Cantellated 6-simplexes
6-simplex |
Cantellated 6-simplex |
Bicantellated 6-simplex |
Birectified 6-simplex |
Cantitruncated 6-simplex |
Bicantitruncated 6-simplex |
Orthogonal projections in A6 Coxeter plane |
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In six-dimensional geometry, a cantellated 6-simplex is a convex uniform 6-polytope, being a cantellation of the regular 6-simplex.
There are unique 4 degrees of cantellation for the 6-simplex, including truncations.
Cantellated 6-simplex
Cantellated 6-simplex | |
---|---|
Type | uniform 6-polytope |
Schläfli symbol | rr{3,3,3,3,3} or |
Coxeter-Dynkin diagrams | |
5-faces | 35 |
4-faces | 210 |
Cells | 560 |
Faces | 805 |
Edges | 525 |
Vertices | 105 |
Vertex figure | 5-cell prism |
Coxeter group | A6, [35], order 5040 |
Properties | convex |
Alternate names
- Small rhombated heptapeton (Acronym: sril) (Jonathan Bowers)[1]
Coordinates
The vertices of the cantellated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,0,1,1,2). This construction is based on facets of the cantellated 7-orthoplex.
Images
Ak Coxeter plane | A6 | A5 | A4 |
---|---|---|---|
Graph | |||
Dihedral symmetry | [7] | [6] | [5] |
Ak Coxeter plane | A3 | A2 | |
Graph | |||
Dihedral symmetry | [4] | [3] |
Bicantellated 6-simplex
Bicantellated 6-simplex | |
---|---|
Type | uniform 6-polytope |
Schläfli symbol | 2rr{3,3,3,3,3} or |
Coxeter-Dynkin diagrams | |
5-faces | 49 |
4-faces | 329 |
Cells | 980 |
Faces | 1540 |
Edges | 1050 |
Vertices | 210 |
Vertex figure | |
Coxeter group | A6, [35], order 5040 |
Properties | convex |
Alternate names
- Small prismated heptapeton (Acronym: sabril) (Jonathan Bowers)[3]
Coordinates
The vertices of the bicantellated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,1,1,2,2). This construction is based on facets of the bicantellated 7-orthoplex.
Images
Ak Coxeter plane | A6 | A5 | A4 |
---|---|---|---|
Graph | |||
Dihedral symmetry | [7] | [6] | [5] |
Ak Coxeter plane | A3 | A2 | |
Graph | |||
Dihedral symmetry | [4] | [3] |
Cantitruncated 6-simplex
cantitruncated 6-simplex | |
---|---|
Type | uniform 6-polytope |
Schläfli symbol | tr{3,3,3,3,3} or |
Coxeter-Dynkin diagrams | |
5-faces | 35 |
4-faces | 210 |
Cells | 560 |
Faces | 805 |
Edges | 630 |
Vertices | 210 |
Vertex figure | |
Coxeter group | A6, [35], order 5040 |
Properties | convex |
Alternate names
- Great rhombated heptapeton (Acronym: gril) (Jonathan Bowers)[4]
Coordinates
The vertices of the cantitruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,0,1,2,3). This construction is based on facets of the cantitruncated 7-orthoplex.
Images
Ak Coxeter plane | A6 | A5 | A4 |
---|---|---|---|
Graph | |||
Dihedral symmetry | [7] | [6] | [5] |
Ak Coxeter plane | A3 | A2 | |
Graph | |||
Dihedral symmetry | [4] | [3] |
Bicantitruncated 6-simplex
bicantitruncated 6-simplex | |
---|---|
Type | uniform 6-polytope |
Schläfli symbol | 2tr{3,3,3,3,3} or |
Coxeter-Dynkin diagrams | |
5-faces | 49 |
4-faces | 329 |
Cells | 980 |
Faces | 1540 |
Edges | 1260 |
Vertices | 420 |
Vertex figure | |
Coxeter group | A6, [35], order 5040 |
Properties | convex |
Alternate names
- Great birhombated heptapeton (Acronym: gabril) (Jonathan Bowers)[5]
Coordinates
The vertices of the bicantitruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,1,2,3,3). This construction is based on facets of the bicantitruncated 7-orthoplex.
Images
Ak Coxeter plane | A6 | A5 | A4 |
---|---|---|---|
Graph | |||
Dihedral symmetry | [7] | [6] | [5] |
Ak Coxeter plane | A3 | A2 | |
Graph | |||
Dihedral symmetry | [4] | [3] |
Related uniform 6-polytopes
The truncated 6-simplex is one of 35 uniform 6-polytopes based on the [3,3,3,3,3] Coxeter group, all shown here in A6 Coxeter plane orthographic projections.
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
- (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. "6D uniform polytopes (polypeta)". x3o3x3o3o3o - sril, o3x3o3x3o3o - sabril, x3x3x3o3o3o - gril, o3x3x3x3o3o - gabril
External links
- Olshevsky, George. "Cross polytope". Glossary for Hyperspace. Archived from the original on 4 February 2007.
- Polytopes of Various Dimensions
- Multi-dimensional Glossary
Fundamental convex regular and uniform polytopes in dimensions 2–10 | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Family | An | Bn | I2(p) / Dn | E6 / E7 / E8 / E9 / E10 / F4 / G2 | Hn | |||||||
Regular polygon | Triangle | Square | p-gon | Hexagon | Pentagon | |||||||
Uniform polyhedron | Tetrahedron | Octahedron • Cube | Demicube | Dodecahedron • Icosahedron | ||||||||
Uniform 4-polytope | 5-cell | 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 |