# Rectified 5-orthoplexes

| 5-cube | Rectified 5-cube | Birectified 5-cube Birectified 5-orthoplex |
| 5-orthoplex | Rectified 5-orthoplex | Birectified 5-cube Birectified 5-orthoplex |
| Orthogonal projections in A$_5$ Coxeter plane |||
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.

## Rectified 5-orthoplex

Rectified pentacross :
| Type | uniform 5-polytope |
| Schläfli symbol | t$_1${3,3,3,4} |
| Coxeter-Dynkin diagrams | |
| Hypercells | 42 total: 10 {3,3,4} 32 t$_1${3,3,3} |
| Cells | 240 total: 80 {3,4} 160 {3,3} |
| Faces | 400 total: 80+320 {3} |
| Edges | 240 |
| Vertices | 40 |
| Vertex figure | Octahedral prism |
| Petrie polygon | Decagon |
| Coxeter groups | BC$_5$, [3,3,3,4] D$_5$, [3$^{2,1,1}$] |
| Properties | convex |
Its 40 vertices represent the root vectors of the simple Lie group D$_5$. 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 B$_5$ and C$_5$ simple Lie groups.
E. L. Elte identified it in 1912 as a semiregular polytope, identifying it as Cr$_5$$^1$ as a first rectification of a 5-dimensional cross polytope.

### Alternate names

* rectified pentacross
* rectified triacontiditeron (32-faceted 5-polytope)

### Construction

There are two Coxeter groups associated with the rectified pentacross, one with the C$_5$ or [4,3,3,3] Coxeter group, and a lower symmetry with two copies of 16-cell facets, alternating, with the D$_5$ or [3$^{2,1,1}$] Coxeter group.

## Cartesian coordinates

Cartesian coordinates for the vertices of a rectified pentacross, centered at the origin, edge length ${\sqrt {2}}\$ are all permutations of:
    (±1,±1,0,0,0)

### Images

orthographic projections :
| Coxeter plane | B$_5$ | B$_4$ / D$_5$ | B$_3$ / D$_4$ / A$_2$ |
| Graph | | | |
| Dihedral symmetry | [10] | [8] | [6] |
| Coxeter plane | B$_2$ | A$_3$ |
| Graph | | |
| Dihedral symmetry | [4] | [4] |

## Related polytopes

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.
B5 polytopes :
| β$_5$ | t$_1$β$_5$ | t$_2$γ$_5$ | t$_1$γ$_5$ | γ$_5$ | t$_{0,1}$β$_5$ | t$_{0,2}$β$_5$ | t$_{1,2}$β$_5$ |
| t$_{0,3}$β$_5$ | t$_{1,3}$γ$_5$ | t$_{1,2}$γ$_5$ | t$_{0,4}$γ$_5$ | t$_{0,3}$γ$_5$ | t$_{0,2}$γ$_5$ | t$_{0,1}$γ$_5$ | t$_{0,1,2}$β$_5$ |
| t$_{0,1,3}$β$_5$ | t$_{0,2,3}$β$_5$ | t$_{1,2,3}$γ$_5$ | t$_{0,1,4}$β$_5$ | t$_{0,2,4}$γ$_5$ | t$_{0,2,3}$γ$_5$ | t$_{0,1,4}$γ$_5$ | t$_{0,1,3}$γ$_5$ |
| t$_{0,1,2}$γ$_5$ | t$_{0,1,2,3}$β$_5$ | t$_{0,1,2,4}$β$_5$ | t$_{0,1,3,4}$γ$_5$ | t$_{0,1,2,4}$γ$_5$ | t$_{0,1,2,3}$γ$_5$ | t$_{0,1,2,3,4}$γ$_5$ |