Truncated 7-cubes
Uniform 7- polytope
7-cube |
Truncated 7-cube |
Bitruncated 7-cube |
Tritruncated 7-cube |
7-orthoplex |
Truncated 7-orthoplex |
Bitruncated 7-orthoplex |
Tritruncated 7-orthoplex |
| Orthogonal projections in B7 Coxeter plane | |||
|---|---|---|---|
In seven-dimensional geometry, a truncated 7-cube is a convex uniform 7-polytope, being a truncation of the regular 7-cube.
There are 6 truncations for the 7-cube. Vertices of the truncated 7-cube are located as pairs on the edge of the 7-cube. Vertices of the bitruncated 7-cube are located on the square faces of the 7-cube. Vertices of the tritruncated 7-cube are located inside the cubic cells of the 7-cube. The final three truncations are best expressed relative to the 7-orthoplex.
01Truncated 7-cube
| Truncated 7-cube | |
|---|---|
| Type | uniform 7-polytope |
| Schläfli symbol | t{4,35} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 3136 |
| Vertices | 896 |
| Vertex figure | Elongated 5-simplex pyramid |
| Coxeter groups | B7, [35,4] |
| Properties | convex |
Alternate names
- Truncated hepteract (Jonathan Bowers)
Coordinates
Cartesian coordinates for the vertices of a truncated 7-cube, centered at the origin, are all sign and coordinate permutations of
- (1,1+√2,1+√2,1+√2,1+√2,1+√2,1+√2)
Images
| Coxeter plane | B7 / A6 | B6 / D7 | B5 / D6 / A4 |
|---|---|---|---|
| Graph | |||
| Dihedral symmetry | [14] | [12] | [10] |
| Coxeter plane | B4 / D5 | B3 / D4 / A2 | B2 / D3 |
| Graph | |||
| Dihedral symmetry | [8] | [6] | [4] |
| Coxeter plane | A5 | A3 | |
| Graph | |||
| Dihedral symmetry | [6] | [4] |
Related polytopes
The truncated 7-cube, is sixth 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} |
02Bitruncated 7-cube
| Bitruncated 7-cube | |
|---|---|
| Type | uniform 7-polytope |
| Schläfli symbol | 2t{4,35} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 9408 |
| Vertices | 2688 |
| Vertex figure | { }v{3,3,3} |
| Coxeter groups | B7, [35,4] D7, [34,1,1] |
| Properties | convex |
Alternate names
- Bitruncated hepteract (Jonathan Bowers)
Coordinates
Cartesian coordinates for the vertices of a bitruncated 7-cube, centered at the origin, are all sign and coordinate permutations of
- (±2,±2,±2,±2,±2,±1,0)
Images
| Coxeter plane | B7 / A6 | B6 / D7 | B5 / D6 / A4 |
|---|---|---|---|
| Graph | |||
| Dihedral symmetry | [14] | [12] | [10] |
| Coxeter plane | B4 / D5 | B3 / D4 / A2 | B2 / D3 |
| Graph | |||
| Dihedral symmetry | [8] | [6] | [4] |
| Coxeter plane | A5 | A3 | |
| Graph | |||
| Dihedral symmetry | [6] | [4] |
Related polytopes
The bitruncated 7-cube is fifth 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} |
03Tritruncated 7-cube
| Tritruncated 7-cube | |
|---|---|
| Type | uniform 7-polytope |
| Schläfli symbol | 3t{4,35} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 13440 |
| Vertices | 3360 |
| Vertex figure | {4}v{3,3} |
| Coxeter groups | B7, [35,4] D7, [34,1,1] |
| Properties | convex |
Alternate names
- Tritruncated hepteract (Jonathan Bowers)
Coordinates
Cartesian coordinates for the vertices of a tritruncated 7-cube, centered at the origin, are all sign and coordinate permutations of
- (±2,±2,±2,±2,±1,0,0)
Images
| Coxeter plane | B7 / A6 | B6 / D7 | B5 / D6 / A4 |
|---|---|---|---|
| Graph | |||
| Dihedral symmetry | [14] | [12] | [10] |
| Coxeter plane | B4 / D5 | B3 / D4 / A2 | B2 / D3 |
| Graph | |||
| Dihedral symmetry | [8] | [6] | [4] |
| Coxeter plane | A5 | A3 | |
| Graph | |||
| Dihedral symmetry | [6] | [4] |
Sources and credits
This article is adapted from the Wikipedia article “Truncated 7-cubes”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.
Images, from Wikimedia Commons:
- 7-cube t0.svg by self, Public domain
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