Truncated 7-cubes
 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.
Truncated 7-cube
[edit]| 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
[edit]- Truncated hepteract (Jonathan Bowers)[1]
Coordinates
[edit]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
[edit]| 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
[edit]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} |
Bitruncated 7-cube
[edit]| 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
[edit]- Bitruncated hepteract (Jonathan Bowers)[2]
Coordinates
[edit]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
[edit]| 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
[edit]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} |
Tritruncated 7-cube
[edit]| 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
[edit]- Tritruncated hepteract (Jonathan Bowers)[3]
Coordinates
[edit]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
[edit]| 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] |
Notes
[edit]- ↑ Klitzing, (x4x3o3o3o3o3o - tasa).
- ↑ Klitzing (o4x3x3o3o3o3o - betsa)
- ↑ Klitzing, (o4o3x3x3o3o3o - tatsa).
References
[edit]- H.S.M. Coxeter:
- H.S.M. Coxeter, Regular Polytopes, 3rd edition, Dover, New York, 1973
- Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivić Weiss, Wiley-Interscience Publication, 1995, wiley.com, ISBN 978-0-471-01003-6
- (Paper 22) H.S.M. Coxeter, Regular and Semi-Regular Polytopes I, [Math. Zeit. 46 (1940) 380–407, MR 2,10]
- (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559–591]
- (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3–45]
- Norman Johnson Uniform Polytopes, Manuscript (1991)
- N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D.
- Klitzing, Richard. "7D uniform polytopes (polyexa) with acronyms". x4x3o3o3o3o3o - tasa, o4x3x3o3o3o3o - betsa, o4o3x3x3o3o3o - tatsa