# Pentellated 7-cubes

In seven-dimensional geometry, a pentellated 7-cube is a convex uniform 7-polytope with 5th order truncations (pentellation) of the regular 7-cube. There are 32 unique pentellations of the 7-cube with permutations of truncations, cantellations, runcinations, and sterications. 16 are more simply constructed relative to the 7-orthoplex.
| 7-cube | Pentellated 7-cube | Pentitruncated 7-cube | Penticantellated 7-cube |
| Penticantitruncated 7-cube | Pentiruncinated 7-cube | Pentiruncitruncated 7-cube | Pentiruncicantellated 7-cube |
| Pentiruncicantitruncated 7-cube | Pentistericated 7-cube | Pentisteritruncated 7-cube | Pentistericantellated 7-cube |
| Pentistericantitruncated 7-cube | Pentisteriruncinated 7-cube | Pentisteriruncitruncated 7-cube | Pentisteriruncicantellated 7-cube |
| Pentisteriruncicantitruncated 7-cube |

## Pentellated 7-cube

Pentellated 7-cube :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,5}${4,3$^5$} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | |
| Vertices | |
| Vertex figure | |
| Coxeter groups | B$_7$, [4,3$^5$] |
| Properties | convex |

### Alternate names

* Small terated hepteract (acronym:) (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] |

## Pentitruncated 7-cube

pentitruncated 7-cube :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,1,5}${4,3$^5$} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | |
| Vertices | |
| Vertex figure | |
| Coxeter groups | B$_7$, [4,3$^5$] |
| Properties | convex |

### Alternate names

* Teritruncated hepteract (acronym: ) (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] |

## Penticantellated 7-cube

Penticantellated 7-cube :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,2,5}${4,3$^5$} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | |
| Vertices | |
| Vertex figure | |
| Coxeter groups | B$_7$, [4,3$^5$] |
| Properties | convex |

### Alternate names

* Terirhombated hepteract (acronym: ) (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] |

## Penticantitruncated 7-cube

penticantitruncated 7-cube :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,1,2,5}${4,3$^5$} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | |
| Vertices | |
| Vertex figure | |
| Coxeter groups | B$_7$, [4,3$^5$] |
| Properties | convex |

### Alternate names

* Terigreatorhombated hepteract (acronym: ) (Jonathan Bowers)
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] |

## Pentiruncinated 7-cube

pentiruncinated 7-cube :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,3,5}${4,3$^5$} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | |
| Vertices | |
| Vertex figure | |
| Coxeter groups | B$_7$, [4,3$^5$] |
| Properties | convex |

### Alternate names

* Teriprismated hepteract (acronym: ) (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] |

## Pentiruncitruncated 7-cube

pentiruncitruncated 7-cube :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,1,3,5}${4,3$^5$} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | |
| Vertices | |
| Vertex figure | |
| Coxeter groups | B$_7$, [4,3$^5$] |
| Properties | convex |

### Alternate names

* Teriprismatotruncated hepteract (acronym: ) (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] |

## Pentiruncicantellated 7-cube

pentiruncicantellated 7-cube :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,2,3,5}${4,3$^5$} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | |
| Vertices | |
| Vertex figure | |
| Coxeter groups | B$_7$, [4,3$^5$] |
| Properties | convex |

### Alternate names

* Teriprismatorhombated hepteract (acronym: ) (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] |

## Pentiruncicantitruncated 7-cube

pentiruncicantitruncated 7-cube :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,1,2,3,5}${4,3$^5$} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | |
| Vertices | |
| Vertex figure | |
| Coxeter groups | B$_7$, [4,3$^5$] |
| Properties | convex |

### Alternate names

* Terigreatoprismated hepteract (acronym: ) (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 | too complex | too complex |
| Dihedral symmetry | [6] | [4] |

## Pentistericated 7-cube

pentistericated 7-cube :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,4,5}${4,3$^5$} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | |
| Vertices | |
| Vertex figure | |
| Coxeter groups | B$_7$, [4,3$^5$] |
| Properties | convex |

### Alternate names

* Tericellated hepteract (acronym: ) (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] |

## Pentisteritruncated 7-cube

pentisteritruncated 7-cube :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,1,4,5}${4,3$^5$} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | |
| Vertices | |
| Vertex figure | |
| Coxeter groups | B$_7$, [4,3$^5$] |
| Properties | convex |

### Alternate names

* Tericellitruncated hepteract (acronym: ) (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] |

## Pentistericantellated 7-cube

pentistericantellated 7-cube :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,2,4,5}${4,3$^5$} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | |
| Vertices | |
| Vertex figure | |
| Coxeter groups | B$_7$, [4,3$^5$] |
| Properties | convex |

### Alternate names

* Tericellirhombated hepteract (acronym: ) (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] |

## Pentistericantitruncated 7-cube

pentistericantitruncated 7-cube :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,1,2,4,5}${4,3$^5$} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | |
| Vertices | |
| Vertex figure | |
| Coxeter groups | B$_7$, [4,3$^5$] |
| Properties | convex |

### Alternate names

* Tericelligreatorhombated hepteract (acronym: ) (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] |

## Pentisteriruncinated 7-cube

Pentisteriruncinated 7-cube :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,3,4,5}${4,3$^5$} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | |
| Vertices | |
| Vertex figure | |
| Coxeter groups | B$_7$, [4,3$^5$] |
| Properties | convex |

### Alternate names

* Bipenticantitruncated 7-cube as t$_{1,2,3,6}${4,3$^5$}
* Tericelliprismated hepteract (acronym: ) (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] |

## Pentisteriruncitruncated 7-cube

pentisteriruncitruncated 7-cube :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,1,3,4,5}${4,3$^5$} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 40320 |
| Vertices | 10080 |
| Vertex figure | |
| Coxeter groups | B$_7$, [4,3$^5$] |
| Properties | convex |

### Alternate names

* Tericelliprismatotruncated hepteract (acronym: ) (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] |

## Pentisteriruncicantellated 7-cube

pentisteriruncicantellated 7-cube :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,2,3,4,5}${4,3$^5$} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 40320 |
| Vertices | 10080 |
| Vertex figure | |
| Coxeter groups | B$_7$, [4,3$^5$] |
| Properties | convex |

### Alternate names

* Bipentiruncicantitruncated 7-cube as t$_{1,2,3,4,6}${4,3$^5$}
* Tericelliprismatorhombated hepteract (acronym: ) (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] |

## Pentisteriruncicantitruncated 7-cube

pentisteriruncicantitruncated 7-cube :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,1,2,3,4,5}${4,3$^5$} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | |
| Vertices | |
| Vertex figure | |
| Coxeter groups | B$_7$, [4,3$^5$] |
| Properties | convex |

### Alternate names

* Great terated hepteract (acronym:) (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 a part of a set of 127 uniform 7-polytopes with B$_7$ symmetry.