# 10-simplex

10-simplex
Rank10
TypeRegular
Notation
Bowers style acronymUx
Coxeter diagramx3o3o3o3o3o3o3o3o3o ()
Schläfli symbol{3,3,3,3,3,3,3,3,3}
Tapertopic notation19
Elements
Xenna11 decayotta
Yotta55 enneazetta
Zetta165 octaexa
Exa330 heptapeta
Peta462 hexatera
Tera462 pentachora
Cells330 tetrahedra
Faces165 triangles
Edges55
Vertices11
Vertex figureDecayotton, edge length 1
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {55}}{11}}\approx 0.67420}$
Inradius${\displaystyle {\frac {\sqrt {55}}{110}}\approx 0.067420}$
Hypervolume${\displaystyle {\frac {\sqrt {11}}{116121600}}\approx 2.8562\times 10^{-8}}$
Dixennal angle${\displaystyle \arccos \left({\frac {1}{10}}\right)\approx 84.26083^{\circ }}$
Height${\displaystyle {\frac {\sqrt {55}}{10}}\approx 0.74162}$
Central density1
Number of external pieces11
Level of complexity1
Related polytopes
ArmyUx
RegimentUx
Dual10-simplex
ConjugateNone
Abstract & topological properties
Flag count39916800
Euler characteristic0
OrientableYes
SkeletonK11
Properties
SymmetryA10, order 39916800
Flag orbits1
ConvexYes
NatureTame

The 10-simplex, also called the hendecaxennon, is the simplest possible non-degenerate 10-polytope. The full symmetry version has 11 regular 9-simplices as facets, joining 3 to an 7-simplex peak and 10 to a vertex, and is one of the 3 regular 1-polytopes. It is the 10-dimensional simplex.

## Vertex coordinates

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

• ${\displaystyle \left(\pm {\frac {1}{2}},\,-{\frac {\sqrt {3}}{6}},\,-{\frac {\sqrt {6}}{12}},\,-{\frac {\sqrt {10}}{20}},\,-{\frac {\sqrt {15}}{30}},\,-{\frac {\sqrt {21}}{42}},\,-{\frac {\sqrt {7}}{28}},\,-{\frac {1}{12}},\,-{\frac {\sqrt {5}}{30}},\,-{\frac {\sqrt {55}}{110}}\right)}$,
• ${\displaystyle \left(0,\,{\frac {\sqrt {3}}{3}},\,-{\frac {\sqrt {6}}{12}},\,-{\frac {\sqrt {10}}{20}},\,-{\frac {\sqrt {15}}{30}},\,-{\frac {\sqrt {21}}{42}},\,-{\frac {\sqrt {7}}{28}},\,-{\frac {1}{12}},\,-{\frac {\sqrt {5}}{30}},\,-{\frac {\sqrt {55}}{110}}\right)}$,
• ${\displaystyle \left(0,\,0,\,{\frac {\sqrt {6}}{4}},\,-{\frac {\sqrt {10}}{20}},\,-{\frac {\sqrt {15}}{30}},\,-{\frac {\sqrt {21}}{42}},\,-{\frac {\sqrt {7}}{28}},\,-{\frac {1}{12}},\,-{\frac {\sqrt {5}}{30}},\,-{\frac {\sqrt {55}}{110}}\right)}$,
• ${\displaystyle \left(0,\,0,\,0,\,{\frac {\sqrt {10}}{5}},\,-{\frac {\sqrt {15}}{30}},\,-{\frac {\sqrt {21}}{42}},\,-{\frac {\sqrt {7}}{28}},\,-{\frac {1}{12}},\,-{\frac {\sqrt {5}}{30}},\,-{\frac {\sqrt {55}}{110}}\right)}$,
• ${\displaystyle \left(0,\,0,\,0,\,0,\,{\frac {\sqrt {15}}{6}},\,-{\frac {\sqrt {21}}{42}},\,-{\frac {\sqrt {7}}{28}},\,-{\frac {1}{12}},\,-{\frac {\sqrt {5}}{30}},\,-{\frac {\sqrt {55}}{110}}\right)}$,
• ${\displaystyle \left(0,\,0,\,0,\,0,\,0,\,{\frac {\sqrt {21}}{7}},\,-{\frac {\sqrt {7}}{28}},\,-{\frac {1}{12}},\,-{\frac {\sqrt {5}}{30}},\,-{\frac {\sqrt {55}}{110}}\right)}$,
• ${\displaystyle \left(0,\,0,\,0,\,0,\,0,\,0,\,{\frac {\sqrt {7}}{4}},\,-{\frac {1}{12}},\,-{\frac {\sqrt {5}}{30}},\,-{\frac {\sqrt {55}}{110}}\right)}$,
• ${\displaystyle \left(0,\,0,\,0,\,0,\,0,\,0,\,0,\,{\frac {2}{3}},\,-{\frac {\sqrt {5}}{30}},\,-{\frac {\sqrt {55}}{110}}\right)}$,
• ${\displaystyle \left(0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,{\frac {3{\sqrt {5}}}{10}},\,-{\frac {\sqrt {55}}{110}}\right)}$,
• ${\displaystyle \left(0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,{\frac {\sqrt {55}}{11}}\right)}$.

Much simpler coordinates can be given in 11 dimensions, as all permutations of:

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