Rectified 8-cubes
8-cube |
Rectified 8-cube |
Birectified 8-cube |
Trirectified 8-cube |
Trirectified 8-orthoplex |
Birectified 8-orthoplex |
Rectified 8-orthoplex |
8-orthoplex |
Orthogonal projections in BC8 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
Rectified 8-cube | |
---|---|
Type | uniform 8-polytope |
Schläfli symbol | t1{4,3,3,3,3,3,3} |
Coxeter-Dynkin diagrams | |
7-faces | |
6-faces | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | |
Vertices | |
Vertex figure | 6-simplex prism |
Coxeter groups | C8, [36,4] D8, [35,1,1] |
Properties | convex |
Alternate names
- rectified octeract
Images
B8 | B7 | ||||
---|---|---|---|---|---|
[16] | [14] | ||||
B6 | B5 | ||||
[12] | [10] | ||||
B4 | B3 | B2 | |||
[8] | [6] | [4] | |||
A7 | A5 | A3 | |||
[8] | [6] | [4] |
Birectified 8-cube
Birectified 8-cube | |
---|---|
Type | uniform 8-polytope |
Coxeter symbol | 0511 |
Schläfli symbol | t2{4,3,3,3,3,3,3} |
Coxeter-Dynkin diagrams | |
7-faces | |
6-faces | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | |
Vertices | |
Vertex figure | {3,3,3,3}x{4} |
Coxeter groups | C8, [36,4] D8, [35,1,1] |
Properties | convex |
Alternate names
- birectified octeract
Images
B8 | B7 | ||||
---|---|---|---|---|---|
[16] | [14] | ||||
B6 | B5 | ||||
[12] | [10] | ||||
B4 | B3 | B2 | |||
[8] | [6] | [4] | |||
A7 | A5 | A3 | |||
[8] | [6] | [4] |
Trirectified 8-cube
Triectified 8-cube | |
---|---|
Type | uniform 8-polytope |
Schläfli symbol | t3{4,3,3,3,3,3,3} |
Coxeter-Dynkin diagrams | |
7-faces | |
6-faces | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | |
Vertices | |
Vertex figure | {3,3,3}x{3,4} |
Coxeter groups | C8, [36,4] D8, [35,1,1] |
Properties | convex |
Alternate names
- trirectified octeract
Images
B8 | B7 | ||||
---|---|---|---|---|---|
[16] | [14] | ||||
B6 | B5 | ||||
[12] | [10] | ||||
B4 | B3 | B2 | |||
[8] | [6] | [4] | |||
A7 | A5 | A3 | |||
[8] | [6] | [4] |
Notes
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 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
External links
- Olshevsky, George. "Cross polytope". Glossary for Hyperspace. Archived from the original on 4 February 2007.
- Polytopes of Various Dimensions
- Multi-dimensional Glossary