Regular hendecaxennon (10-simplex) | |
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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.
The Cartesian coordinates of the vertices of an origin-centered regular 10-simplex having edge length 2 are:
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.
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] |
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).