# Steric 6-cubes

| 6-demicube = | Steric 6-cube = | Stericantic 6-cube = |
| | Steriruncic 6-cube = | Stericruncicantic 6-cube = |
| Orthogonal projections in D$_6$ Coxeter plane |||
In six-dimensional geometry, a steric 6-cube is a convex uniform 6-polytope. There are unique 4 steric forms of the 6-cube.

## Steric 6-cube

Steric 6-cube :
| Type | uniform 6-polytope |
| Schläfli symbol | t$_{0,3}${3,3$^{3,1}$} h$_4${4,3$^4$} |
| Coxeter-Dynkin diagram | = |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 3360 |
| Vertices | 480 |
| Vertex figure | |
| Coxeter groups | D$_6$, [3$^{3,1,1}$] |
| Properties | convex |

### Alternate names

* Runcinated demihexeract/6-demicube
* Small prismated hemihexeract (Acronym sophax) (Jonathan Bowers)

### Cartesian coordinates

The Cartesian coordinates for the 480 vertices of a steric 6-cube centered at the origin are coordinate permutations:
    (±1,±1,±1,±1,±1,±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

Dimensional family of steric n-cubes :
| n | 5 | 6 | 7 | 8 |
| [1$^+$,4,3$^{n-2}$] = [3,3$^{n-3,1}$] | [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$^{5,1}$] |
| Steric figure | | | | |
| Coxeter | = | = | = | = |
| Schläfli | h$_4${4,3$^3$} | h$_4${4,3$^4$} | h$_4${4,3$^5$} | h$_4${4,3$^6$} |

## Stericantic 6-cube

Stericantic 6-cube :
| Type | uniform 6-polytope |
| Schläfli symbol | t$_{0,1,3}${3,3$^{3,1}$} h$_{2,4}${4,3$^4$} |
| Coxeter-Dynkin diagram | = |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 12960 |
| Vertices | 2880 |
| Vertex figure | |
| Coxeter groups | D$_6$, [3$^{3,1,1}$] |
| Properties | convex |

### Alternate names

* Runcitruncated demihexeract/6-demicube
* Prismatotruncated hemihexeract (Acronym pithax) (Jonathan Bowers)

### Cartesian coordinates

The Cartesian coordinates for the 2880 vertices of a stericantic 6-cube centered at the origin are coordinate permutations:
    (±1,±1,±1,±3,±3,±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] |

## Steriruncic 6-cube

Steriruncic 6-cube :
| Type | uniform 6-polytope |
| Schläfli symbol | t$_{0,2,3}${3,3$^{3,1}$} h$_{3,4}${4,3$^4$} |
| Coxeter-Dynkin diagram | = |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 7680 |
| Vertices | 1920 |
| Vertex figure | |
| Coxeter groups | D$_6$, [3$^{3,1,1}$] |
| Properties | convex |

### Alternate names

* Runcicantellated demihexeract/6-demicube
* Prismatorhombated hemihexeract (Acronym prohax) (Jonathan Bowers)

### Cartesian coordinates

The Cartesian coordinates for the 1920 vertices of a steriruncic 6-cube centered at the origin are coordinate permutations:
    (±1,±1,±1,±1,±3,±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] |

## Steriruncicantic 6-cube

Steriruncicantic 6-cube :
| Type | uniform 6-polytope |
| Schläfli symbol | t$_{0,1,2,3}${3,3$^{2,1}$} h$_{2,3,4}${4,3$^4$} |
| Coxeter-Dynkin diagram | = |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 17280 |
| Vertices | 5760 |
| Vertex figure | |
| Coxeter groups | D$_6$, [3$^{3,1,1}$] |
| Properties | convex |

### Alternate names

* Runcicantitruncated demihexeract/6-demicube
* Great prismated hemihexeract (Acronym gophax) (Jonathan Bowers)

### Cartesian coordinates

The Cartesian coordinates for the 5760 vertices of a steriruncicantic 6-cube centered at the origin are coordinate permutations:
    (±1,±1,±1,±3,±5,±7)
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

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$^4$} |