Truncated 8-cubes
# Truncated 8-cubes

| 8-cube | Truncated 8-cube | Bitruncated 8-cube |
| - | - | - |
| Quadritruncated 8-cube | Tritruncated 8-cube | Tritruncated 8-orthoplex |
| Bitruncated 8-orthoplex | Truncated 8-orthoplex | 8-orthoplex |
| Orthogonal projections in $B_8$ Coxeter plane |||

In eight-dimensional geometry, a truncated 8-cube is a convex uniform 8-polytope, being a truncation of the regular 8-cube.

There are unique 7 degrees of truncation for the 8-cube. Vertices of the truncation 8-cube are located as pairs on the edge of the 8-cube. Vertices of the bitruncated 8-cube are located on the square faces of the 8-cube. Vertices of the tritruncated 7-cube are located inside the cubic cells of the 8-cube. The final truncations are best expressed relative to the 8-orthoplex.

## Truncated 8-cube

Truncated 8-cube :

| Type | uniform 8-polytope |
| - | - |
| Schläfli symbol | t{4,3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | |
| Vertices | |
| Vertex figure | ( )v{3,3,3,3,3} |
| Coxeter groups | $B_8,$ [3,3,3,3,3,3,4] |
| Properties | convex |

### Alternate names

* Truncated octeract (acronym tocto) (Jonathan Bowers)

### Coordinates

Cartesian coordinates for the vertices of a truncated 8-cube, centered at the origin, are all 224 vertices are sign (4) and coordinate (56) permutations of

    (±2,±2,±2,±2,±2,±2,±1,0)

### Images

orthographic projections :

| $B_8$ | $B_8$ | $B_8$ | $B_7$ | $B_7$ | $B_7$ |
| - | - | - | - | - | - |
| [16] | [16] | [16] | [14] | [14] | [14] |
| $B_6$ | $B_6$ | $B_6$ | $B_5$ | $B_5$ | $B_5$ |
| [12] | [12] | [12] | [10] | [10] | [10] |
| $B_4$ | $B_4$ | $B_3$ | $B_3$ | $B_2$ | $B_2$ |
| [8] | [8] | [6] | [6] | [4] | [4] |
| $A_7$ | $A_7$ | $A_5$ | $A_5$ | $A_3$ | $A_3$ |
| [8] | [8] | [6] | [6] | [4] | [4] |

### Related polytopes

The truncated 8-cube, is seventh 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 8-cube

Bitruncated 8-cube :

| Type | uniform 8-polytope |
| - | - |
| Schläfli symbol | 2t{4,3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | |
| Vertices | |
| Vertex figure | { }v{3,3,3,3} |
| Coxeter groups | $B_8,$ [3,3,3,3,3,3,4] |
| Properties | convex |

### Alternate names

* Bitruncated octeract (acronym bato) (Jonathan Bowers)

### Coordinates

Cartesian coordinates for the vertices of a truncated 8-cube, centered at the origin, are all the sign coordinate permutations of

    (±2,±2,±2,±2,±2,±1,0,0)

### Images

orthographic projections :

| $B_8$ | $B_8$ | $B_8$ | $B_7$ | $B_7$ | $B_7$ |
| - | - | - | - | - | - |
| [16] | [16] | [16] | [14] | [14] | [14] |
| $B_6$ | $B_6$ | $B_6$ | $B_5$ | $B_5$ | $B_5$ |
| [12] | [12] | [12] | [10] | [10] | [10] |
| $B_4$ | $B_4$ | $B_3$ | $B_3$ | $B_2$ | $B_2$ |
| [8] | [8] | [6] | [6] | [4] | [4] |
| $A_7$ | $A_7$ | $A_5$ | $A_5$ | $A_3$ | $A_3$ |
| [8] | [8] | [6] | [6] | [4] | [4] |

### Related polytopes

The bitruncated 8-cube is sixth 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 8-cube

Tritruncated 8-cube :

| Type | uniform 8-polytope |
| - | - |
| Schläfli symbol | 3t{4,3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | |
| Vertices | |
| Vertex figure | {4}v{3,3,3} |
| Coxeter groups | $B_8,$ [3,3,3,3,3,3,4] |
| Properties | convex |

### Alternate names

* Tritruncated octeract (acronym tato) (Jonathan Bowers)

### Coordinates

Cartesian coordinates for the vertices of a truncated 8-cube, centered at the origin, are all the sign coordinate permutations of

    (±2,±2,±2,±2,±1,0,0,0)

### Images

orthographic projections :

| $B_8$ | $B_8$ | $B_8$ | $B_7$ | $B_7$ | $B_7$ |
| - | - | - | - | - | - |
| [16] | [16] | [16] | [14] | [14] | [14] |
| $B_6$ | $B_6$ | $B_6$ | $B_5$ | $B_5$ | $B_5$ |
| [12] | [12] | [12] | [10] | [10] | [10] |
| $B_4$ | $B_4$ | $B_3$ | $B_3$ | $B_2$ | $B_2$ |
| [8] | [8] | [6] | [6] | [4] | [4] |
| $A_7$ | $A_7$ | $A_5$ | $A_5$ | $A_3$ | $A_3$ |
| [8] | [8] | [6] | [6] | [4] | [4] |

## Quadritruncated 8-cube

Quadritruncated 8-cube :

| Type | uniform 8-polytope |
| - | - |
| Schläfli symbol | 4t{3,3,3,3,3,3,4} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | |
| Vertices | |
| Vertex figure | {3,4}v{3,3} |
| Coxeter groups | $B_8,$ [3,3,3,3,3,3,4] $D_8,$ $[3^{5,1,1}]$ |
| Properties | convex |

### Alternate names

* Quadritruncated octeract (acronym oke) (Jonathan Bowers)

### Coordinates

Cartesian coordinates for the vertices of a bitruncated 8-orthoplex, centered at the origin, are all sign and coordinate permutations of

    (±2,±2,±2,±2,±1,0,0,0)

### Images

orthographic projections :

| $B_8$ | $B_8$ | $B_8$ | $B_7$ | $B_7$ | $B_7$ |
| - | - | - | - | - | - |
| [16] | [16] | [16] | [14] | [14] | [14] |
| $B_6$ | $B_6$ | $B_6$ | $B_5$ | $B_5$ | $B_5$ |
| [12] | [12] | [12] | [10] | [10] | [10] |
| $B_4$ | $B_4$ | $B_3$ | $B_3$ | $B_2$ | $B_2$ |
| [8] | [8] | [6] | [6] | [4] | [4] |
| $A_7$ | $A_7$ | $A_5$ | $A_5$ | $A_3$ | $A_3$ |
| [8] | [8] | [6] | [6] | [4] | [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} | ... |
