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10-simplex

Convex regular 10-polytope

Wikipedia contributorsAdapted for Fathomly · View authors
1 min read
Source updated June 4, 2026
Image credit is listed at the end of this article.
Regular hendecaxennon
(10-simplex)

Orthogonal projection
inside Petrie polygon
TypeRegular 10-polytope
Familysimplex
Schläfli symbol{3,3,3,3,3,3,3,3,3}
Coxeter-Dynkin
diagram
9-faces11 9-simplex
8-faces55 8-simplex
7-faces165 7-simplex
6-faces330 6-simplex
5-faces462 5-simplex
4-faces462 5-cell
Cells330 tetrahedron
Faces165 triangle
Edges55
Vertices11
Vertex figure9-simplex
Petrie polygonhendecagon
Coxeter groupA10 [3,3,3,3,3,3,3,3,3]
DualSelf-dual
Propertiesconvex

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. Acronym: ux

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.

01Coordinates

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

\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)
\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)
\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)
\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)
\left({\sqrt {1/55}},\ {\sqrt {1/45}},\ 1/6,\ {\sqrt {1/28}},\ {\sqrt {1/21}},\ -{\sqrt {5/3}},\ 0,\ 0,\ 0,\ 0\right)
\left({\sqrt {1/55}},\ {\sqrt {1/45}},\ 1/6,\ {\sqrt {1/28}},\ -{\sqrt {12/7}},\ 0,\ 0,\ 0,\ 0,\ 0\right)
\left({\sqrt {1/55}},\ {\sqrt {1/45}},\ 1/6,\ -{\sqrt {7/4}},\ 0,\ 0,\ 0,\ 0,\ 0,\ 0\right)
\left({\sqrt {1/55}},\ {\sqrt {1/45}},\ -4/3,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0\right)
\left({\sqrt {1/55}},\ -3{\sqrt {1/5}},\ 0,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0\right)
\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.

02Images

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]
Watch videos about 10-simplexExplainers and documentaries on YouTube (opens in a new tab)

Sources and credits

This article is adapted from the Wikipedia article “10-simplex”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

Images, from Wikimedia Commons:

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