# Pentic 7-cubes

| 7-demicube (half 7-cube, h{4,3$^5$}) | Pentic 7-cube h$_5${4,3$^5$} | Penticantic 7-cube h$_{2,5}${4,3$^5$} |
| Pentiruncic 7-cube h$_{3,5}${4,3$^5$} | Pentiruncicantic 7-cube h$_{2,3,5}${4,3$^5$} | Pentisteric 7-cube h$_{4,5}${4,3$^5$} |
| Pentistericantic 7-cube h$_{2,4,5}${4,3$^5$} | Pentisteriruncic 7-cube h$_{3,4,5}${4,3$^5$} | Penticsteriruncicantic 7-cube h$_{2,3,4,5}${4,3$^5$} |
| Orthogonal projections in D$_7$ Coxeter plane |||
In seven-dimensional geometry, a pentic 7-cube is a convex uniform 7-polytope, related to the uniform 7-demicube. There are 8 unique forms.

## Pentic 7-cube

Pentic 7-cube :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,4}${3,3$^{4,1}$} h$_5${4,3$^5$} |
| Coxeter-Dynkin diagram | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 13440 |
| Vertices | 1344 |
| Vertex figure | |
| Coxeter groups | D$_7$, [3$^{4,1,1}$] |
| Properties | convex |

### Cartesian coordinates

The Cartesian coordinates for the vertices of a pentic 7-cube centered at the origin are coordinate permutations:
    (±1,±1,±1,±1,±1,±3,±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] |

### Related polytopes

Dimensional family of pentic n-cubes :
| n | 6 | 7 | 8 |
| [1$^+$,4,3$^{n-2}$] = [3,3$^{n-3,1}$] | [1$^+$,4,3$^4$] = [3,3$^{3,1}$] | [1$^+$,4,3$^5$] = [3,3$^{4,1}$] | [1$^+$,4,3$^6$] = [3,3$^{5,1}$] |
| Cantic figure | | | |
| Coxeter | = | = | = |
| Schläfli | h$_5${4,3$^4$} | h$_5${4,3$^5$} | h$_5${4,3$^6$} |

## Penticantic 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] |

## Pentiruncic 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] |

## Pentiruncicantic 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] |

## Pentisteric 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] |

## Pentistericantic 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] |

## Pentisteriruncic 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] |

## Pentisteriruncicantic 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$_{41}$) |