Truncated 7-cubes: Difference between revisions
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== External links == |
== External links == |
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*{{GlossaryForHyperspace | anchor=Cross | title=Cross polytope }} |
*{{GlossaryForHyperspace | anchor=Cross | title=Cross polytope }} |
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* [http:// |
* [http://www.polytope.net/hedrondude/topes.htm Polytopes of Various Dimensions] |
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* [http://tetraspace.alkaline.org/glossary.htm Multi-dimensional Glossary] |
* [http://tetraspace.alkaline.org/glossary.htm Multi-dimensional Glossary] |
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Revision as of 02:10, 2 June 2011
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 BC7 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
Truncated 7-cube | |
---|---|
Type | uniform polyexon |
Schläfli symbol | t0,1{4,3,3,3,3,3} |
Coxeter-Dynkin diagrams | |
6-faces | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | 3136 |
Vertices | 896 |
Vertex figure | |
Coxeter groups | BC7, [3,3,3,3,3,4] |
Properties | convex |
Alternate names
- Truncated hepteract (Jonathan Bowers)[1]
Coordinates
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
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] |
Bitruncated 7-cube
Bitruncated 7-cube | |
---|---|
Type | uniform polyexon |
Schläfli symbol | t1,2{4,3,3,3,3,3} |
Coxeter-Dynkin diagrams | |
6-faces | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | 9408 |
Vertices | 2688 |
Vertex figure | |
Coxeter groups | BC7, [3,3,3,3,3,4] D7 |
Properties | convex |
Alternate names
- Bitruncated hepteract (Jonathan Bowers)[2]
Coordinates
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
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] |
Tritruncated 7-cube
Tritruncated 7-cube | |
---|---|
Type | uniform polyexon |
Schläfli symbol | t2,3{4,3,3,3,3,3} |
Coxeter-Dynkin diagrams | |
6-faces | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | 13440 |
Vertices | 3360 |
Vertex figure | |
Coxeter groups | BC7, [3,3,3,3,3,4] D7 |
Properties | convex |
Alternate names
- Tritruncated hepteract (Jonathan Bowers)[3]
Coordinates
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
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
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, editied 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. "7D uniform polytopes (polyexa)". o3o3o3o3o3x4x - taz, o3o3o3o3x3x4o - botaz, o3o3o3x3x3o4o - totaz
External links
- Olshevsky, George. "Cross polytope". Glossary for Hyperspace. Archived from the original on 4 February 2007.
- Polytopes of Various Dimensions
- Multi-dimensional Glossary