Trirectified_7-cube
In seven-dimensional geometry, a rectified 7-cube is a convex uniform 7-polytope, being a rectification of the regular 7-cube.
More information Orthogonal projections in B7 Coxeter plane ...
 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 | |||
|---|---|---|---|
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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 | |
|---|---|
| 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]
Images
More information Coxeter plane, B7 / A6 ...
| 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] |
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Cartesian coordinates
Cartesian coordinates for the vertices of a rectified 7-cube, centered at the origin, edge length 
- (±1,±1,±1,±1,±1,±1,0)
More information 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 |
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Alternate names
- Birectified hepteract (Acronym bersa) (Jonathan Bowers)[2]
Images
More information Coxeter plane, B7 / A6 ...
| 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] |
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Cartesian coordinates
Cartesian coordinates for the vertices of a birectified 7-cube, centered at the origin, edge length
- (±1,±1,±1,±1,±1,0,0)
More information 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 |
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Alternate names
- Trirectified hepteract
- Trirectified 7-orthoplex
- Trirectified heptacross (Acronym sez) (Jonathan Bowers)[3]
Images
More information Coxeter plane, B7 / A6 ...
| 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] |
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Cartesian coordinates
Cartesian coordinates for the vertices of a trirectified 7-cube, centered at the origin, edge length
- (±1,±1,±1,±1,0,0,0)
Related polytopes
More information Dim., n ...
| 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 |
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| Images |  |
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| 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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- 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 Ivic Weiss, Wiley-Interscience Publication, 1995, 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)". o3o3o3x3o3o4o - sez, o3o3o3o3x3o4o - bersa, o3o3o3o3o3x4o - rasa
More information Family, Regular polygon ...
Fundamental convex regular and uniform polytopes in dimensions 2–10
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|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Family | An | Bn | I2(p) / Dn | E6 / E7 / E8 / F4 / G2 | Hn | |||||||
| Regular polygon | Triangle | Square | p-gon | Hexagon | Pentagon | |||||||
| Uniform polyhedron | Tetrahedron | Octahedron • Cube | Demicube | Dodecahedron • Icosahedron | ||||||||
| Uniform polychoron | Pentachoron | 16-cell • Tesseract | Demitesseract | 24-cell | 120-cell • 600-cell | |||||||
| Uniform 5-polytope | 5-simplex | 5-orthoplex • 5-cube | 5-demicube | |||||||||
| Uniform 6-polytope | 6-simplex | 6-orthoplex • 6-cube | 6-demicube | 122 • 221 | ||||||||
| Uniform 7-polytope | 7-simplex | 7-orthoplex • 7-cube | 7-demicube | 132 • 231 • 321 | ||||||||
| Uniform 8-polytope | 8-simplex | 8-orthoplex • 8-cube | 8-demicube | 142 • 241 • 421 | ||||||||
| Uniform 9-polytope | 9-simplex | 9-orthoplex • 9-cube | 9-demicube | |||||||||
| Uniform 10-polytope | 10-simplex | 10-orthoplex • 10-cube | 10-demicube | |||||||||
| Uniform n-polytope | n-simplex | n-orthoplex • n-cube | n-demicube | 1k2 • 2k1 • k21 | n-pentagonal polytope | |||||||
| Topics: Polytope families • Regular polytope • List of regular polytopes and compounds | ||||||||||||
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