# Cantellated 6-orthoplexes

| 6-orthoplex | 6-orthoplex | 6-orthoplex | 6-orthoplex | Cantellated 6-orthoplex | Cantellated 6-orthoplex | Cantellated 6-orthoplex | Cantellated 6-orthoplex | Bicantellated 6-orthoplex | Bicantellated 6-orthoplex | Bicantellated 6-orthoplex | Bicantellated 6-orthoplex |
| 6-cube | 6-cube | 6-cube | 6-cube | Cantellated 6-cube | Cantellated 6-cube | Cantellated 6-cube | Cantellated 6-cube | Bicantellated 6-cube | Bicantellated 6-cube | Bicantellated 6-cube | Bicantellated 6-cube |
| Cantitruncated 6-orthoplex | Cantitruncated 6-orthoplex | Cantitruncated 6-orthoplex | Bicantitruncated 6-orthoplex | Bicantitruncated 6-orthoplex | Bicantitruncated 6-orthoplex | Bicantitruncated 6-cube | Bicantitruncated 6-cube | Bicantitruncated 6-cube | Cantitruncated 6-cube | Cantitruncated 6-cube | Cantitruncated 6-cube |
| Orthogonal projections in B$_6$ Coxeter plane ||||||||||||
In six-dimensional geometry, a cantellated 6-orthoplex is a convex uniform 6-polytope, being a cantellation of the regular 6-orthoplex.
There are 8 cantellation for the 6-orthoplex including truncations. Half of them are more easily constructed from the dual 5-cube

## Cantellated 6-orthoplex

Cantellated 6-orthoplex :
| Type | uniform 6-polytope |
| Schläfli symbol | t$_{0,2}${3,3,3,3,4} rr{3,3,3,3,4} |
| Coxeter-Dynkin diagrams | = |
| 5-faces | 136 |
| 4-faces | 1656 |
| Cells | 5040 |
| Faces | 6400 |
| Edges | 3360 |
| Vertices | 480 |
| Vertex figure | |
| Coxeter groups | B$_6$, [3,3,3,3,4] D$_6$, [3$^{3,1,1}$] |
| Properties | convex |

### Alternate names

* Cantellated hexacross
* Small rhombated hexacontatetrapeton (acronym: srog) (Jonathan Bowers)

### Construction

There are two Coxeter groups associated with the cantellated 6-orthoplex, one with the B$_6$ or [4,3,3,3,3] Coxeter group, and a lower symmetry with the D$_6$ or [3$^{3,1,1}$] Coxeter group.

### Coordinates

Cartesian coordinates for the 480 vertices of a cantellated 6-orthoplex, centered at the origin, are all the sign and coordinate permutations of
    (2,1,1,0,0,0)

### Images

orthographic projections :
| Coxeter plane | B$_6$ | B$_5$ | B$_4$ |
| Graph | | | |
| Dihedral symmetry | [12] | [10] | [8] |
| Coxeter plane | B$_3$ | B$_2$ |
| Graph | | |
| Dihedral symmetry | [6] | [4] |
| Coxeter plane | A$_5$ | A$_3$ |
| Graph | | |
| Dihedral symmetry | [6] | [4] |

## Bicantellated 6-orthoplex

Bicantellated 6-orthoplex :
| Type | uniform 6-polytope |
| Schläfli symbol | t$_{1,3}${3,3,3,3,4} 2rr{3,3,3,3,4} |
| Coxeter-Dynkin diagrams | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 8640 |
| Vertices | 1440 |
| Vertex figure | |
| Coxeter groups | B$_6$, [3,3,3,3,4] D$_6$, [3$^{3,1,1}$] |
| Properties | convex |

### Alternate names

* Bicantellated hexacross, bicantellated hexacontatetrapeton
* Small birhombated hexacontatetrapeton (acronym: siborg) (Jonathan Bowers)

### Construction

There are two Coxeter groups associated with the bicantellated 6-orthoplex, one with the B$_6$ or [4,3,3,3,3] Coxeter group, and a lower symmetry with the D$_6$ or [3$^{3,1,1}$] Coxeter group.

### Coordinates

Cartesian coordinates for the 1440 vertices of a bicantellated 6-orthoplex, centered at the origin, are all the sign and coordinate permutations of
    (2,2,1,1,0,0)

### Images

orthographic projections :
| Coxeter plane | B$_6$ | B$_5$ | B$_4$ |
| Graph | | | |
| Dihedral symmetry | [12] | [10] | [8] |
| Coxeter plane | B$_3$ | B$_2$ |
| Graph | | |
| Dihedral symmetry | [6] | [4] |
| Coxeter plane | A$_5$ | A$_3$ |
| Graph | | |
| Dihedral symmetry | [6] | [4] |

## Cantitruncated 6-orthoplex

Cantitruncated 6-orthoplex :
| Type | uniform 6-polytope |
| Schläfli symbol | t$_{0,1,2}${3,3,3,3,4} tr{3,3,3,3,4} |
| Coxeter-Dynkin diagrams | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 3840 |
| Vertices | 960 |
| Vertex figure | |
| Coxeter groups | B$_6$, [3,3,3,3,4] D$_6$, [3$^{3,1,1}$] |
| Properties | convex |

### Alternate names

* Cantitruncated hexacross, cantitruncated hexacontatetrapeton
* Great rhombihexacontatetrapeton (acronym: grog) (Jonathan Bowers)

### Construction

There are two Coxeter groups associated with the cantitruncated 6-orthoplex, one with the B$_6$ or [4,3,3,3,3] Coxeter group, and a lower symmetry with the D$_6$ or [3$^{3,1,1}$] Coxeter group.

### Coordinates

Cartesian coordinates for the 960 vertices of a cantitruncated 6-orthoplex, centered at the origin, are all the sign and coordinate permutations of
    (3,2,1,0,0,0)

### Images

orthographic projections :
| Coxeter plane | B$_6$ | B$_5$ | B$_4$ |
| Graph | | | |
| Dihedral symmetry | [12] | [10] | [8] |
| Coxeter plane | B$_3$ | B$_2$ |
| Graph | | |
| Dihedral symmetry | [6] | [4] |
| Coxeter plane | A$_5$ | A$_3$ |
| Graph | | |
| Dihedral symmetry | [6] | [4] |

## Bicantitruncated 6-orthoplex

Bicantitruncated 6-orthoplex :
| Type | uniform 6-polytope |
| Schläfli symbol | t$_{1,2,3}${3,3,3,3,4} 2tr{3,3,3,3,4} |
| Coxeter-Dynkin diagrams | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 10080 |
| Vertices | 2880 |
| Vertex figure | |
| Coxeter groups | B$_6$, [3,3,3,3,4] D$_6$, [3$^{3,1,1}$] |
| Properties | convex |

### Alternate names

* Bicantitruncated hexacross, bicantitruncated hexacontatetrapeton
* Great birhombihexacontatetrapeton (acronym: gaborg) (Jonathan Bowers)

### Construction

There are two Coxeter groups associated with the bicantitruncated 6-orthoplex, one with the B$_6$ or [4,3,3,3,3] Coxeter group, and a lower symmetry with the D$_6$ or [3$^{3,1,1}$] Coxeter group.

### Coordinates

Cartesian coordinates for the 2880 vertices of a bicantitruncated 6-orthoplex, centered at the origin, are all the sign and coordinate permutations of
    (3,3,2,1,0,0)

### Images

orthographic projections :
| Coxeter plane | B$_6$ | B$_5$ | B$_4$ |
| Graph | | | |
| Dihedral symmetry | [12] | [10] | [8] |
| Coxeter plane | B$_3$ | B$_2$ |
| Graph | | |
| Dihedral symmetry | [6] | [4] |
| Coxeter plane | A$_5$ | A$_3$ |
| Graph | | |
| Dihedral symmetry | [6] | [4] |

## Related polytopes

These polytopes are part of a set of 63 uniform 6-polytopes generated from the B$_6$ Coxeter plane, including the regular 6-cube or 6-orthoplex.
B6 polytopes :
| β$_6$ | t$_1$β$_6$ | t$_2$β$_6$ | t$_2$γ$_6$ | t$_1$γ$_6$ | γ$_6$ | t$_{0,1}$β$_6$ | t$_{0,2}$β$_6$ |
| t$_{1,2}$β$_6$ | t$_{0,3}$β$_6$ | t$_{1,3}$β$_6$ | t$_{2,3}$γ$_6$ | t$_{0,4}$β$_6$ | t$_{1,4}$γ$_6$ | t$_{1,3}$γ$_6$ | t$_{1,2}$γ$_6$ |
| t$_{0,5}$γ$_6$ | t$_{0,4}$γ$_6$ | t$_{0,3}$γ$_6$ | t$_{0,2}$γ$_6$ | t$_{0,1}$γ$_6$ | t$_{0,1,2}$β$_6$ | t$_{0,1,3}$β$_6$ | t$_{0,2,3}$β$_6$ |
| t$_{1,2,3}$β$_6$ | t$_{0,1,4}$β$_6$ | t$_{0,2,4}$β$_6$ | t$_{1,2,4}$β$_6$ | t$_{0,3,4}$β$_6$ | t$_{1,2,4}$γ$_6$ | t$_{1,2,3}$γ$_6$ | t$_{0,1,5}$β$_6$ |
| t$_{0,2,5}$β$_6$ | t$_{0,3,4}$γ$_6$ | t$_{0,2,5}$γ$_6$ | t$_{0,2,4}$γ$_6$ | t$_{0,2,3}$γ$_6$ | t$_{0,1,5}$γ$_6$ | t$_{0,1,4}$γ$_6$ | t$_{0,1,3}$γ$_6$ |
| t$_{0,1,2}$γ$_6$ | t$_{0,1,2,3}$β$_6$ | t$_{0,1,2,4}$β$_6$ | t$_{0,1,3,4}$β$_6$ | t$_{0,2,3,4}$β$_6$ | t$_{1,2,3,4}$γ$_6$ | t$_{0,1,2,5}$β$_6$ | t$_{0,1,3,5}$β$_6$ |
| t$_{0,2,3,5}$γ$_6$ | t$_{0,2,3,4}$γ$_6$ | t$_{0,1,4,5}$γ$_6$ | t$_{0,1,3,5}$γ$_6$ | t$_{0,1,3,4}$γ$_6$ | t$_{0,1,2,5}$γ$_6$ | t$_{0,1,2,4}$γ$_6$ | t$_{0,1,2,3}$γ$_6$ |
| t$_{0,1,2,3,4}$β$_6$ | t$_{0,1,2,3,5}$β$_6$ | t$_{0,1,2,4,5}$β$_6$ | t$_{0,1,2,4,5}$γ$_6$ | t$_{0,1,2,3,5}$γ$_6$ | t$_{0,1,2,3,4}$γ$_6$ | t$_{0,1,2,3,4,5}$γ$_6$ |