# Runcic 6-cubes

| 6-demicube = | Runcic 6-cube = | Runcicantic 6-cube = |
| Orthogonal projections in D$_6$ Coxeter plane |||
In six-dimensional geometry, a runcic 6-cube is a convex uniform 6-polytope. There are 2 unique runcic for the 6-cube.

## Runcic 6-cube

Runcic 6-cube :
| Type | uniform 6-polytope |
| Schläfli symbol | t$_{0,2}${3,3$^{3,1}$} h$_3${4,3$^4$} |
| Coxeter-Dynkin diagram | = |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 3840 |
| Vertices | 640 |
| Vertex figure | |
| Coxeter groups | D$_6$, [3$^{3,1,1}$] |
| Properties | convex |

### Alternate names

* Cantellated 6-demicube/demihexeract
* Small rhombated hemihexeract (Acronym sirhax) (Jonathan Bowers)

### Cartesian coordinates

The Cartesian coordinates for the vertices of a runcic 6-cube centered at the origin are coordinate permutations:
    (±1,±1,±1,±3,±3,±3)
with an odd number of plus signs.

### Images

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

### Related polytopes

Runcic n-cubes :
| n | 4 | 5 | 6 | 7 | 8 |
| [1$^+$,4,3$^{n-2}$] = [3,3$^{n-3,1}$] | [1$^+$,4,3$^2$] = [3,3$^{1,1}$] | [1$^+$,4,3$^3$] = [3,3$^{2,1}$] | [1$^+$,4,3$^4$] = [3,3$^{3,1}$] | [1$^+$,4,3$^5$] = [3,3$^{4,1}$] | [1$^+$,4,3$^6$] = [3,3$^{5,1}$] |
| Runcic figure | | | | | |
| Coxeter | = | = | = | = | = |
| Schläfli | h$_3${4,3$^2$} | h$_3${4,3$^3$} | h$_3${4,3$^4$} | h$_3${4,3$^5$} | h$_3${4,3$^6$} |

## Runcicantic 6-cube

Runcicantic 6-cube :
| Type | uniform 6-polytope |
| Schläfli symbol | t$_{0,1,2}${3,3$^{3,1}$} h$_{2,3}${4,3$^4$} |
| Coxeter-Dynkin diagram | = |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 5760 |
| Vertices | 1920 |
| Vertex figure | |
| Coxeter groups | D$_6$, [3$^{3,1,1}$] |
| Properties | convex |

### Alternate names

* Cantitruncated 6-demicube/demihexeract
* Great rhombated hemihexeract (Acronym girhax) (Jonathan Bowers)

### Cartesian coordinates

The Cartesian coordinates for the vertices of a runcicantic 6-cube centered at the origin are coordinate permutations:
    (±1,±1,±3,±5,±5,±5)
with an odd number of plus signs.

### Images

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

## Related polytopes

This polytope is based on the 6-demicube, a part of a dimensional family of uniform polytopes called demihypercubes for being alternation of the hypercube family.
There are 47 uniform polytopes with D$_6$ symmetry, 31 are shared by the B$_6$ symmetry, and 16 are unique:
D6 polytopes :
| h{4,3$^4$} | h$_2${4,3$^4$} | h$_3${4,3$^4$} | h$_4${4,3$^4$} | h$_5${4,3$^4$} | h$_{2,3}${4,3$^4$} | h$_{2,4}${4,3$^4$} | h$_{2,5}${4,3$^4$} |
| h$_{3,4}${4,3$^4$} | h$_{3,5}${4,3$^4$} | h$_{4,5}${4,3$^4$} | h$_{2,3,4}${4,3$^4$} | h$_{2,3,5}${4,3$^4$} | h$_{2,4,5}${4,3$^4$} | h$_{3,4,5}${4,3$^4$} | h$_{2,3,4,5}${4,3$^4$} |