There are unique 7 degrees of rectifications, the zeroth being the 7-orthoplex, and the 6th and last being the 7-cube. Vertices of the rectified 7-orthoplex are located at the edge-centers of the 7-orthoplex. Vertices of the birectified 7-orthoplex are located in the triangular face centers of the 7-orthoplex. Vertices of the trirectified 7-orthoplex are located in the tetrahedral cell centers of the 7-orthoplex.
| bgcolor=#e7dcc3 colspan=2 | Rectified 7-orthoplex | |
|---|---|---|
| Type | uniform 7-polytope | |
| Schläfli symbol | r | |
| Coxeter-Dynkin diagrams | ||
| 6-faces | 142 | |
| 5-faces | 1344 | |
| 4-faces | 3360 | |
| Cells | 3920 | |
| Faces | 2520 | |
| Edges | 840 | |
| Vertices | 84 | |
| Vertex figure | 5-orthoplex prism | |
| Coxeter groups | B7, [3,3,3,3,3,4] D7, [3<sup>4,1,1</sup>] | |
| Properties | convex |
The rectified 7-orthoplex is the vertex figure for the demihepteractic honeycomb. The rectified 7-orthoplex's 84 vertices represent the kissing number of a sphere-packing constructed from this honeycomb.
or
There are two Coxeter groups associated with the rectified heptacross, one with the C7 or [4,3,3,3,3,3] Coxeter group, and a lower symmetry with two copies of pentacross facets, alternating, with the D7 or [3<sup>4,1,1</sup>] Coxeter group.
Cartesian coordinates for the vertices of a rectified heptacross, centered at the origin, edge length
\sqrt{2}
(±1,±1,0,0,0,0,0)
Its 84 vertices represent the root vectors of the simple Lie group D7. The vertices can be seen in 3 hyperplanes, with the 21 vertices rectified 6-simplexs cells on opposite sides, and 42 vertices of an expanded 6-simplex passing through the center. When combined with the 14 vertices of the 7-orthoplex, these vertices represent the 98 root vectors of the B7 and C7 simple Lie groups.
| bgcolor=#e7dcc3 colspan=2 | Birectified 7-orthoplex | |
|---|---|---|
| Type | uniform 7-polytope | |
| Schläfli symbol | 2r | |
| Coxeter-Dynkin diagrams | ||
| 6-faces | 142 | |
| 5-faces | 1428 | |
| 4-faces | 6048 | |
| Cells | 10640 | |
| Faces | 8960 | |
| Edges | 3360 | |
| Vertices | 280 | |
| Vertex figure | × | |
| Coxeter groups | B7, [3,3,3,3,3,4] D7, [3<sup>4,1,1</sup>] | |
| Properties | convex |
Cartesian coordinates for the vertices of a birectified 7-orthoplex, centered at the origin, edge length
\sqrt{2}
(±1,±1,±1,0,0,0,0)
A trirectified 7-orthoplex is the same as a trirectified 7-cube.