# Hendecaxennon

Hendecaxennon Rank10
TypeRegular
SpaceSpherical
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$\frac{\sqrt{55}}{11} ≈ 0.67420$ Inradius$\frac{\sqrt{55}}{110} ≈ 0.067420$ Hypervolume$\frac{\sqrt{11}}{116121600} ≈ 2.8562×10^{-8}$ Dixennal angle$\arccos\left(\frac{1}{10}\right) ≈ 84.26083°$ Height$\frac{\sqrt{55}}{10} ≈ 0.74162$ Central density1
Number of external pieces11
Level of complexity1
Related polytopes
ArmyUx
RegimentUx
DualHendecaxennon
ConjugateNone
Abstract & topological properties
Euler characteristic0
OrientableYes
Properties
SymmetryA10, order 39916800
ConvexYes
NatureTame

The hendecaxennon, or ux, also commonly called the 10-simplex, is the simplest possible non-degenerate polyxennon. The full symmetry version has 11 regular decayotta as facets, joining 3 to an octaexon peak and 10 to a vertex, and is one of the 3 regular polyxenna. 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:

• $\left(±\frac{1}{2},\,-\frac{\sqrt{3}}{6},\,-\frac{\sqrt{6}}{12},\,-\frac{\sqrt{10}}{20},\,-\frac{\sqrt{15}}{30},\,-\frac{\sqrt{21}}{42},\,-\frac{\sqrt7}{28},\,-\frac{1}{12},\,-\frac{\sqrt5}{30},\,-\frac{\sqrt{55}}{110}\right),$ • $\left(0,\,\frac{\sqrt{3}}{3},\,-\frac{\sqrt{6}}{12},\,-\frac{\sqrt{10}}{20},\,-\frac{\sqrt{15}}{30},\,-\frac{\sqrt{21}}{42},\,-\frac{\sqrt7}{28},\,-\frac{1}{12},\,-\frac{\sqrt5}{30},\,-\frac{\sqrt{55}}{110}\right),$ • $\left(0,\,0,\,\frac{\sqrt{6}}{4},\,-\frac{\sqrt{10}}{20},\,-\frac{\sqrt{15}}{30},\,-\frac{\sqrt{21}}{42},\,-\frac{\sqrt7}{28},\,-\frac{1}{12},\,-\frac{\sqrt5}{30},\,-\frac{\sqrt{55}}{110}\right),$ • $\left(0,\,0,\,0,\,\frac{\sqrt{10}}{5},\,-\frac{\sqrt{15}}{30},\,-\frac{\sqrt{21}}{42},\,-\frac{\sqrt7}{28},\,-\frac{1}{12},\,-\frac{\sqrt5}{30},\,-\frac{\sqrt{55}}{110}\right),$ • $\left(0,\,0,\,0,\,0,\,\frac{\sqrt{15}}{6},\,-\frac{\sqrt{21}}{42},\,-\frac{\sqrt7}{28},\,-\frac{1}{12},\,-\frac{\sqrt5}{30},\,-\frac{\sqrt{55}}{110}\right),$ • $\left(0,\,0,\,0,\,0,\,0,\,\frac{\sqrt{21}}{7},\,-\frac{\sqrt7}{28},\,-\frac{1}{12},\,-\frac{\sqrt5}{30},\,-\frac{\sqrt{55}}{110}\right),$ • $\left(0,\,0,\,0,\,0,\,0,\,0,\,\frac{\sqrt7}{4},\,-\frac{1}{12},\,-\frac{\sqrt5}{30},\,-\frac{\sqrt{55}}{110}\right),$ • $\left(0,\,0,\,0,\,0,\,0,\,0,\,0,\,\frac23,\,-\frac{\sqrt5}{30},\,-\frac{\sqrt{55}}{110}\right),$ • $\left(0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,\frac{3\sqrt5}{10},\,-\frac{\sqrt{55}}{110}\right),$ • $\left(0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,0,\,\frac{\sqrt{55}}{11}\right).$ Much simpler coordinates can be given in 11 dimensions, as all permutations of:

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