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In eight-dimensional geometry, a truncated 8-cube is a convex uniform 8-polytope, being a truncation of the regular 8-cube.
 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 B8 Coxeter plane | ||||
|---|---|---|---|---|
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 | |
|---|---|
| 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 | B8, [3,3,3,3,3,3,4] |
| Properties | convex |
Alternate names
- Truncated octeract (acronym tocto) (Jonathan Bowers)[1]
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
| B8 | B7 | ||||
|---|---|---|---|---|---|
|
 | ||||
| [16] | [14] | ||||
| B6 | B5 | ||||
 |
 | ||||
| [12] | [10] | ||||
| B4 | B3 | B2 | |||
 |
 |
 | |||
| [8] | [6] | [4] | |||
| A7 | A5 | A3 | |||
 |
 |
 | |||
| [8] | [6] | [4] | |||
Related polytopes
The truncated 8-cube, is seventh in a sequence of 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 | |
|---|---|
| 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 | B8, [3,3,3,3,3,3,4] |
| Properties | convex |
Alternate names
- Bitruncated octeract (acronym bato) (Jonathan Bowers)[2]
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
| B8 | B7 | ||||
|---|---|---|---|---|---|
|
 | ||||
| [16] | [14] | ||||
| B6 | B5 | ||||
 |
 | ||||
| [12] | [10] | ||||
| B4 | B3 | B2 | |||
 |
 |
 | |||
| [8] | [6] | [4] | |||
| A7 | A5 | A3 | |||
 |
 |
 | |||
| [8] | [6] | [4] | |||
Related polytopes
The bitruncated 8-cube is sixth in a sequence of 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 | |
|---|---|
| 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 | B8, [3,3,3,3,3,3,4] |
| Properties | convex |
Alternate names
- Tritruncated octeract (acronym tato) (Jonathan Bowers)[3]
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
| B8 | B7 | ||||
|---|---|---|---|---|---|
|
 | ||||
| [16] | [14] | ||||
| B6 | B5 | ||||
 |
 | ||||
| [12] | [10] | ||||
| B4 | B3 | B2 | |||
 |
 |
 | |||
| [8] | [6] | [4] | |||
| A7 | A5 | A3 | |||
 |
 |
 | |||
| [8] | [6] | [4] | |||
| 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 | B8, [3,3,3,3,3,3,4] D8, [35,1,1] |
| Properties | convex |
Alternate names
- Quadritruncated octeract (acronym oke) (Jonathan Bowers)[4]
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
| B8 | B7 | ||||
|---|---|---|---|---|---|
|
 | ||||
| [16] | [14] | ||||
| B6 | B5 | ||||
 |
 | ||||
| [12] | [10] | ||||
| B4 | B3 | B2 | |||
 |
 |
 | |||
| [8] | [6] | [4] | |||
| A7 | A5 | A3 | |||
 |
 |
 | |||
| [8] | [6] | [4] | |||
Related polytopes
| 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} |
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