Pentic 6-cubes
# Pentic 6-cubes

| 6-demicube (half 6-cube) = | Pentic 6-cube = | Penticantic 6-cube = |
| - | - | - |
| Pentiruncic 6-cube = | Pentiruncicantic 6-cube = | Pentisteric 6-cube = |
| Pentistericantic 6-cube = | Pentisteriruncic 6-cube = | Pentisteriruncicantic 6-cube = |
| Orthogonal projections in $D_6$ Coxeter plane |||

In six-dimensional geometry, a pentic 6-cube is a convex uniform 6-polytope.

There are 8 pentic forms of the 6-cube.

## Pentic 6-cube

Pentic 6-cube :

| Type | uniform 6-polytope |
| - | - |
| Schläfli symbol | $t_{0,4}{3,3^{4,1}}$ $h_5{4,3^4}$ |
| Coxeter-Dynkin diagram | = |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 1440 |
| Vertices | 192 |
| Vertex figure | |
| Coxeter groups | $D_6,$ $[3^{3,1,1}]$ |
| Properties | convex |

The pentic 6-cube, , has half of the vertices of a pentellated 6-cube, .

### Alternate names

* Stericated 6-demicube/demihexeract
* Small cellated hemihexeract (Acronym: sochax) (Jonathan Bowers)

### Cartesian coordinates

The Cartesian coordinates for the vertices, 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] |

## Penticantic 6-cube

Penticantic 6-cube :

| Type | uniform 6-polytope |
| - | - |
| Schläfli symbol | $t_{0,1,4}{3,3^{4,1}}$ $h_{2,5}{4,3^4}$ |
| Coxeter-Dynkin diagram | = |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 9600 |
| Vertices | 1920 |
| Vertex figure | |
| Coxeter groups | $D_6,$ $[3^{3,1,1}]$ |
| Properties | convex |

The penticantic 6-cube, , has half of the vertices of a penticantellated 6-cube, .

### Alternate names

* Steritruncated 6-demicube/demihexeract
* cellitruncated hemihexeract (Acronym: cathix) (Jonathan Bowers)

### Cartesian coordinates

The Cartesian coordinates for the vertices, centered at the origin are coordinate permutations:

    (±1,±1,±3,±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] |

## Pentiruncic 6-cube

Pentiruncic 6-cube :

| Type | uniform 6-polytope |
| - | - |
| Schläfli symbol | $t_{0,2,4}{3,3^{4,1}}$ $h_{3,5}{4,3^4}$ |
| Coxeter-Dynkin diagram | = |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 10560 |
| Vertices | 1920 |
| Vertex figure | |
| Coxeter groups | $D_6,$ $[3^{3,1,1}]$ |
| Properties | convex |

The pentiruncic 6-cube, , has half of the vertices of a pentiruncinated 6-cube (penticantellated 6-orthoplex), .

### Alternate names

* Stericantellated 6-demicube/demihexeract
* cellirhombated hemihexeract (Acronym: crohax) (Jonathan Bowers)

### Cartesian coordinates

The Cartesian coordinates for the vertices, 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] |

## Pentiruncicantic 6-cube

Pentiruncicantic 6-cube :

| Type | uniform 6-polytope |
| - | - |
| Schläfli symbol | $t_{0,1,2,4}{3,3^{2,1}}$ $h_{2,3,5}{4,3^4}$ |
| Coxeter-Dynkin diagram | = |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 20160 |
| Vertices | 5760 |
| Vertex figure | |
| Coxeter groups | $D_6,$ $[3^{3,1,1}]$ |
| Properties | convex |

The pentiruncicantic 6-cube, , has half of the vertices of a pentiruncicantellated 6-cube or (pentiruncicantellated 6-orthoplex),

### Alternate names

* Stericantitruncated demihexeract, stericantitruncated 7-demicube
* Great cellated hemihexeract (Acronym: cagrohax) (Jonathan Bowers)

### Cartesian coordinates

The Cartesian coordinates for the vertices, centered at the origin are coordinate permutations:

    (±1,±1,±3,±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] |

## Pentisteric 6-cube

Pentisteric 6-cube :

| Type | uniform 6-polytope |
| - | - |
| Schläfli symbol | $t_{0,3,4}{3,3^{4,1}}$ $h_{4,5}{4,3^4}$ |
| Coxeter-Dynkin diagram | = |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 5280 |
| Vertices | 960 |
| Vertex figure | |
| Coxeter groups | $D_6,$ $[3^{3,1,1}]$ |
| Properties | convex |

The pentisteric 6-cube, , has half of the vertices of a pentistericated 6-cube (pentitruncated 6-orthoplex),

### Alternate names

* Steriruncinated 6-demicube/demihexeract
* Small cellipriamated hemihexeract (Acronym: cophix) (Jonathan Bowers)

### Cartesian coordinates

The Cartesian coordinates for the vertices, 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] |

## Pentistericantic 6-cube

Pentistericantic 6-cube :

| Type | uniform 6-polytope |
| - | - |
| Schläfli symbol | $t_{0,1,3,4}{3,3^{4,1}}$ $h_{2,4,5}{4,3^4}$ |
| Coxeter-Dynkin diagram | = |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 23040 |
| Vertices | 5760 |
| Vertex figure | |
| Coxeter groups | $D_6,$ $[3^{3,1,1}]$ |
| Properties | convex |

The pentistericantic 6-cube, , has half of the vertices of a pentistericantellated 6-cube (pentiruncitruncated 6-orthoplex), .

### Alternate names

* Steriruncitruncated demihexeract/7-demicube
* cellitruncated hemihexeract (Acronym: capthix) (Jonathan Bowers)

### Cartesian coordinates

The Cartesian coordinates for the vertices, centered at the origin are coordinate permutations:

    (±1,±1,±3,±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] |

## Pentisteriruncic 6-cube

Pentisteriruncic 6-cube :

| Type | uniform 6-polytope |
| - | - |
| Schläfli symbol | $t_{0,2,3,4}{3,3^{4,1}}$ $h_{3,4,5}{4,3^4}$ |
| Coxeter-Dynkin diagram | = |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 15360 |
| Vertices | 3840 |
| Vertex figure | |
| Coxeter groups | $D_6,$ $[3^{3,1,1}]$ |
| Properties | convex |

The pentisteriruncic 6-cube, , has half of the vertices of a pentisteriruncinated 6-cube (penticantitruncated 6-orthoplex), .

### Alternate names

* Steriruncicantellated 6-demicube/demihexeract
* Celliprismatorhombated hemihexeract (Acronym: caprohax) (Jonathan Bowers)

### Cartesian coordinates

The Cartesian coordinates for the vertices, 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] |

## Pentisteriruncicantic 6-cube

Pentisteriruncicantic 6-cube :

| Type | uniform 6-polytope |
| - | - |
| Schläfli symbol | $t_{0,1,2,3,4}{3,3^{2,1}}$ $h_{2,3,4,5}{4,3^4}$ |
| Coxeter-Dynkin diagram | = |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 34560 |
| Vertices | 11520 |
| Vertex figure | |
| Coxeter groups | $D_6,$ $[3^{3,1,1}]$ |
| Properties | convex |

The pentisteriruncicantic 6-cube, , has half of the vertices of a pentisteriruncicantellated 6-cube (pentisteriruncicantitruncated 6-orthoplex), .

### Alternate names

* Steriruncicantitruncated 6-demicube/demihexeract
* Great cellated hemihexeract (Acronym: gochax) (Jonathan Bowers)

### Cartesian coordinates

The Cartesian coordinates for the vertices, centered at the origin are coordinate permutations:

    (±1,±1,±3,±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 :

| $h{4,3^4}$ | $h_2{4,3^4}$ | $h_3{4,3^4}$ | $h_4{4,3^4}$ | $h_5{4,3^4}$ | $h_{2,3}{4,3^4}$ | $h_{2,4}{4,3^4}$ | $h_{2,5}{4,3^4}$ |
| - | - | - | - | - | - | - | - |
| $h_{3,4}{4,3^4}$ | $h_{3,5}{4,3^4}$ | $h_{4,5}{4,3^4}$ | $h_{2,3,4}{4,3^4}$ | $h_{2,3,5}{4,3^4}$ | $h_{2,4,5}{4,3^4}$ | $h_{3,4,5}{4,3^4}$ | $h_{2,3,4,5}{4,3^4}$ |
