# Rectified 8-cubes

| 8-cube | Rectified 8-cube | Birectified 8-cube | Trirectified 8-cube |
| Trirectified 8-orthoplex | Birectified 8-orthoplex | Rectified 8-orthoplex | 8-orthoplex |
| Orthogonal projections in B$_8$ Coxeter plane ||||
In eight-dimensional geometry, a rectified 8-cube is a convex uniform 8-polytope, being a rectification of the regular 8-cube.
There are unique 8 degrees of rectifications, the zeroth being the 8-cube, and the 7th and last being the 8-orthoplex. Vertices of the rectified 8-cube are located at the edge-centers of the 8-cube. Vertices of the birectified 8-cube are located in the square face centers of the 8-cube. Vertices of the trirectified 8-cube are located in the 7-cube cell centers of the 8-cube.

## Rectified 8-cube

Rectified 8-cube :
| Type | uniform 8-polytope |
| Schläfli symbol | t$_1${4,3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 7-faces | 256 + 16 |
| 6-faces | 2048 + 112 |
| 5-faces | 7168 + 448 |
| 4-faces | 14336 + 1120 |
| Cells | 17920 +* 1792 |
| Faces | 4336 + 1792 |
| Edges | 7168 |
| Vertices | 1024 |
| Vertex figure | 6-simplex prism {3,3,3,3,3}× |
| Coxeter groups | B$_8$, [3$^6$,4] D$_8$, [3$^{5,1,1}$] |
| Properties | convex |

### Alternate names

* rectified octeract

### Images

orthographic projections :
| B$_8$ | B$_8$ | B$_8$ | B$_7$ | B$_7$ | B$_7$ |
| [16] | [16] | [16] | [14] | [14] | [14] |
| B$_6$ | B$_6$ | B$_6$ | B$_5$ | B$_5$ | B$_5$ |
| [12] | [12] | [12] | [10] | [10] | [10] |
| B$_4$ | B$_4$ | B$_3$ | B$_3$ | B$_2$ | B$_2$ |
| [8] | [8] | [6] | [6] | [4] | [4] |
| A$_7$ | A$_7$ | A$_5$ | A$_5$ | A$_3$ | A$_3$ |
| [8] | [8] | [6] | [6] | [4] | [4] |

## Birectified 8-cube

Birectified 8-cube :
| Type | uniform 8-polytope |
| Coxeter symbol | 0$_{511}$ |
| Schläfli symbol | t$_2${4,3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 7-faces | 256 + 16 |
| 6-faces | 1024 + 2048 + 112 |
| 5-faces | 7168 + 7168 + 448 |
| 4-faces | 21504 + 14336 + 1120 |
| Cells | 35840 + 17920 + 1792 |
| Faces | 35840 + 14336 |
| Edges | 21504 |
| Vertices | 1792 |
| Vertex figure | {3,3,3,3}x{4} |
| Coxeter groups | B$_8$, [3$^6$,4] D$_8$, [3$^{5,1,1}$] |
| Properties | convex |

### Alternate names

* Birectified octeract
* Rectified 8-demicube

### Images

orthographic projections :
| B$_8$ | B$_8$ | B$_8$ | B$_7$ | B$_7$ | B$_7$ |
| [16] | [16] | [16] | [14] | [14] | [14] |
| B$_6$ | B$_6$ | B$_6$ | B$_5$ | B$_5$ | B$_5$ |
| [12] | [12] | [12] | [10] | [10] | [10] |
| B$_4$ | B$_4$ | B$_3$ | B$_3$ | B$_2$ | B$_2$ |
| [8] | [8] | [6] | [6] | [4] | [4] |
| A$_7$ | A$_7$ | A$_5$ | A$_5$ | A$_3$ | A$_3$ |
| [8] | [8] | [6] | [6] | [4] | [4] |

## Trirectified 8-cube

Triectified 8-cube :
| Type | uniform 8-polytope |
| Schläfli symbol | t$_3${4,3,3,3,3,3,3} |
| Coxeter diagrams | |
| 7-faces | 16+256 |
| 6-faces | 1024 + 2048 + 112 |
| 5-faces | 1792 + 7168 + 7168 + 448 |
| 4-faces | 1792 + 10752 + 21504 +14336 |
| Cells | 8960 + 26880 + 35840 |
| Faces | 17920+35840 |
| Edges | 17920 |
| Vertices | 1152 |
| Vertex figure | {3,3,3}x{3,4} |
| Coxeter groups | B$_8$, [3$^6$,4] D$_8$, [3$^{5,1,1}$] |
| Properties | convex |

### Alternate names

* trirectified octeract

### Images

orthographic projections :
| B$_8$ | B$_8$ | B$_8$ | B$_7$ | B$_7$ | B$_7$ |
| [16] | [16] | [16] | [14] | [14] | [14] |
| B$_6$ | B$_6$ | B$_6$ | B$_5$ | B$_5$ | B$_5$ |
| [12] | [12] | [12] | [10] | [10] | [10] |
| B$_4$ | B$_4$ | B$_3$ | B$_3$ | B$_2$ | B$_2$ |
| [8] | [8] | [6] | [6] | [4] | [4] |
| A$_7$ | A$_7$ | A$_5$ | A$_5$ | A$_3$ | A$_3$ |
| [8] | [8] | [6] | [6] | [4] | [4] |