# Rectified 6-cubes

| 6-cube | Rectified 6-cube | Birectified 6-cube |
| Birectified 6-orthoplex | Rectified 6-orthoplex | 6-orthoplex |
| Orthogonal projections in A$_6$ Coxeter plane |||
In six-dimensional geometry, a rectified 6-cube is a convex uniform 6-polytope, being a rectification of the regular 6-cube.
There are unique 6 degrees of rectifications, the zeroth being the 6-cube, and the 6th and last being the 6-orthoplex. Vertices of the rectified 6-cube are located at the edge-centers of the 6-cube. Vertices of the birectified 6-cube are located in the square face centers of the 6-cube.

## Rectified 6-cube

Rectified 6-cube :
| Type | uniform 6-polytope |
| Schläfli symbol | t$_1${4,3$^4$} or r{4,3$^4$} $\left\{{\begin{array}{l}4\\3,3,3,3\end{array}}\right\}$ |
| Coxeter-Dynkin diagrams | = |
| 5-faces | 76 |
| 4-faces | 444 |
| Cells | 1120 |
| Faces | 1520 |
| Edges | 960 |
| Vertices | 192 |
| Vertex figure | 5-cell prism |
| Petrie polygon | Dodecagon |
| Coxeter groups | B$_6$, [3,3,3,3,4] D$_6$, [3$^{3,1,1}$] |
| Properties | convex |

### Alternate names

* Rectified hexeract (acronym: rax) (Jonathan Bowers)

### Construction

The rectified 6-cube may be constructed from the 6-cube by truncating its vertices at the midpoints of its edges.

### Coordinates

The Cartesian coordinates of the vertices of the rectified 6-cube with edge length √2 are all permutations of:
    $(0,\ \pm 1,\ \pm 1,\ \pm 1,\ \pm 1,\ \pm 1)$

### Images

orthographic projections :
| Coxeter plane | B$_6$ | B$_5$ | B$_4$ |
| Graph | | | |
| Dihedral symmetry | [12] | [10] | [8] |
| Coxeter plane | B$_3$ | B$_2$ |
| Graph | | |
| Dihedral symmetry | [6] | [4] |
| Coxeter plane | A$_5$ | A$_3$ |
| Graph | | |
| Dihedral symmetry | [6] | [4] |

## Birectified 6-cube

Birectified 6-cube :
| Type | uniform 6-polytope |
| Coxeter symbol | 0$_{311}$ |
| Schläfli symbol | t$_2${4,3$^4$} or 2r{4,3$^4$} $\left\{{\begin{array}{l}3,4\\3,3,3\end{array}}\right\}$ |
| Coxeter-Dynkin diagrams | = = |
| 5-faces | 76 |
| 4-faces | 636 |
| Cells | 2080 |
| Faces | 3200 |
| Edges | 1920 |
| Vertices | 240 |
| Vertex figure | {4}x{3,3} duoprism |
| Coxeter groups | B$_6$, [3,3,3,3,4] D$_6$, [3$^{3,1,1}$] |
| Properties | convex |

### Alternate names

* Birectified hexeract (acronym: brox) (Jonathan Bowers)
* Rectified 6-demicube

### Construction

The birectified 6-cube may be constructed from the 6-cube by truncating its vertices at the midpoints of its edges.

### Coordinates

The Cartesian coordinates of the vertices of the rectified 6-cube with edge length √2 are all permutations of:
    $(0,\ 0,\ \pm 1,\ \pm 1,\ \pm 1,\ \pm 1)$

### Images

orthographic projections :
| Coxeter plane | B$_6$ | B$_5$ | B$_4$ |
| Graph | | | |
| Dihedral symmetry | [12] | [10] | [8] |
| Coxeter plane | B$_3$ | B$_2$ |
| Graph | | |
| Dihedral symmetry | [6] | [4] |
| Coxeter plane | A$_5$ | A$_3$ |
| Graph | | |
| Dihedral symmetry | [6] | [4] |

## Related polytopes

These polytopes are part of a set of 63 uniform 6-polytopes generated from the B$_6$ Coxeter plane, including the regular 6-cube or 6-orthoplex.
B6 polytopes :
| β$_6$ | t$_1$β$_6$ | t$_2$β$_6$ | t$_2$γ$_6$ | t$_1$γ$_6$ | γ$_6$ | t$_{0,1}$β$_6$ | t$_{0,2}$β$_6$ |
| t$_{1,2}$β$_6$ | t$_{0,3}$β$_6$ | t$_{1,3}$β$_6$ | t$_{2,3}$γ$_6$ | t$_{0,4}$β$_6$ | t$_{1,4}$γ$_6$ | t$_{1,3}$γ$_6$ | t$_{1,2}$γ$_6$ |
| t$_{0,5}$γ$_6$ | t$_{0,4}$γ$_6$ | t$_{0,3}$γ$_6$ | t$_{0,2}$γ$_6$ | t$_{0,1}$γ$_6$ | t$_{0,1,2}$β$_6$ | t$_{0,1,3}$β$_6$ | t$_{0,2,3}$β$_6$ |
| t$_{1,2,3}$β$_6$ | t$_{0,1,4}$β$_6$ | t$_{0,2,4}$β$_6$ | t$_{1,2,4}$β$_6$ | t$_{0,3,4}$β$_6$ | t$_{1,2,4}$γ$_6$ | t$_{1,2,3}$γ$_6$ | t$_{0,1,5}$β$_6$ |
| t$_{0,2,5}$β$_6$ | t$_{0,3,4}$γ$_6$ | t$_{0,2,5}$γ$_6$ | t$_{0,2,4}$γ$_6$ | t$_{0,2,3}$γ$_6$ | t$_{0,1,5}$γ$_6$ | t$_{0,1,4}$γ$_6$ | t$_{0,1,3}$γ$_6$ |
| t$_{0,1,2}$γ$_6$ | t$_{0,1,2,3}$β$_6$ | t$_{0,1,2,4}$β$_6$ | t$_{0,1,3,4}$β$_6$ | t$_{0,2,3,4}$β$_6$ | t$_{1,2,3,4}$γ$_6$ | t$_{0,1,2,5}$β$_6$ | t$_{0,1,3,5}$β$_6$ |
| t$_{0,2,3,5}$γ$_6$ | t$_{0,2,3,4}$γ$_6$ | t$_{0,1,4,5}$γ$_6$ | t$_{0,1,3,5}$γ$_6$ | t$_{0,1,3,4}$γ$_6$ | t$_{0,1,2,5}$γ$_6$ | t$_{0,1,2,4}$γ$_6$ | t$_{0,1,2,3}$γ$_6$ |
| t$_{0,1,2,3,4}$β$_6$ | t$_{0,1,2,3,5}$β$_6$ | t$_{0,1,2,4,5}$β$_6$ | t$_{0,1,2,4,5}$γ$_6$ | t$_{0,1,2,3,5}$γ$_6$ | t$_{0,1,2,3,4}$γ$_6$ | t$_{0,1,2,3,4,5}$γ$_6$ |