# Runcic 7-cubes

| 7-demicube | Runcic 7-cube | Runcicantic 7-cube |
| Orthogonal projections in D$_7$ Coxeter plane |||
In seven-dimensional geometry, a runcic 7-cube is a convex uniform 7-polytope, related to the uniform 7-demicube. There are 2 unique forms.

## Runcic 7-cube

Runcic 7-cube :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,2}${3,3$^{4,1}$} h$_3${4,3$^5$} |
| Coxeter-Dynkin diagram | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 16800 |
| Vertices | 2240 |
| Vertex figure | |
| Coxeter groups | D$_7$, [3$^{4,1,1}$] |
| Properties | convex |
A runcic 7-cube, h$_3${4,3$^5$}, has half the vertices of a runcinated 7-cube, t$_{0,3}${4,3$^5$}.

### Alternate names

* Small rhombated hemihepteract (Acronym sirhesa) (Jonathan Bowers)

### Cartesian coordinates

The Cartesian coordinates for the vertices of a cantellated demihepteract centered at the origin are coordinate permutations:
    (±1,±1,±1,±3,±3,±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] |

## Runcicantic 7-cube

Runcicantic 7-cube :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,1,2}${3,3$^{4,1}$} h$_{2,3}${4,3$^5$} |
| Coxeter-Dynkin diagram | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 23520 |
| Vertices | 6720 |
| Vertex figure | |
| Coxeter groups | D$_6$, [3$^{3,1,1}$] |
| Properties | convex |
A runcicantic 7-cube, h$_{2,3}${4,3$^5$}, has half the vertices of a runcicantellated 7-cube, t$_{0,1,3}${4,3$^5$}.

### Alternate names

* Great rhombated hemihepteract (Acronym girhesa) (Jonathan Bowers)

### Cartesian coordinates

The Cartesian coordinates for the vertices of a runcicantic 7-cube centered at the origin are coordinate permutations:
    (±1,±1,±1,±1,±3,±5,±5)
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

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$_6$ 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$_{41}$) |