# Hexicated 7-orthoplexes

Orthogonal projections in B$_4$ Coxeter plane :
| 7-orthoplex | Hexicated 7-orthoplex Hexicated 7-cube | Hexi-truncated 7-orthoplex | Hexi-cantellated 7-orthoplex | Hexicanti-truncated 7-orthoplex |
| Hexirunci-truncated 7-orthoplex | Hexirunci-cantellated 7-orthoplex | Hexisteri-truncated 7-orthoplex | Hexiruncicanti-truncated 7-orthoplex | Hexistericanti-truncated 7-orthoplex |
| Hexisterirunci-truncated 7-orthoplex | Hexipenticanti-truncated 7-orthoplex | Hexisteriruncicanti-truncated 7-orthoplex | Hexipentiruncicanti-truncated 7-orthoplex |
In seven-dimensional geometry, a hexicated 7-orthoplex (also hexicated 7-cube) is a convex uniform 7-polytope, including 6th-order truncations (hexication) from the regular 7-orthoplex.
There are 32 hexications for the 7-orthoplex, including all permutations of truncations, cantellations, runcinations, sterications, and pentellations. 12 are represented here, while 20 are more easily constructed from the 7-cube.

## Hexitruncated 7-orthoplex

Hexitruncated 7-orthoplex :
| Type | Uniform 7-polytope |
| Schläfli symbol | t$_{0,1,6}${3$^5$,4 |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 29568 |
| Vertices | 5376 |
| Vertex figure | |
| Coxeter groups | B$_7$, [4,3$^5$] |
| Properties | convex |

### Alternate names

* Petitruncated heptacross

### 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] |

## Hexicantellated 7-orthoplex

Hexicantellated 7-orthoplex :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,2,6}${3$^5$,4} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 94080 |
| Vertices | 13440 |
| Vertex figure | |
| Coxeter groups | B$_7$, [4,3$^5$] |
| Properties | convex |

### Alternate names

* Petirhombated heptacross

### 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] |

## Hexicantitruncated 7-orthoplex

Hexicantitruncated 7-orthoplex :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,1,2,6}${3$^5$,4} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 134400 |
| Vertices | 26880 |
| Vertex figure | |
| Coxeter groups | B$_7$, [4,3$^5$] |
| Properties | convex |

### Alternate names

* Petigreatorhombated heptacross

### 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] |

## Hexiruncitruncated 7-orthoplex

Hexiruncitruncated 7-orthoplex :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,1,3,6}${3$^5$,3} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 322560 |
| Vertices | 53760 |
| Vertex figure | |
| Coxeter groups | B$_7$, [4,3$^5$] |
| Properties | convex |

### Alternate names

* Petiprismatotruncated heptacross

### 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] |

## Hexiruncicantellated 7-orthoplex

Hexiruncicantellated 7-orthoplex :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,2,3,6}${3$^5$,4} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 268800 |
| Vertices | 53760 |
| Vertex figure | |
| Coxeter groups | B$_7$, [4,3$^5$] |
| Properties | convex |
In seven-dimensional geometry, a hexiruncicantellated 7-orthoplex is a uniform 7-polytope.

### Alternate names

* Petiprismatorhombated heptacross

### 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] |

## Hexisteritruncated 7-orthoplex

hexisteritruncated 7-orthoplex :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,1,4,6}${3$^5$,4} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 322560 |
| Vertices | 53760 |
| Vertex figure | |
| Coxeter groups | B$_7$, [4,3$^5$] |
| Properties | convex |

### Alternate names

* Peticellitruncated heptacross

### 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] |

## Hexiruncicantitruncated 7-orthoplex

Hexiruncicantitruncated 7-orthoplex :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,1,2,3,6}${3$^5$,4} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 483840 |
| Vertices | 107520 |
| Vertex figure | |
| Coxeter groups | B$_7$, [4,3$^5$] |
| Properties | convex |

### Alternate names

* Petigreatoprismated heptacross

### 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] |

## Hexistericantitruncated 7-orthoplex

Hexistericantitruncated 7-orthoplex :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,1,2,4,6}${3$^5$,4} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 806400 |
| Vertices | 161280 |
| Vertex figure | |
| Coxeter groups | B$_7$, [4,3$^5$] |
| Properties | convex |

### Alternate names

* Peticelligreatorhombated heptacross

### 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] |

## Hexisteriruncitruncated 7-orthoplex

Hexisteriruncitruncated 7-orthoplex :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,1,3,4,6}${3$^5$,4} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 725760 |
| Vertices | 161280 |
| Vertex figure | |
| Coxeter groups | B$_7$, [4,3$^5$] |
| Properties | convex |

### Alternate names

* Peticelliprismatotruncated heptacross

### 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] |

## Hexipenticantitruncated 7-orthoplex

hexipenticantitruncated 7-orthoplex :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,1,2,5,6}${3$^5$,4} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 483840 |
| Vertices | 107520 |
| Vertex figure | |
| Coxeter groups | B$_7$, [4,3$^5$] |
| Properties | convex |

### Alternate names

* Petiterigreatorhombated heptacross

### 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] |

## Hexisteriruncicantitruncated 7-orthoplex

Hexisteriruncicantitruncated 7-orthoplex :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,1,2,3,4,6}${4,3$^5$} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 1290240 |
| Vertices | 322560 |
| Vertex figure | |
| Coxeter groups | B$_7$, [4,3$^5$] |
| Properties | convex |

### Alternate names

* Great petacellated heptacross

### 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] |

## Hexipentiruncicantitruncated 7-orthoplex

Hexipentiruncicantitruncated 7-orthoplex :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,1,2,3,5,6}${3$^5$,3} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 1290240 |
| Vertices | 322560 |
| Vertex figure | |
| Coxeter groups | B$_7$, [4,3$^5$] |
| Properties | convex |

### Alternate names

* Petiterigreatoprismated heptacross

### 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] |