| bgcolor=#e7dcc3 colspan=2 | Regular hendecaxennon (10-simplex) | |
|---|---|---|
| bgcolor=#ffffff align=center colspan=2 | Orthogonal projection inside Petrie polygon | |
| Type | Regular 10-polytope | |
| Family | simplex | |
| Schläfli symbol | ||
| 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 polytope | Self-dual |
| Properties | convex |
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:
\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}, \pm1\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.
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).