 1)
Circumradius${\displaystyle {\frac {2{\sqrt {34}}}{17}}\approx 0.68599}$
Inradius${\displaystyle {\frac {\sqrt {34}}{136}}\approx 0.042875}$
Hypervolume${\displaystyle {\frac {\sqrt {17}}{5356234211328000}}\approx 7.6978\times 10^{-16}}$
Dihedral angle${\displaystyle \arccos \left({\frac {1}{16}}\right)\approx 86.41668^{\circ }}$
Height${\displaystyle {\frac {\sqrt {34}}{8}}\approx 0.72887}$
Central density1
Number of external pieces17
Level of complexity1
Related polytopes
Army*
Regiment*
Dual16-simplex
ConjugateNone
Abstract & topological properties
Flag count355687428096000
Euler characteristic0
OrientableYes
Properties
SymmetryA16, order 355687428096000
ConvexYes
NatureTame

The 16-simplex (also called the heptadecapedakon) is the simplest possible non-degenerate 16-polytope. The full symmetry version has 17 regular 15-simplices as facets, joining 3 to a peak and 16 to a vertex, and is regular.

## Vertex coordinates

The vertices of a regular 16-simplex of edge length 1, centered at the origin, are given by:

• ${\displaystyle \left(\pm {\frac {1}{2}},\,-{\frac {\sqrt {3}}{6}},\,-{\frac {\sqrt {6}}{12}},\,-{\frac {\sqrt {10}}{20}},\,-{\frac {\sqrt {15}}{30}},\,-{\frac {\sqrt {21}}{42}},\,-{\frac {\sqrt {7}}{28}},\,-{\frac {1}{12}},\,-{\frac {\sqrt {5}}{30}},\,-{\frac {\sqrt {55}}{110}},\,-{\frac {\sqrt {66}}{132}},\,-{\frac {\sqrt {78}}{156}},\,-{\frac {\sqrt {91}}{182}},\,-{\frac {\sqrt {105}}{210}},\,-{\frac {\sqrt {30}}{120}},\,-{\frac {\sqrt {34}}{136}}\right)}$,
• ${\displaystyle \left(0,\,{\frac {\sqrt {3}}{3}},\,-{\frac {\sqrt {6}}{12}},\,-{\frac {\sqrt {10}}{20}},\,-{\frac {\sqrt {15}}{30}},\,-{\frac {\sqrt {21}}{42}},\,-{\frac {\sqrt {7}}{28}},\,-{\frac {1}{12}},\,-{\frac {\sqrt {5}}{30}},\,-{\frac {\sqrt {55}}{110}},\,-{\frac {\sqrt {66}}{132}},\,-{\frac {\sqrt {78}}{156}},\,-{\frac {\sqrt {91}}{182}},\,-{\frac {\sqrt {105}}{210}},\,-{\frac {\sqrt {30}}{120}},\,-{\frac {\sqrt {34}}{136}}\right)}$,
• ${\displaystyle \left(0,\,0,\,{\frac {\sqrt {6}}{4}},\,-{\frac {\sqrt {10}}{20}},\,-{\frac {\sqrt {15}}{30}},\,-{\frac {\sqrt {21}}{42}},\,-{\frac {\sqrt {7}}{28}},\,-{\frac {1}{12}},\,-{\frac {\sqrt {5}}{30}},\,-{\frac {\sqrt {55}}{110}},\,-{\frac {\sqrt {66}}{132}},\,-{\frac {\sqrt {78}}{156}},\,-{\frac {\sqrt {91}}{182}},\,-{\frac {\sqrt {105}}{210}},\,-{\frac {\sqrt {30}}{120}},\,-{\frac {\sqrt {34}}{136}}\right)}$,
• ${\displaystyle \left(0,\,0,\,0,\,{\frac {\sqrt {10}}{5}},\,-{\frac {\sqrt {15}}{30}},\,-{\frac {\sqrt {21}}{42}},\,-{\frac {\sqrt {7}}{28}},\,-{\frac {1}{12}},\,-{\frac {\sqrt {5}}{30}},\,-{\frac {\sqrt {55}}{110}},\,-{\frac {\sqrt {66}}{132}},\,-{\frac {\sqrt {78}}{156}},\,-{\frac {\sqrt {91}}{182}},\,-{\frac {\sqrt {105}}{210}},\,-{\frac {\sqrt {30}}{120}},\,-{\frac {\sqrt {34}}{136}}\right)}$,
• ${\displaystyle \left(0,\,0,\,0,\,0,\,{\frac {\sqrt {15}}{6}},\,-{\frac {\sqrt {21}}{42}},\,-{\frac {\sqrt {7}}{28}},\,-{\frac {1}{12}},\,-{\frac {\sqrt {5}}{30}},\,-{\frac {\sqrt {55}}{110}},\,-{\frac {\sqrt {66}}{132}},\,-{\frac {\sqrt {78}}{156}},\,-{\frac {\sqrt {91}}{182}},\,-{\frac {\sqrt {105}}{210}},\,-{\frac {\sqrt {30}}{120}},\,-{\frac {\sqrt {34}}{136}}\right)}$,
• ${\displaystyle \left(0,\,0,\,0,\,0,\,0,\,{\frac {\sqrt {21}}{7}},\,-{\frac {\sqrt {7}}{28}},\,-{\frac {1}{12}},\,-{\frac {\sqrt {5}}{30}},\,-{\frac {\sqrt {55}}{110}},\,-{\frac {\sqrt {66}}{132}},\,-{\frac {\sqrt {78}}{156}},\,-{\frac {\sqrt {91}}{182}},\,-{\frac {\sqrt {105}}{210}},\,-{\frac {\sqrt {30}}{120}},\,-{\frac {\sqrt {34}}{136}}\right)}$,
• ${\displaystyle \left(0,\,0,\,0,\,0,\,0,\,0,\,{\frac {\sqrt {7}}{4}},\,-{\frac {1}{12}},\,-{\frac {\sqrt {5}}{30}},\,-{\frac {\sqrt {55}}{110}},\,-{\frac {\sqrt {66}}{132}},\,-{\frac {\sqrt {78}}{156}},\,-{\frac {\sqrt {91}}{182}},\,-{\frac {\sqrt {105}}{210}},\,-{\frac {\sqrt {30}}{120}},\,-{\frac {\sqrt {34}}{136}}\right)}$,
• ${\displaystyle \left(0,\,0,\,0,\,0,\,0,\,0,\,0,\,{\frac {2}{3}},\,-{\frac {\sqrt {5}}{30}},\,-{\frac {\sqrt {55}}{110}},\,-{\frac {\sqrt {66}}{132}},\,-{\frac {\sqrt {78}}{156}},\,-{\frac {\sqrt {91}}{182}},\,-{\frac {\sqrt {105}}{210}},\,-{\frac {\sqrt {30}}{120}},\,-{\frac {\sqrt {34}}{136}}\right)}$,
• ${\displaystyle \left(0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,{\frac {3{\sqrt {5}}}{10}},\,-{\frac {\sqrt {55}}{110}},\,-{\frac {\sqrt {66}}{132}},\,-{\frac {\sqrt {78}}{156}},\,-{\frac {\sqrt {91}}{182}},\,-{\frac {\sqrt {105}}{210}},\,-{\frac {\sqrt {30}}{120}},\,-{\frac {\sqrt {34}}{136}}\right)}$,
• ${\displaystyle \left(0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,{\frac {\sqrt {55}}{11}},\,-{\frac {\sqrt {66}}{132}},\,-{\frac {\sqrt {78}}{156}},\,-{\frac {\sqrt {91}}{182}},\,-{\frac {\sqrt {105}}{210}},\,-{\frac {\sqrt {30}}{120}},\,-{\frac {\sqrt {34}}{136}}\right)}$,
• ${\displaystyle \left(0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,{\frac {\sqrt {66}}{12}},\,-{\frac {\sqrt {78}}{156}},\,-{\frac {\sqrt {91}}{182}},\,-{\frac {\sqrt {105}}{210}},\,-{\frac {\sqrt {30}}{120}},\,-{\frac {\sqrt {34}}{136}}\right)}$,
• ${\displaystyle \left(0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,{\frac {\sqrt {78}}{13}},\,-{\frac {\sqrt {91}}{182}},\,-{\frac {\sqrt {105}}{210}},\,-{\frac {\sqrt {30}}{120}},\,-{\frac {\sqrt {34}}{136}}\right)}$,
• ${\displaystyle \left(0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,{\frac {\sqrt {91}}{14}},\,-{\frac {\sqrt {105}}{210}},\,-{\frac {\sqrt {30}}{120}},\,-{\frac {\sqrt {34}}{136}}\right)}$,
• ${\displaystyle \left(0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,{\frac {\sqrt {105}}{15}},\,-{\frac {\sqrt {30}}{120}},\,-{\frac {\sqrt {34}}{136}}\right)}$.
• ${\displaystyle \left(0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,{\frac {\sqrt {30}}{8}},\,-{\frac {\sqrt {34}}{136}}\right)}$,
• ${\displaystyle \left(0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,{\frac {2{\sqrt {34}}}{17}}\right)}$.

Much simpler coordinates can be given in 17 dimensions, as all permutations of:

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