Rectified 8-cubes
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 8-cube    
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 Rectified 8-cube
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 Birectified 8-cube
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 Trirectified 8-cube
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 Trirectified 8-orthoplex
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 Birectified 8-orthoplex
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 Rectified 8-orthoplex
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 8-orthoplex
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| Orthogonal projections in B8 Coxeter plane | |||
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In eight-dimensional geometry, a rectified 8-cube is a convex uniform 8-polytope, being a rectification of the regular 8-cube.
There are unique 8 degrees of rectifications, the zeroth being the 8-cube, and the 7th and last being the 8-orthoplex. Vertices of the rectified 8-cube are located at the edge-centers of the 8-cube. Vertices of the birectified 8-cube are located in the square face centers of the 8-cube. Vertices of the trirectified 8-cube are located in the 7-cube cell centers of the 8-cube.
Rectified 8-cube
[edit]| Rectified 8-cube | |
|---|---|
| Type | uniform 8-polytope |
| Schläfli symbol | t1{4,3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams |  
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| 7-faces | 256 + 16 |
| 6-faces | 2048 + 112 |
| 5-faces | 7168 + 448 |
| 4-faces | 14336 + 1120 |
| Cells | 17920 +* 1792 |
| Faces | 4336 + 1792 |
| Edges | 7168 |
| Vertices | 1024 |
| Vertex figure | 6-simplex prism {3,3,3,3,3}×{} |
| Coxeter groups | B8, [36,4] D8, [35,1,1] |
| Properties | convex |
Alternate names
[edit]- rectified octeract
Images
[edit]| B8 | B7 | ||||
|---|---|---|---|---|---|
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| [16] | [14] | ||||
| B6 | B5 | ||||
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| [12] | [10] | ||||
| B4 | B3 | B2 | |||
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| [8] | [6] | [4] | |||
| A7 | A5 | A3 | |||
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| [8] | [6] | [4] | |||
Birectified 8-cube
[edit]| Birectified 8-cube | |
|---|---|
| Type | uniform 8-polytope |
| Coxeter symbol | 0511 |
| Schläfli symbol | t2{4,3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams | 
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| 7-faces | 256 + 16 |
| 6-faces | 1024 + 2048 + 112 |
| 5-faces | 7168 + 7168 + 448 |
| 4-faces | 21504 + 14336 + 1120 |
| Cells | 35840 + 17920 + 1792 |
| Faces | 35840 + 14336 |
| Edges | 21504 |
| Vertices | 1792 |
| Vertex figure | {3,3,3,3}x{4} |
| Coxeter groups | B8, [36,4] D8, [35,1,1] |
| Properties | convex |
Alternate names
[edit]- Birectified octeract
- Rectified 8-demicube
Images
[edit]| B8 | B7 | ||||
|---|---|---|---|---|---|
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| [16] | [14] | ||||
| B6 | B5 | ||||
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| [12] | [10] | ||||
| B4 | B3 | B2 | |||
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| [8] | [6] | [4] | |||
| A7 | A5 | A3 | |||
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| [8] | [6] | [4] | |||
Trirectified 8-cube
[edit]| Triectified 8-cube | |
|---|---|
| Type | uniform 8-polytope |
| Schläfli symbol | t3{4,3,3,3,3,3,3} |
| Coxeter diagrams |
|
| 7-faces | 16+256 |
| 6-faces | 1024 + 2048 + 112 |
| 5-faces | 1792 + 7168 + 7168 + 448 |
| 4-faces | 1792 + 10752 + 21504 +14336 |
| Cells | 8960 + 26880 + 35840 |
| Faces | 17920+35840 |
| Edges | 17920 |
| Vertices | 1152 |
| Vertex figure | {3,3,3}x{3,4} |
| Coxeter groups | B8, [36,4] D8, [35,1,1] |
| Properties | convex |
Alternate names
[edit]- trirectified octeract
Images
[edit]| B8 | B7 | ||||
|---|---|---|---|---|---|
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| [16] | [14] | ||||
| B6 | B5 | ||||
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| [12] | [10] | ||||
| B4 | B3 | B2 | |||
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| [8] | [6] | [4] | |||
| A7 | A5 | A3 | |||
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| [8] | [6] | [4] | |||
Notes
[edit]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 Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [1]
- (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. "8D uniform polytopes (polyzetta)". o3o3o3o3o3o3x4o, o3o3o3o3o3x3o4o, o3o3o3o3x3o3o4o