# Cantellated 7-orthoplexes

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

## Cantellated 7-orthoplex

Cantellated 7-orthoplex :
| Type | uniform 7-polytope |
| Schläfli symbol | rr{3,3,3,3,3,4} |
| Coxeter-Dynkin diagram | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 7560 |
| Vertices | 840 |
| Vertex figure | |
| Coxeter groups | B$_7$, [4,3,3,3,3,3] |
| Properties | convex |

### Alternate names

* Small rhombated hecatonicosoctaexon (acronym: sarz) (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-orthoplex

Bicantellated 7-orthoplex :
| Type | uniform 7-polytope |
| Schläfli symbol | 2rr{3,3,3,3,3,4} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | |
| Vertices | |
| Vertex figure | |
| Coxeter groups | B$_7$, [4,3,3,3,3,3] |
| Properties | convex |

### Alternate names

* Small birhombated hecatonicosoctaexon (acronym: sebraz) (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-orthoplex

Cantitruncated 7-orthoplex :
| Type | uniform 7-polytope |
| Schläfli symbol | tr{3,3,3,3,3,4} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 8400 |
| Vertices | 1680 |
| Vertex figure | |
| Coxeter groups | B$_7$, [4,3,3,3,3,3] |
| Properties | convex |

### Alternate names

* Great rhombated hecatonicosoctaexon (acronym: garz) (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] |

## Bicantitruncated 7-orthoplex

Bicantitruncated 7-orthoplex :
| Type | uniform 7-polytope |
| Schläfli symbol | 2tr{3,3,3,3,3,4} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 30240 |
| Vertices | 6720 |
| Vertex figure | |
| Coxeter groups | B$_7$, [4,3,3,3,3,3] |
| Properties | convex |

### Alternate names

* Great birhombated hecatonicosoctaexon (acronym: gebraz) (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.