Truncated 6-cubes
# Truncated 6-cubes

| 6-cube | Truncated 6-cube | Bitruncated 6-cube | Tritruncated 6-cube |
| - | - | - | - |
| 6-orthoplex | Truncated 6-orthoplex | Bitruncated 6-orthoplex | Tritruncated 6-cube |
| Orthogonal projections in $B_6$ Coxeter plane ||||

In six-dimensional geometry, a truncated 6-cube (or truncated hexeract) is a convex uniform 6-polytope, being a truncation of the regular 6-cube.

There are 5 truncations for the 6-cube. Vertices of the truncated 6-cube are located as pairs on the edge of the 6-cube. Vertices of the bitruncated 6-cube are located on the square faces of the 6-cube. Vertices of the tritruncated 6-cube are located inside the cubic cells of the 6-cube.

## Truncated 6-cube

Truncated 6-cube :

| Type | uniform 6-polytope |
| - | - |
| Class | B6 polytope |
| Schläfli symbol | t{4,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 5-faces | 76 |
| 4-faces | 464 |
| Cells | 1120 |
| Faces | 1520 |
| Edges | 1152 |
| Vertices | 384 |
| Vertex figure | ( )v{3,3,3} |
| Coxeter groups | $B_6,$ [3,3,3,3,4] |
| Properties | convex |

### Alternate names

* Truncated hexeract (Acronym: tox) (Jonathan Bowers)

### Construction and coordinates

The truncated 6-cube may be constructed by truncating the vertices of the 6-cube at $1/({\sqrt {2}}+2)$ of the edge length. A regular 5-simplex replaces each original vertex.

The Cartesian coordinates of the vertices of a truncated 6-cube having edge length 2 are the permutations of:

    $\left(\pm 1,\ \pm (1+{\sqrt {2}}),\ \pm (1+{\sqrt {2}}),\ \pm (1+{\sqrt {2}}),\ \pm (1+{\sqrt {2}}),\ \pm (1+{\sqrt {2}})\right)$

### 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

The truncated 6-cube, is fifth in a sequence of truncated hypercubes:

Truncated hypercubes :

| Image | | | | | | | | ... |
| - | - | - | - | - | - | - | - | - |
| Name | Octagon | Truncated cube | Truncated tesseract | Truncated 5-cube | Truncated 6-cube | Truncated 7-cube | Truncated 8-cube | ... |
| Coxeter diagram | | | | | | | | ... |
| Vertex figure | ( )v( ) | ( )v{ } | ( )v{3} | ( )v{3,3} | ( )v{3,3,3} | ( )v{3,3,3,3} | ( )v{3,3,3,3,3} |

## Bitruncated 6-cube

Bitruncated 6-cube :

| Type | uniform 6-polytope |
| - | - |
| Class | B6 polytope |
| Schläfli symbol | 2t{4,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | |
| Vertices | |
| Vertex figure | { }v{3,3} |
| Coxeter groups | $B_6,$ [3,3,3,3,4] |
| Properties | convex |

### Alternate names

* Bitruncated hexeract (Acronym: botox) (Jonathan Bowers)

### Construction and coordinates

The Cartesian coordinates of the vertices of a bitruncated 6-cube having edge length 2 are the permutations of:

    $\left(0,\ \pm 1,\ \pm 2,\ \pm 2,\ \pm 2,\ \pm 2\right)$

### 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

The bitruncated 6-cube is fourth in a sequence of bitruncated hypercubes:

Bitruncated hypercubes :

| Image | | | | | | | ... |
| - | - | - | - | - | - | - | - |
| Name | Bitruncated cube | Bitruncated tesseract | Bitruncated 5-cube | Bitruncated 6-cube | Bitruncated 7-cube | Bitruncated 8-cube | ... |
| Coxeter | | | | | | | ... |
| Vertex figure | ( )v{ } | { }v{ } | { }v{3} | { }v{3,3} | { }v{3,3,3} | { }v{3,3,3,3} |

## Tritruncated 6-cube

Tritruncated 6-cube :

| Type | uniform 6-polytope |
| - | - |
| Class | B6 polytope |
| Schläfli symbol | 3t{4,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | |
| Vertices | |
| Vertex figure | {3}v{4} |
| Coxeter groups | $B_6,$ [3,3,3,3,4] |
| Properties | convex |

### Alternate names

* Tritruncated hexeract (Acronym: xog) (Jonathan Bowers)

### Construction and coordinates

The Cartesian coordinates of the vertices of a tritruncated 6-cube having edge length 2 are the permutations of:

    $\left(0,\ 0,\ \pm 1,\ \pm 2,\ \pm 2,\ \pm 2\right)$

### 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

2-isotopic hypercubes :

| Dim. | 2 | 3 | 4 | 5 | 6 | 7 | 8 | n |
| - | - | - | - | - | - | - | - | - |
| Name | t{4} | r{4,3} | 2t{4,3,3} | 2r{4,3,3,3} | 3t{4,3,3,3,3} | 3r{4,3,3,3,3,3} | 4t{4,3,3,3,3,3,3} | ... |
| Coxeter diagram | | | | | | | | ... |
| Images | | | | | | | | ... |
| Facets | | {3} {4} | t{3,3} t{3,4} | r{3,3,3} r{3,3,4} | 2t{3,3,3,3} 2t{3,3,3,4} | 2r{3,3,3,3,3} 2r{3,3,3,3,4} | 3t{3,3,3,3,3,3} 3t{3,3,3,3,3,4} | ... |
| Vertex figure | ( )v( ) | { }×{ } | { }v{ } | {3}×{4} | {3}v{4} | {3,3}×{3,4} | {3,3}v{3,4} | ... |

These polytopes are from a set of 63 Uniform 6-polytopes generated from the $B_6$ Coxeter plane, including the regular 6-cube or 6-orthoplex.

B6 polytopes :

| $β_6$ | $t_1β_6$ | $t_2β_6$ | $t_2γ_6$ | $t_1γ_6$ | $γ_6$ | $t_{0,1}β_6$ | $t_{0,2}β_6$ |
| - | - | - | - | - | - | - | - |
| $t_{1,2}β_6$ | $t_{0,3}β_6$ | $t_{1,3}β_6$ | $t_{2,3}γ_6$ | $t_{0,4}β_6$ | $t_{1,4}γ_6$ | $t_{1,3}γ_6$ | $t_{1,2}γ_6$ |
| $t_{0,5}γ_6$ | $t_{0,4}γ_6$ | $t_{0,3}γ_6$ | $t_{0,2}γ_6$ | $t_{0,1}γ_6$ | $t_{0,1,2}β_6$ | $t_{0,1,3}β_6$ | $t_{0,2,3}β_6$ |
| $t_{1,2,3}β_6$ | $t_{0,1,4}β_6$ | $t_{0,2,4}β_6$ | $t_{1,2,4}β_6$ | $t_{0,3,4}β_6$ | $t_{1,2,4}γ_6$ | $t_{1,2,3}γ_6$ | $t_{0,1,5}β_6$ |
| $t_{0,2,5}β_6$ | $t_{0,3,4}γ_6$ | $t_{0,2,5}γ_6$ | $t_{0,2,4}γ_6$ | $t_{0,2,3}γ_6$ | $t_{0,1,5}γ_6$ | $t_{0,1,4}γ_6$ | $t_{0,1,3}γ_6$ |
| $t_{0,1,2}γ_6$ | $t_{0,1,2,3}β_6$ | $t_{0,1,2,4}β_6$ | $t_{0,1,3,4}β_6$ | $t_{0,2,3,4}β_6$ | $t_{1,2,3,4}γ_6$ | $t_{0,1,2,5}β_6$ | $t_{0,1,3,5}β_6$ |
| $t_{0,2,3,5}γ_6$ | $t_{0,2,3,4}γ_6$ | $t_{0,1,4,5}γ_6$ | $t_{0,1,3,5}γ_6$ | $t_{0,1,3,4}γ_6$ | $t_{0,1,2,5}γ_6$ | $t_{0,1,2,4}γ_6$ | $t_{0,1,2,3}γ_6$ |
| $t_{0,1,2,3,4}β_6$ | $t_{0,1,2,3,5}β_6$ | $t_{0,1,2,4,5}β_6$ | $t_{0,1,2,4,5}γ_6$ | $t_{0,1,2,3,5}γ_6$ | $t_{0,1,2,3,4}γ_6$ | $t_{0,1,2,3,4,5}γ_6$ |
