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).

## References

• Coxeter, H.S.M.:
• — (1973). "Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)". Regular Polytopes (3rd ed.). Dover. pp. 296. ISBN 0-486-61480-8.
• Sherk, F. Arthur; McMullen, Peter; Thompson, Anthony C.; Weiss, Asia Ivic, eds. (1995). Kaleidoscopes: Selected Writings of H.S.M. Coxeter. Wiley. ISBN 978-0-471-01003-6.
• Conway, John H.; Burgiel, Heidi; Goodman-Strass, Chaim (2008). "26. Hemicubes: 1n1". The Symmetries of Things. p. 409. ISBN 978-1-56881-220-5.
• Johnson, Norman (1991). "Uniform Polytopes" (Manuscript). ((cite journal)): Cite journal requires |journal= (help)
• Klitzing, Richard. "10D uniform polytopes (polyxenna) x3o3o3o3o3o3o3o3o3o — ux".
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform polychoron Pentachoron 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds