bgcolor=#e7dcc3 colspan=2 | Regular decayotton (9-simplex) | |
---|---|---|
bgcolor=#ffffff align=center colspan=2 | Orthogonal projection inside Petrie polygon | |
Type | Regular 9-polytope | |
Family | simplex | |
Schläfli symbol | ||
Coxeter-Dynkin diagram | ||
8-faces | 10 8-simplex | |
7-faces | 45 7-simplex | |
6-faces | 120 6-simplex | |
5-faces | 210 5-simplex | |
4-faces | 252 5-cell | |
Cells | 210 tetrahedron | |
Faces | 120 triangle | |
Edges | 45 | |
Vertices | 10 | |
Vertex figure | 8-simplex | |
Petrie polygon | decagon | |
Coxeter group | A9 [3,3,3,3,3,3,3,3] | |
Dual | Self-dual | |
Properties | convex |
It can also be called a decayotton, or deca-9-tope, as a 10-facetted polytope in 9-dimensions.. The name decayotton is derived from deca for ten facets in Greek and yotta (a variation of "oct" for eight), having 8-dimensional facets, and -on.
The Cartesian coordinates of the vertices of an origin-centered regular decayotton having edge length 2 are:
\left(\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{1/45}, 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{1/45}, 1/6, \sqrt{1/28}, \sqrt{1/21}, \sqrt{1/15}, \sqrt{1/10}, -\sqrt{3/2}, 0, 0\right)
\left(\sqrt{1/45}, 1/6, \sqrt{1/28}, \sqrt{1/21}, \sqrt{1/15}, -2\sqrt{2/5}, 0, 0, 0\right)
\left(\sqrt{1/45}, 1/6, \sqrt{1/28}, \sqrt{1/21}, -\sqrt{5/3}, 0, 0, 0, 0\right)
\left(\sqrt{1/45}, 1/6, \sqrt{1/28}, -\sqrt{12/7}, 0, 0, 0, 0, 0\right)
\left(\sqrt{1/45}, 1/6, -\sqrt{7/4}, 0, 0, 0, 0, 0, 0\right)
\left(\sqrt{1/45}, -4/3, 0, 0, 0, 0, 0, 0, 0\right)
\left(-3\sqrt{1/5}, 0, 0, 0, 0, 0, 0, 0, 0\right)
More simply, the vertices of the 9-simplex can be positioned in 10-space as permutations of (0,0,0,0,0,0,0,0,0,1). These are the vertices of one Facet of the 10-orthoplex.