There are 10 rectifications of the 10-orthoplex. Vertices of the rectified 10-orthoplex are located at the edge-centers of the 9-orthoplex. Vertices of the birectified 10-orthoplex are located in the triangular face centers of the 10-orthoplex. Vertices of the trirectified 10-orthoplex are located in the tetrahedral cell centers of the 10-orthoplex.
These polytopes are part of a family 1023 uniform 10-polytopes with BC10 symmetry.
| bgcolor=#e7dcc3 colspan=2 | Rectified 10-orthoplex | |
|---|---|---|
| Type | uniform 10-polytope | |
| Schläfli symbol | t1 | |
| Coxeter-Dynkin diagrams | ||
| 7-faces | ||
| 6-faces | ||
| 5-faces | ||
| 4-faces | ||
| Cells | ||
| Faces | ||
| Edges | 2880 | |
| Vertices | 180 | |
| Vertex figure | 8-orthoplex prism | |
| Petrie polygon | icosagon | |
| Coxeter groups | C10, [4,3<sup>8</sup>] D10, [3<sup>7,1,1</sup>] | |
| Properties | convex |
The rectified 10-orthoplex is the vertex figure for the demidekeractic honeycomb.
or
There are two Coxeter groups associated with the rectified 10-orthoplex, one with the C10 or [4,3<sup>8</sup>] Coxeter group, and a lower symmetry with two copies of 9-orthoplex facets, alternating, with the D10 or [3<sup>7,1,1</sup>] Coxeter group.
Cartesian coordinates for the vertices of a rectified 10-orthoplex, centered at the origin, edge length
\sqrt{2}
(±1,±1,0,0,0,0,0,0,0,0)
Its 180 vertices represent the root vectors of the simple Lie group D10. The vertices can be seen in 3 hyperplanes, with the 45 vertices rectified 9-simplices facets on opposite sides, and 90 vertices of an expanded 9-simplex passing through the center. When combined with the 20 vertices of the 9-orthoplex, these vertices represent the 200 root vectors of the simple Lie group B10.
| bgcolor=#e7dcc3 colspan=2 | Birectified 10-orthoplex | |
|---|---|---|
| Type | uniform 10-polytope | |
| Schläfli symbol | t2 | |
| Coxeter-Dynkin diagrams | ||
| 7-faces | ||
| 6-faces | ||
| 5-faces | ||
| 4-faces | ||
| Cells | ||
| Faces | ||
| Edges | ||
| Vertices | ||
| Vertex figure | ||
| Coxeter groups | C10, [4,3<sup>8</sup>] D10, [3<sup>7,1,1</sup>] | |
| Properties | convex |
Cartesian coordinates for the vertices of a birectified 10-orthoplex, centered at the origin, edge length
\sqrt{2}
(±1,±1,±1,0,0,0,0,0,0,0)
| bgcolor=#e7dcc3 colspan=2 | Trirectified 10-orthoplex | |
|---|---|---|
| Type | uniform 10-polytope | |
| Schläfli symbol | t3 | |
| Coxeter-Dynkin diagrams | ||
| 7-faces | ||
| 6-faces | ||
| 5-faces | ||
| 4-faces | ||
| Cells | ||
| Faces | ||
| Edges | ||
| Vertices | ||
| Vertex figure | ||
| Coxeter groups | C10, [4,3<sup>8</sup>] D10, [3<sup>7,1,1</sup>] | |
| Properties | convex |
Cartesian coordinates for the vertices of a trirectified 10-orthoplex, centered at the origin, edge length
\sqrt{2}
(±1,±1,±1,±1,0,0,0,0,0,0)
| bgcolor=#e7dcc3 colspan=2 | Quadrirectified 10-orthoplex | |
|---|---|---|
| Type | uniform 10-polytope | |
| Schläfli symbol | t4 | |
| Coxeter-Dynkin diagrams | ||
| 7-faces | ||
| 6-faces | ||
| 5-faces | ||
| 4-faces | ||
| Cells | ||
| Faces | ||
| Edges | ||
| Vertices | ||
| Vertex figure | ||
| Coxeter groups | C10, [4,3<sup>8</sup>] D10, [3<sup>7,1,1</sup>] | |
| Properties | convex |
Cartesian coordinates for the vertices of a quadrirectified 10-orthoplex, centered at the origin, edge length
\sqrt{2}
(±1,±1,±1,±1,±1,0,0,0,0,0)