In five-dimensional geometry, a rectified 5-orthoplex is a convex uniform 5-polytope, being a rectification of the regular 5-orthoplex.
There are 5 degrees of rectifications for any 5-polytope, the zeroth here being the 5-orthoplex itself, and the 4th and last being the 5-cube. Vertices of the rectified 5-orthoplex are located at the edge-centers of the 5-orthoplex. Vertices of the birectified 5-orthoplex are located in the triangular face centers of the 5-orthoplex.
| bgcolor=#e7dcc3 colspan=2 | Rectified pentacross | |
|---|---|---|
| Type | uniform 5-polytope | |
| Schläfli symbol | t1 | |
| Coxeter-Dynkin diagrams | ||
| Hypercells | 42 total: 10 32 t1 | |
| Cells | 240 total: 80 160 | |
| Faces | 400 total: 80+320 | |
| Edges | 240 | |
| Vertices | 40 | |
| Vertex figure | Octahedral prism | |
| Petrie polygon | Decagon | |
| Coxeter groups | BC5, [3,3,3,4] D5, [3<sup>2,1,1</sup>] | |
| Properties | convex |
Its 40 vertices represent the root vectors of the simple Lie group D5. The vertices can be seen in 3 hyperplanes, with the 10 vertices rectified 5-cells cells on opposite sides, and 20 vertices of a runcinated 5-cell passing through the center. When combined with the 10 vertices of the 5-orthoplex, these vertices represent the 50 root vectors of the B5 and C5 simple Lie groups.
E. L. Elte identified it in 1912 as a semiregular polytope, identifying it as Cr51 as a first rectification of a 5-dimensional cross polytope.
There are two Coxeter groups associated with the rectified pentacross, one with the C5 or [4,3,3,3] Coxeter group, and a lower symmetry with two copies of 16-cell facets, alternating, with the D5 or [3<sup>2,1,1</sup>] Coxeter group.
Cartesian coordinates for the vertices of a rectified pentacross, centered at the origin, edge length
\sqrt{2}
(±1,±1,0,0,0)
The rectified 5-orthoplex is the vertex figure for the 5-demicube honeycomb:
or
This polytope is one of 31 uniform 5-polytope generated from the regular 5-cube or 5-orthoplex.