# Hexic 7-cubes

| 7-demicube | Hexic 7-cube | Hexicantic 7-cube | Hexiruncic 7-cube | Hexiruncicantic 7-cube |
| Hexisteric 7-cube | Hexistericantic 7-cube | Hexisteriruncic 7-cube | Hexisteriruncicantic 7-cube | Hexipentic 7-cube |
| Hexipenticantic 7-cube | Hexipentiruncic 7-cube | Hexipentiruncicantic 7-cube | Hexipentisteric 7-cube | Hexipentistericantic 7-cube |
| Hexipentisteriruncic 7-cube | Hexipentisteriruncicantic 7-cube | | | |
| Orthogonal projections in D$_7$ Coxeter plane |||||
In seven-dimensional geometry, a hexic 7-cube is a convex uniform 7-polytope, constructed from the uniform 7-demicube. There are 16 unique forms.

## Hexic 7-cube

Hexic 7-cube :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,5}${3,3$^{4,1}$} h$_6${4,3$^5$} |
| Coxeter-Dynkin diagram | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 4704 |
| Vertices | 448 |
| Vertex figure | |
| Coxeter groups | D$_7$, [3$^{4,1,1}$] |
| Properties | convex |

### Cartesian coordinates

The Cartesian coordinates for the vertices of a hexic 7-cube centered at the origin are coordinate permutations:
    (±1,±1,±1,±1,±1,±1,±3)
with an odd number of plus signs.

### Images

orthographic projections :
| Coxeter plane | B$_7$ | D$_7$ | D$_6$ |
| Graph | | | |
| Dihedral symmetry | [14/2] | [12] | [10] |
| Coxeter plane | D$_5$ | D$_4$ | D$_3$ |
| Graph | | | |
| Dihedral symmetry | [8] | [6] | [4] |
| Coxeter plane | A$_5$ | A$_3$ |
| Graph | | |
| Dihedral symmetry | [6] | [4] |

## Hexicantic 7-cube
### Images

orthographic projections :
| Coxeter plane | B$_7$ | D$_7$ | D$_6$ |
| Graph | | | |
| Dihedral symmetry | [14/2] | [12] | [10] |
| Coxeter plane | D$_5$ | D$_4$ | D$_3$ |
| Graph | | | |
| Dihedral symmetry | [8] | [6] | [4] |
| Coxeter plane | A$_5$ | A$_3$ |
| Graph | | |
| Dihedral symmetry | [6] | [4] |

## Hexiruncic 7-cube
### Images

orthographic projections :
| Coxeter plane | B$_7$ | D$_7$ | D$_6$ |
| Graph | | | |
| Dihedral symmetry | [14/2] | [12] | [10] |
| Coxeter plane | D$_5$ | D$_4$ | D$_3$ |
| Graph | | | |
| Dihedral symmetry | [8] | [6] | [4] |
| Coxeter plane | A$_5$ | A$_3$ |
| Graph | | |
| Dihedral symmetry | [6] | [4] |

## Hexisteric 7-cube
### Images

orthographic projections :
| Coxeter plane | B$_7$ | D$_7$ | D$_6$ |
| Graph | | | |
| Dihedral symmetry | [14/2] | [12] | [10] |
| Coxeter plane | D$_5$ | D$_4$ | D$_3$ |
| Graph | | | |
| Dihedral symmetry | [8] | [6] | [4] |
| Coxeter plane | A$_5$ | A$_3$ |
| Graph | | |
| Dihedral symmetry | [6] | [4] |

## Hexipentic 7-cube
### Images

orthographic projections :
| Coxeter plane | B$_7$ | D$_7$ | D$_6$ |
| Graph | | | |
| Dihedral symmetry | [14/2] | [12] | [10] |
| Coxeter plane | D$_5$ | D$_4$ | D$_3$ |
| Graph | | | |
| Dihedral symmetry | [8] | [6] | [4] |
| Coxeter plane | A$_5$ | A$_3$ |
| Graph | | |
| Dihedral symmetry | [6] | [4] |

## Hexiruncicantic 7-cube
### Images

orthographic projections :
| Coxeter plane | B$_7$ | D$_7$ | D$_6$ |
| Graph | | | |
| Dihedral symmetry | [14/2] | [12] | [10] |
| Coxeter plane | D$_5$ | D$_4$ | D$_3$ |
| Graph | | | |
| Dihedral symmetry | [8] | [6] | [4] |
| Coxeter plane | A$_5$ | A$_3$ |
| Graph | | |
| Dihedral symmetry | [6] | [4] |

## Hexistericantic 7-cube
### Images

orthographic projections :
| Coxeter plane | B$_7$ | D$_7$ | D$_6$ |
| Graph | | | |
| Dihedral symmetry | [14/2] | [12] | [10] |
| Coxeter plane | D$_5$ | D$_4$ | D$_3$ |
| Graph | | | |
| Dihedral symmetry | [8] | [6] | [4] |
| Coxeter plane | A$_5$ | A$_3$ |
| Graph | | |
| Dihedral symmetry | [6] | [4] |

## Hexipenticantic 7-cube
### Images

orthographic projections :
| Coxeter plane | B$_7$ | D$_7$ | D$_6$ |
| Graph | | | |
| Dihedral symmetry | [14/2] | [12] | [10] |
| Coxeter plane | D$_5$ | D$_4$ | D$_3$ |
| Graph | | | |
| Dihedral symmetry | [8] | [6] | [4] |
| Coxeter plane | A$_5$ | A$_3$ |
| Graph | | |
| Dihedral symmetry | [6] | [4] |

## Hexisteriruncic 7-cube
### Images

orthographic projections :
| Coxeter plane | B$_7$ | D$_7$ | D$_6$ |
| Graph | | | |
| Dihedral symmetry | [14/2] | [12] | [10] |
| Coxeter plane | D$_5$ | D$_4$ | D$_3$ |
| Graph | | | |
| Dihedral symmetry | [8] | [6] | [4] |
| Coxeter plane | A$_5$ | A$_3$ |
| Graph | | |
| Dihedral symmetry | [6] | [4] |

## Hexipentiruncic 7-cube
### Images

orthographic projections :
| Coxeter plane | B$_7$ | D$_7$ | D$_6$ |
| Graph | | | |
| Dihedral symmetry | [14/2] | [12] | [10] |
| Coxeter plane | D$_5$ | D$_4$ | D$_3$ |
| Graph | | | |
| Dihedral symmetry | [8] | [6] | [4] |
| Coxeter plane | A$_5$ | A$_3$ |
| Graph | | |
| Dihedral symmetry | [6] | [4] |

## Hexipentisteric 7-cube
### Images

orthographic projections :
| Coxeter plane | B$_7$ | D$_7$ | D$_6$ |
| Graph | | | |
| Dihedral symmetry | [14/2] | [12] | [10] |
| Coxeter plane | D$_5$ | D$_4$ | D$_3$ |
| Graph | | | |
| Dihedral symmetry | [8] | [6] | [4] |
| Coxeter plane | A$_5$ | A$_3$ |
| Graph | | |
| Dihedral symmetry | [6] | [4] |

## Hexisteriruncicantic 7-cube
### Images

orthographic projections :
| Coxeter plane | B$_7$ | D$_7$ | D$_6$ |
| Graph | | | |
| Dihedral symmetry | [14/2] | [12] | [10] |
| Coxeter plane | D$_5$ | D$_4$ | D$_3$ |
| Graph | | | |
| Dihedral symmetry | [8] | [6] | [4] |
| Coxeter plane | A$_5$ | A$_3$ |
| Graph | | |
| Dihedral symmetry | [6] | [4] |

## Hexipentiruncicantic 7-cube
### Images

orthographic projections :
| Coxeter plane | B$_7$ | D$_7$ | D$_6$ |
| Graph | | | |
| Dihedral symmetry | [14/2] | [12] | [10] |
| Coxeter plane | D$_5$ | D$_4$ | D$_3$ |
| Graph | | | |
| Dihedral symmetry | [8] | [6] | [4] |
| Coxeter plane | A$_5$ | A$_3$ |
| Graph | | |
| Dihedral symmetry | [6] | [4] |

## Hexipentisteriruncic 7-cube
### Images

orthographic projections :
| Coxeter plane | B$_7$ | D$_7$ | D$_6$ |
| Graph | | | |
| Dihedral symmetry | [14/2] | [12] | [10] |
| Coxeter plane | D$_5$ | D$_4$ | D$_3$ |
| Graph | | | |
| Dihedral symmetry | [8] | [6] | [4] |
| Coxeter plane | A$_5$ | A$_3$ |
| Graph | | |
| Dihedral symmetry | [6] | [4] |

## Hexipentistericantic 7-cube
### Images

orthographic projections :
| Coxeter plane | B$_7$ | D$_7$ | D$_6$ |
| Graph | | | |
| Dihedral symmetry | [14/2] | [12] | [10] |
| Coxeter plane | D$_5$ | D$_4$ | D$_3$ |
| Graph | | | |
| Dihedral symmetry | [8] | [6] | [4] |
| Coxeter plane | A$_5$ | A$_3$ |
| Graph | | |
| Dihedral symmetry | [6] | [4] |

## Hexipentisteriruncicantic 7-cube
### Images

orthographic projections :
| Coxeter plane | B$_7$ | D$_7$ | D$_6$ |
| Graph | | | |
| Dihedral symmetry | [14/2] | [12] | [10] |
| Coxeter plane | D$_5$ | D$_4$ | D$_3$ |
| Graph | | | |
| Dihedral symmetry | [8] | [6] | [4] |
| Coxeter plane | A$_5$ | A$_3$ |
| Graph | | |
| Dihedral symmetry | [6] | [4] |

## Related polytopes

This polytope is based on the 7-demicube, a part of a dimensional family of uniform polytopes called demihypercubes for being alternation of the hypercube family.
There are 95 uniform polytopes with D$_7$ symmetry, 63 are shared by the BC$_7$ symmetry, and 32 are unique:
D7 polytopes :
| t$_0$(1$_{41}$) | t$_{0,1}$(1$_{41}$) | t$_{0,2}$(1$_{41}$) | t$_{0,3}$(1$_{41}$) | t$_{0,4}$(1$_{41}$) | t$_{0,5}$(1$_{41}$) | t$_{0,1,2}$(1$_{41}$) | t$_{0,1,3}$(1$_{41}$) |
| t$_{0,1,4}$(1$_{41}$) | t$_{0,1,5}$(1$_{41}$) | t$_{0,2,3}$(1$_{41}$) | t$_{0,2,4}$(1$_{41}$) | t$_{0,2,5}$(1$_{41}$) | t$_{0,3,4}$(1$_{41}$) | t$_{0,3,5}$(1$_{41}$) | t$_{0,4,5}$(1$_{41}$) |
| t$_{0,1,2,3}$(1$_{41}$) | t$_{0,1,2,4}$(1$_{41}$) | t$_{0,1,2,5}$(1$_{41}$) | t$_{0,1,3,4}$(1$_{41}$) | t$_{0,1,3,5}$(1$_{41}$) | t$_{0,1,4,5}$(1$_{41}$) | t$_{0,2,3,4}$(1$_{41}$) | t$_{0,2,3,5}$(1$_{41}$) |
| t$_{0,2,4,5}$(1$_{41}$) | t$_{0,3,4,5}$(1$_{41}$) | t$_{0,1,2,3,4}$(1$_{41}$) | t$_{0,1,2,3,5}$(1$_{41}$) | t$_{0,1,2,4,5}$(1$_{41}$) | t$_{0,1,3,4,5}$(1$_{41}$) | t$_{0,2,3,4,5}$(1$_{41}$) | t$_{0,1,2,3,4,5}$(1$_{41}$) |