# Cantellated 7-cubes

| 7-cube | Cantellated 7-cube | Bicantellated 7-cube | Tricantellated 7-cube |
| Birectified 7-cube | Cantitruncated 7-cube | Bicantitruncated 7-cube | Tricantitruncated 7-cube |
| Cantellated 7-orthoplex | Bicantellated 7-orthoplex | Cantitruncated 7-orthoplex | Bicantitruncated 7-orthoplex |
| Orthogonal projections in B$_6$ Coxeter plane ||||
In seven-dimensional geometry, a cantellated 7-cube is a convex uniform 7-polytope, being a cantellation of the regular 7-cube.
There are 10 degrees of cantellation for the 7-cube, including truncations. 4 are most simply constructible from the dual 7-orthoplex.

## Cantellated 7-cube

Cantellated 7-cube :
| Type | uniform 7-polytope |
| Schläfli symbol | rr{4,3,3,3,3,3} |
| Coxeter diagram | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 16128 |
| Vertices | 2688 |
| Vertex figure | |
| Coxeter groups | B$_7$, [4,3,3,3,3,3] |
| Properties | convex |

### Alternate names

* Small rhombated hepteract (acronym: sersa) (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] |

## Bicantellated 7-cube

Bicantellated 7-cube :
| Type | uniform 7-polytope |
| Schläfli symbol | r2r{4,3,3,3,3,3} |
| Coxeter diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 40320 |
| Vertices | 6720 |
| Vertex figure | |
| Coxeter groups | B$_7$, [4,3,3,3,3,3] |
| Properties | convex |

### Alternate names

* Small birhombated hepteract (acronym: sibrosa) (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] |

## Tricantellated 7-cube

Tricantellated 7-cube :
| Type | uniform 7-polytope |
| Schläfli symbol | r3r{4,3,3,3,3,3} |
| Coxeter diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 47040 |
| Vertices | 6720 |
| Vertex figure | |
| Coxeter groups | B$_7$, [4,3,3,3,3,3] |
| Properties | convex |

### Alternate names

* Small trirhombihepteractihecatonicosoctaexon (acronym: strasaz) (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] |

## Cantitruncated 7-cube

Cantitruncated 7-cube :
| Type | uniform 7-polytope |
| Schläfli symbol | tr{4,3,3,3,3,3} |
| Coxeter diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 18816 |
| Vertices | 5376 |
| Vertex figure | |
| Coxeter groups | B$_7$, [4,3,3,3,3,3] |
| Properties | convex |

### Alternate names

* Great rhombated hepteract (acronym: gersa) (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] |
It is fifth in a series of cantitruncated hypercubes:
Petrie polygon projections :
| Truncated cuboctahedron | Cantitruncated tesseract | Cantitruncated 5-cube | Cantitruncated 6-cube | Cantitruncated 7-cube | Cantitruncated 8-cube |

## Bicantitruncated 7-cube

Bicantitruncated 7-cube :
| Type | uniform 7-polytope |
| Schläfli symbol | r2r{4,3,3,3,3,3} |
| Coxeter diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 47040 |
| Vertices | 13440 |
| Vertex figure | |
| Coxeter groups | B$_7$, [4,3,3,3,3,3] |
| Properties | convex |

### Alternate names

* Great birhombated hepteract (acronym: gibrosa) (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] |

## Tricantitruncated 7-cube

Tricantitruncated 7-cube :
| Type | uniform 7-polytope |
| Schläfli symbol | t3r{4,3,3,3,3,3} |
| Coxeter diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 53760 |
| Vertices | 13440 |
| Vertex figure | |
| Coxeter groups | B$_7$, [4,3,3,3,3,3] |
| Properties | convex |

### Alternate names

* Great trirhombihepteractihecatonicosoctaexon (acronym: gotrasaz) (Jonathan Bowers)

### Images

orthographic projections :
| Coxeter plane | B$_7$ / A$_6$ | B$_6$ / D$_7$ | B$_5$ / D$_6$ / A$_4$ |
| Graph | too complex | | |
| 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] |

## Related polytopes

These polytopes are from a family of 127 uniform 7-polytopes with B$_7$ symmetry.