Regular hendecaxennon (10-simplex)
Orthogonal projection inside Petrie polygon
Type	Regular 10-polytope
Family	simplex
Schläfli symbol	{3,3,3,3,3,3,3,3,3}
Coxeter-Dynkin diagram
9-faces	11 9-simplex
8-faces	55 8-simplex
7-faces	165 7-simplex
6-faces	330 6-simplex
5-faces	462 5-simplex
4-faces	462 5-cell
Cells	330 tetrahedron
Faces	165 triangle
Edges	55
Vertices	11
Vertex figure	9-simplex
Petrie polygon	hendecagon
Coxeter group	A10 [3,3,3,3,3,3,3,3,3]
Dual	Self-dual
Properties	convex

In geometry, a 10-simplex is a self-dual regular 10-polytope. It has 11 vertices, 55 edges, 165 triangle faces, 330 tetrahedral cells, 462 5-cell 4-faces, 462 5-simplex 5-faces, 330 6-simplex 6-faces, 165 7-simplex 7-faces, 55 8-simplex 8-faces, and 11 9-simplex 9-faces. Its dihedral angle is cos⁻¹(1/10), or approximately 84.26°.

It can also be called a hendecaxennon, or hendeca-10-tope, as an 11-facetted polytope in 10-dimensions. The name hendecaxennon is derived from hendeca for 11 facets in Greek and -xenn (variation of ennea for nine), having 9-dimensional facets, and -on.

Coordinates

The Cartesian coordinates of the vertices of an origin-centered regular 10-simplex having edge length 2 are:

${\displaystyle \left({\sqrt {1/55)),\ {\sqrt {1/45)),\ 1/6,\ {\sqrt {1/28)),\ {\sqrt {1/21)),\ {\sqrt {1/15)),\ {\sqrt {1/10)),\ {\sqrt {1/6)),\ {\sqrt {1/3)),\ \pm 1\right)}$

${\displaystyle \left({\sqrt {1/55)),\ {\sqrt {1/45)),\ 1/6,\ {\sqrt {1/28)),\ {\sqrt {1/21)),\ {\sqrt {1/15)),\ {\sqrt {1/10)),\ {\sqrt {1/6)),\ -2{\sqrt {1/3)),\ 0\right)}$

${\displaystyle \left({\sqrt {1/55)),\ {\sqrt {1/45)),\ 1/6,\ {\sqrt {1/28)),\ {\sqrt {1/21)),\ {\sqrt {1/15)),\ {\sqrt {1/10)),\ -{\sqrt {3/2)),\ 0,\ 0\right)}$

${\displaystyle \left({\sqrt {1/55)),\ {\sqrt {1/45)),\ 1/6,\ {\sqrt {1/28)),\ {\sqrt {1/21)),\ {\sqrt {1/15)),\ -2{\sqrt {2/5)),\ 0,\ 0,\ 0\right)}$

${\displaystyle \left({\sqrt {1/55)),\ {\sqrt {1/45)),\ 1/6,\ {\sqrt {1/28)),\ {\sqrt {1/21)),\ -{\sqrt {5/3)),\ 0,\ 0,\ 0,\ 0\right)}$

${\displaystyle \left({\sqrt {1/55)),\ {\sqrt {1/45)),\ 1/6,\ {\sqrt {1/28)),\ -{\sqrt {12/7)),\ 0,\ 0,\ 0,\ 0,\ 0\right)}$

${\displaystyle \left({\sqrt {1/55)),\ {\sqrt {1/45)),\ 1/6,\ -{\sqrt {7/4)),\ 0,\ 0,\ 0,\ 0,\ 0,\ 0\right)}$

${\displaystyle \left({\sqrt {1/55)),\ {\sqrt {1/45)),\ -4/3,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0\right)}$

${\displaystyle \left({\sqrt {1/55)),\ -3{\sqrt {1/5)),\ 0,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0\right)}$

${\displaystyle \left(-{\sqrt {20/11)),\ 0,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0\right)}$

More simply, the vertices of the 10-simplex can be positioned in 11-space as permutations of (0,0,0,0,0,0,0,0,0,0,1). This construction is based on facets of the 11-orthoplex.

Images

orthographic projections

Ak Coxeter plane	A10	A9	A8
Graph
Dihedral symmetry	[11]	[10]	[9]
Ak Coxeter plane	A7	A6	A5
Graph
Dihedral symmetry	[8]	[7]	[6]
Ak Coxeter plane	A4	A3	A2
Graph
Dihedral symmetry	[5]	[4]	[3]

Related polytopes

The 2-skeleton of the 10-simplex is topologically related to the 11-cell abstract regular polychoron which has the same 11 vertices, 55 edges, but only 1/3 the faces (55).