Pentic 7-cubes
# 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 :

| $t_0(1_{41})$ | $t_{0,1}(1_{41})$ | $t_{0,2}(1_{41})$ | $t_{0,3}(1_{41})$ | $t_{0,4}(1_{41})$ | $t_{0,5}(1_{41})$ | $t_{0,1,2}(1_{41})$ | $t_{0,1,3}(1_{41})$ |
| - | - | - | - | - | - | - | - |
| $t_{0,1,4}(1_{41})$ | $t_{0,1,5}(1_{41})$ | $t_{0,2,3}(1_{41})$ | $t_{0,2,4}(1_{41})$ | $t_{0,2,5}(1_{41})$ | $t_{0,3,4}(1_{41})$ | $t_{0,3,5}(1_{41})$ | $t_{0,4,5}(1_{41})$ |
| $t_{0,1,2,3}(1_{41})$ | $t_{0,1,2,4}(1_{41})$ | $t_{0,1,2,5}(1_{41})$ | $t_{0,1,3,4}(1_{41})$ | $t_{0,1,3,5}(1_{41})$ | $t_{0,1,4,5}(1_{41})$ | $t_{0,2,3,4}(1_{41})$ | $t_{0,2,3,5}(1_{41})$ |
| $t_{0,2,4,5}(1_{41})$ | $t_{0,3,4,5}(1_{41})$ | $t_{0,1,2,3,4}(1_{41})$ | $t_{0,1,2,3,5}(1_{41})$ | $t_{0,1,2,4,5}(1_{41})$ | $t_{0,1,3,4,5}(1_{41})$ | $t_{0,2,3,4,5}(1_{41})$ | $t_{0,1,2,3,4,5}(1_{41})$ |
