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Cantellated 7-simplexes
| Orthogonal projections in A7 Coxeter plane |
|---|
In seven-dimensional geometry, a cantellated 7-simplex is a convex uniform 7-polytope, being a cantellation of the regular 7-simplex.
There are unique 6 degrees of cantellation for the 7-simplex, including truncations.
Cantellated 7-simplex
| Cantellated 7-simplex |
|---|
| Type |
| Schläfli symbol |
| Coxeter-Dynkin diagram |
| 6-faces |
| 5-faces |
| 4-faces |
| Cells |
| Faces |
| Edges |
| Vertices |
| Vertex figure |
| Coxeter groups |
| Properties |
Alternate names
- Small rhombated octaexon (acronym: saro) (Jonathan Bowers)
Coordinates
The vertices of the cantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,0,1,1,2). This construction is based on facets of the cantellated 8-orthoplex.
Images
Bicantellated 7-simplex
| Bicantellated 7-simplex |
|---|
| Type |
| Schläfli symbol |
| Coxeter-Dynkin diagrams |
| 6-faces |
| 5-faces |
| 4-faces |
| Cells |
| Faces |
| Edges |
| Vertices |
| Vertex figure |
| Coxeter groups |
| Properties |
Alternate names
- Small birhombated octaexon (acronym: sabro) (Jonathan Bowers)
Coordinates
The vertices of the bicantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,1,1,2,2). This construction is based on facets of the bicantellated 8-orthoplex.
Images
Tricantellated 7-simplex
| Tricantellated 7-simplex |
|---|
| Type |
| Schläfli symbol |
| Coxeter-Dynkin diagrams |
| 6-faces |
| 5-faces |
| 4-faces |
| Cells |
| Faces |
| Edges |
| Vertices |
| Vertex figure |
| Coxeter groups |
| Properties |
Alternate names
- Small trirhombihexadecaexon (stiroh) (Jonathan Bowers)
Coordinates
The vertices of the tricantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,1,2,2,2). This construction is based on facets of the tricantellated 8-orthoplex.
Images
Cantitruncated 7-simplex
| Cantitruncated 7-simplex |
|---|
| Type |
| Schläfli symbol |
| Coxeter-Dynkin diagrams |
| 6-faces |
| 5-faces |
| 4-faces |
| Cells |
| Faces |
| Edges |
| Vertices |
| Vertex figure |
| Coxeter groups |
| Properties |
Alternate names
- Great rhombated octaexon (acronym: garo) (Jonathan Bowers)
Coordinates
The vertices of the cantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,0,1,2,3). This construction is based on facets of the cantitruncated 8-orthoplex.
Images
Bicantitruncated 7-simplex
| Bicantitruncated 7-simplex |
|---|
| Type |
| Schläfli symbol |
| Coxeter-Dynkin diagrams |
| 6-faces |
| 5-faces |
| 4-faces |
| Cells |
| Faces |
| Edges |
| Vertices |
| Vertex figure |
| Coxeter groups |
| Properties |
Alternate names
- Great birhombated octaexon (acronym: gabro) (Jonathan Bowers)
Coordinates
The vertices of the bicantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,1,2,3,3). This construction is based on facets of the bicantitruncated 8-orthoplex.
Images
Tricantitruncated 7-simplex
| Tricantitruncated 7-simplex |
|---|
| Type |
| Schläfli symbol |
| Coxeter-Dynkin diagrams |
| 6-faces |
| 5-faces |
| 4-faces |
| Cells |
| Faces |
| Edges |
| Vertices |
| Vertex figure |
| Coxeter groups |
| Properties |
Alternate names
- Great trirhombihexadecaexon (acronym: gatroh) (Jonathan Bowers)
Coordinates
The vertices of the tricantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,2,3,4,4). This construction is based on facets of the tricantitruncated 8-orthoplex.
Images
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, wiley.com,
- (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.
- x3o3x3o3o3o3o - saro, o3x3o3x3o3o3o - sabro, o3o3x3o3x3o3o - stiroh, x3x3x3o3o3o3o - garo, o3x3x3x3o3o3o - gabro, o3o3x3x3x3o3o - gatroh
References
- Klitizing, (x3o3x3o3o3o3o - saro)
- Klitizing, (o3x3o3x3o3o3o - sabro)
- Klitizing, (o3o3x3o3x3o3o - stiroh)
- Klitizing, (x3x3x3o3o3o3o - garo)
- Klitizing, (o3x3x3x3o3o3o - gabro)
- Klitizing, (o3o3x3x3x3o3o - gatroh)
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