# Rectified 7-cubes

| 7-cube | Rectified 7-cube | Birectified 7-cube | Trirectified 7-cube |
| Birectified 7-orthoplex | Rectified 7-orthoplex | 7-orthoplex |
| Orthogonal projections in B$_7$ Coxeter plane ||||
In seven-dimensional geometry, a rectified 7-cube is a convex uniform 7-polytope, being a rectification of the regular 7-cube.
There are unique 7 degrees of rectifications, the zeroth being the 7-cube, and the 6th and last being the 7-cube. Vertices of the rectified 7-cube are located at the edge-centers of the 7-ocube. Vertices of the birectified 7-cube are located in the square face centers of the 7-cube. Vertices of the trirectified 7-cube are located in the cube cell centers of the 7-cube.

## Rectified 7-cube

Rectified 7-cube :
| Type | uniform 7-polytope |
| Schläfli symbol | r{4,3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 6-faces | 128 + 14 |
| 5-faces | 896 + 84 |
| 4-faces | 2688 + 280 |
| Cells | 4480 + 560 |
| Faces | 4480 + 672 |
| Edges | 2688 |
| Vertices | 448 |
| Vertex figure | 5-simplex prism |
| Coxeter groups | B$_7$, [3,3,3,3,3,4] |
| Properties | convex |

### Alternate names

* rectified hepteract (Acronym rasa) (Jonathan Bowers)

### Images

orthographic projections :
| Coxeter plane | B$_7$ / A$_6$ | B$_6$ / D$_7$ | B$_5$ / D$_6$ / A$_4$ |
| Graph | | | |
| Dihedral symmetry | [14] | [12] | [10] |
| Coxeter plane | B$_4$ / D$_5$ | B$_3$ / D$_4$ / A$_2$ | B$_2$ / D$_3$ |
| Graph | | | |
| Dihedral symmetry | [8] | [6] | [4] |
| Coxeter plane | A$_5$ | A$_3$ |
| Graph | | |
| Dihedral symmetry | [6] | [4] |

### Cartesian coordinates

Cartesian coordinates for the vertices of a rectified 7-cube, centered at the origin, edge length ${\sqrt {2}}\$ are all permutations of:
    (±1,±1,±1,±1,±1,±1,0)

## Birectified 7-cube

Birectified 7-cube :
| Type | uniform 7-polytope |
| Coxeter symbol | 0$_{411}$ |
| Schläfli symbol | 2r{4,3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 6-faces | 128 + 14 |
| 5-faces | 448 + 896 + 84 |
| 4-faces | 2688 + 2688 + 280 |
| Cells | 6720 + 4480 + 560 |
| Faces | 8960 + 4480 |
| Edges | 6720 |
| Vertices | 672 |
| Vertex figure | {3}x{3,3,3} |
| Coxeter groups | B$_7$, [3,3,3,3,3,4] |
| Properties | convex |

### Alternate names

* Birectified hepteract (Acronym bersa) (Jonathan Bowers)

### Images

orthographic projections :
| Coxeter plane | B$_7$ / A$_6$ | B$_6$ / D$_7$ | B$_5$ / D$_6$ / A$_4$ |
| Graph | | | |
| Dihedral symmetry | [14] | [12] | [10] |
| Coxeter plane | B$_4$ / D$_5$ | B$_3$ / D$_4$ / A$_2$ | B$_2$ / D$_3$ |
| Graph | | | |
| Dihedral symmetry | [8] | [6] | [4] |
| Coxeter plane | A$_5$ | A$_3$ |
| Graph | | |
| Dihedral symmetry | [6] | [4] |

### Cartesian coordinates

Cartesian coordinates for the vertices of a birectified 7-cube, centered at the origin, edge length ${\sqrt {2}}\$ are all permutations of:
    (±1,±1,±1,±1,±1,0,0)

## Trirectified 7-cube

Trirectified 7-cube :
| Type | uniform 7-polytope |
| Schläfli symbol | 3r{4,3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 6-faces | 128 + 14 |
| 5-faces | 448 + 896 + 84 |
| 4-faces | 672 + 2688 + 2688 + 280 |
| Cells | 3360 + 6720 + 4480 |
| Faces | 6720 + 8960 |
| Edges | 6720 |
| Vertices | 560 |
| Vertex figure | {3,3}x{3,3} |
| Coxeter groups | B$_7$, [3,3,3,3,3,4] |
| Properties | convex |

### Alternate names

* Trirectified hepteract
* Trirectified 7-orthoplex
* Trirectified heptacross (Acronym sez) (Jonathan Bowers)

### Images

orthographic projections :
| Coxeter plane | B$_7$ / A$_6$ | B$_6$ / D$_7$ | B$_5$ / D$_6$ / A$_4$ |
| Graph | | | |
| Dihedral symmetry | [14] | [12] | [10] |
| Coxeter plane | B$_4$ / D$_5$ | B$_3$ / D$_4$ / A$_2$ | B$_2$ / D$_3$ |
| Graph | | | |
| Dihedral symmetry | [8] | [6] | [4] |
| Coxeter plane | A$_5$ | A$_3$ |
| Graph | | |
| Dihedral symmetry | [6] | [4] |

### Cartesian coordinates

Cartesian coordinates for the vertices of a trirectified 7-cube, centered at the origin, edge length ${\sqrt {2}}\$ are all permutations of:
    (±1,±1,±1,±1,0,0,0)

### Related polytopes

2-isotopic hypercubes :
| Dim. | 2 | 3 | 4 | 5 | 6 | 7 | 8 | n |
| Name | t{4} | r{4,3} | 2t{4,3,3} | 2r{4,3,3,3} | 3t{4,3,3,3,3} | 3r{4,3,3,3,3,3} | 4t{4,3,3,3,3,3,3} | ... |
| Coxeter diagram | | | | | | | | ... |
| Images | | | | | | | | ... |
| Facets | | {3} {4} | t{3,3} t{3,4} | r{3,3,3} r{3,3,4} | 2t{3,3,3,3} 2t{3,3,3,4} | 2r{3,3,3,3,3} 2r{3,3,3,3,4} | 3t{3,3,3,3,3,3} 3t{3,3,3,3,3,4} | ... |
| Vertex figure | ( )v( ) | { }×{ } | { }v{ } | {3}×{4} | {3}v{4} | {3,3}×{3,4} | {3,3}v{3,4} | ... |