7-cube | Rectified 7-cube | Birectified 7-cube | Trirectified 7-cube |
Birectified 7-orthoplex | Rectified 7-orthoplex | 7-orthoplex | |
| Orthogonal projections in B7 Coxeter plane | |||
|---|---|---|---|
In seven-dimensional geometry, a rectified 7-cube is a convex uniform 7-polytope, being a rectification of the regular 7-cube.
There are unique 7 degrees of rectifications, the zeroth being the 7-cube, and the 6th and last being the 7-cube. Vertices of the rectified 7-cube are located at the edge-centers of the 7-ocube. Vertices of the birectified 7-cube are located in the square face centers of the 7-cube. Vertices of the trirectified 7-cube are located in the cube cell centers of the 7-cube.
Rectified 7-cube
| Rectified 7-cube | |
|---|---|
| Type | uniform 7-polytope |
| Schläfli symbol | r{4,3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 6-faces | 128 + 14 |
| 5-faces | 896 + 84 |
| 4-faces | 2688 + 280 |
| Cells | 4480 + 560 |
| Faces | 4480 + 672 |
| Edges | 2688 |
| Vertices | 448 |
| Vertex figure | 5-simplex prism |
| Coxeter groups | B7, [3,3,3,3,3,4] |
| Properties | convex |
Alternate names
- rectified hepteract (acronym: rasa) (Jonathan Bowers)[1]
Cartesian coordinates
Cartesian coordinates for the vertices of a rectified 7-cube, centered at the origin, edge length \sqrt{2}\ are all permutations of:
- (±1,±1,±1,±1,±1,±1,0)
Birectified 7-cube
| Birectified 7-cube | |
|---|---|
| Type | uniform 7-polytope |
| Coxeter symbol | 0411 |
| Schläfli symbol | 2r{4,3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 6-faces | 128 + 14 |
| 5-faces | 448 + 896 + 84 |
| 4-faces | 2688 + 2688 + 280 |
| Cells | 6720 + 4480 + 560 |
| Faces | 8960 + 4480 |
| Edges | 6720 |
| Vertices | 672 |
| Vertex figure | {3}x{3,3,3} |
| Coxeter groups | B7, [3,3,3,3,3,4] |
| Properties | convex |
Alternate names
- Birectified hepteract (acronym: bersa) (Jonathan Bowers)[2]
Cartesian coordinates
Cartesian coordinates for the vertices of a birectified 7-cube, centered at the origin, edge length \sqrt{2}\ are all permutations of:
- (±1,±1,±1,±1,±1,0,0)
Trirectified 7-cube
| Trirectified 7-cube | |
|---|---|
| Type | uniform 7-polytope |
| Schläfli symbol | 3r{4,3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 6-faces | 128 + 14 |
| 5-faces | 448 + 896 + 84 |
| 4-faces | 672 + 2688 + 2688 + 280 |
| Cells | 3360 + 6720 + 4480 |
| Faces | 6720 + 8960 |
| Edges | 6720 |
| Vertices | 560 |
| Vertex figure | {3,3}x{3,3} |
| Coxeter groups | B7, [3,3,3,3,3,4] |
| Properties | convex |
Alternate names
- Trirectified hepteract
- Trirectified 7-orthoplex
- Trirectified heptacross (acronym: sez) (Jonathan Bowers)[3]
Cartesian coordinates
Cartesian coordinates for the vertices of a trirectified 7-cube, centered at the origin, edge length \sqrt{2}\ are all permutations of:
- (±1,±1,±1,±1,0,0,0)
Notes
- ^ Klitzing, pp. (o3o3o3o3o3x4o - rasa).
- ^ Klitzing, pp. (o3o3o3o3x3o4o - bersa).
- ^ Klitzing, pp. (o3o3o3x3o3o4o - sez).
References
- 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.
- o3o3o3o3o3x4o - rasa, o3o3o3o3x3o4o - bersa, o3o3o3x3o3o4o - sez