Biruncitruncated_6-cube
In six-dimensional geometry, a runcinated 6-cube is a convex uniform 6-polytope with 3rd order truncations (runcination) of the regular 6-cube.
More information Orthogonal projections in B6 Coxeter plane ...
 6-cube     |
 Runcinated 6-cube |
 Biruncinated 6-cube |
 Runcinated 6-orthoplex |
 6-orthoplex |
 Runcitruncated 6-cube |
 Biruncitruncated 6-cube |
 Runcicantellated 6-orthoplex |
 Runcicantellated 6-cube |
 Biruncitruncated 6-orthoplex |
 Runcitruncated 6-orthoplex |
 Runcicanti-truncated 6-cube |
 Biruncicanti-truncated 6-cube |
 Runcicanti-truncated 6-orthoplex | |
| Orthogonal projections in B6 Coxeter plane | ||||
|---|---|---|---|---|
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There are 12 unique runcinations of the 6-cube with permutations of truncations, and cantellations. Half are expressed relative to the dual 6-orthoplex.
| Runcinated 6-cube | |
| Type | Uniform 6-polytope |
| Schläfli symbol | t0,3{4,3,3,3,3} |
| Coxeter-Dynkin diagram | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 7680 |
| Vertices | 1280 |
| Vertex figure | |
| Coxeter group | B6 [4,3,3,3,3] |
| Properties | convex |
Alternate names
- Small prismated hexeract (spox) (Jonathan Bowers)[1]
Images
More information Coxeter plane, B6 ...
| Coxeter plane | B6 | B5 | B4 |
|---|---|---|---|
| Graph | |
 |
 |
| Dihedral symmetry | [12] | [10] | [8] |
| Coxeter plane | B3 | B2 | |
| Graph |  |
 | |
| Dihedral symmetry | [6] | [4] | |
| Coxeter plane | A5 | A3 | |
| Graph |  |
 | |
| Dihedral symmetry | [6] | [4] |
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| Biruncinated 6-cube | |
| Type | Uniform 6-polytope |
| Schläfli symbol | t1,4{4,3,3,3,3} |
| Coxeter-Dynkin diagram | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 11520 |
| Vertices | 1920 |
| Vertex figure | |
| Coxeter group | B6 [4,3,3,3,3] |
| Properties | convex |
Alternate names
- Small biprismated hexeractihexacontatetrapeton (sobpoxog) (Jonathan Bowers)[2]
Images
More information Coxeter plane, B6 ...
| Coxeter plane | B6 | B5 | B4 |
|---|---|---|---|
| Graph | |||
| Dihedral symmetry | [12] | [10] | [8] |
| Coxeter plane | B3 | B2 | |
| Graph | |||
| Dihedral symmetry | [6] | [4] | |
| Coxeter plane | A5 | A3 | |
| Graph | |||
| Dihedral symmetry | [6] | [4] |
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| Runcitruncated 6-cube | |
| Type | Uniform 6-polytope |
| Schläfli symbol | t0,1,3{4,3,3,3,3} |
| Coxeter-Dynkin diagram | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 17280 |
| Vertices | 3840 |
| Vertex figure | |
| Coxeter group | B6 [4,3,3,3,3] |
| Properties | convex |
Alternate names
- Prismatotruncated hexeract (potax) (Jonathan Bowers)[3]
Images
More information Coxeter plane, B6 ...
| Coxeter plane | B6 | B5 | B4 |
|---|---|---|---|
| Graph | |||
| Dihedral symmetry | [12] | [10] | [8] |
| Coxeter plane | B3 | B2 | |
| Graph | |||
| Dihedral symmetry | [6] | [4] | |
| Coxeter plane | A5 | A3 | |
| Graph | |||
| Dihedral symmetry | [6] | [4] |
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| Biruncitruncated 6-cube | |
| Type | Uniform 6-polytope |
| Schläfli symbol | t1,2,4{4,3,3,3,3} |
| Coxeter-Dynkin diagram | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 23040 |
| Vertices | 5760 |
| Vertex figure | |
| Coxeter group | B6 [4,3,3,3,3] |
| Properties | convex |
Alternate names
- Biprismatotruncated hexeract (boprag) (Jonathan Bowers)[4]
Images
More information Coxeter plane, B6 ...
| Coxeter plane | B6 | B5 | B4 |
|---|---|---|---|
| Graph | |||
| Dihedral symmetry | [12] | [10] | [8] |
| Coxeter plane | B3 | B2 | |
| Graph | |||
| Dihedral symmetry | [6] | [4] | |
| Coxeter plane | A5 | A3 | |
| Graph | |||
| Dihedral symmetry | [6] | [4] |
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| Runcicantellated 6-cube | |
| Type | Uniform 6-polytope |
| Schläfli symbol | t0,2,3{4,3,3,3,3} |
| Coxeter-Dynkin diagram | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 13440 |
| Vertices | 3840 |
| Vertex figure | |
| Coxeter group | B6 [4,3,3,3,3] |
| Properties | convex |
Alternate names
- Prismatorhombated hexeract (prox) (Jonathan Bowers)[5]
Images
More information Coxeter plane, B6 ...
| Coxeter plane | B6 | B5 | B4 |
|---|---|---|---|
| Graph | |||
| Dihedral symmetry | [12] | [10] | [8] |
| Coxeter plane | B3 | B2 | |
| Graph | |||
| Dihedral symmetry | [6] | [4] | |
| Coxeter plane | A5 | A3 | |
| Graph | |||
| Dihedral symmetry | [6] | [4] |
Close
| Runcicantitruncated 6-cube | |
| Type | Uniform 6-polytope |
| Schläfli symbol | t0,1,2,3{4,3,3,3,3} |
| Coxeter-Dynkin diagram | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 23040 |
| Vertices | 7680 |
| Vertex figure | |
| Coxeter group | B6 [4,3,3,3,3] |
| Properties | convex |
Alternate names
- Great prismated hexeract (gippox) (Jonathan Bowers)[6]
Images
More information Coxeter plane, B6 ...
| Coxeter plane | B6 | B5 | B4 |
|---|---|---|---|
| Graph | |||
| Dihedral symmetry | [12] | [10] | [8] |
| Coxeter plane | B3 | B2 | |
| Graph | |||
| Dihedral symmetry | [6] | [4] | |
| Coxeter plane | A5 | A3 | |
| Graph | |||
| Dihedral symmetry | [6] | [4] |
Close
| Biruncitruncated 6-cube | |
| Type | Uniform 6-polytope |
| Schläfli symbol | t1,2,3,4{4,3,3,3,3} |
| Coxeter-Dynkin diagram | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 23040 |
| Vertices | 5760 |
| Vertex figure | |
| Coxeter group | B6 [4,3,3,3,3] |
| Properties | convex |
Alternate names
- Biprismatotruncated hexeract (boprag) (Jonathan Bowers)[7]
Images
More information Coxeter plane, B6 ...
| Coxeter plane | B6 | B5 | B4 |
|---|---|---|---|
| Graph | |||
| Dihedral symmetry | [12] | [10] | [8] |
| Coxeter plane | B3 | B2 | |
| Graph | |||
| Dihedral symmetry | [6] | [4] | |
| Coxeter plane | A5 | A3 | |
| Graph | |||
| Dihedral symmetry | [6] | [4] |
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These polytopes are from a set of 63 uniform 6-polytopes generated from the B6 Coxeter plane, including the regular 6-cube or 6-orthoplex.
More information B6 polytopes ...
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- 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)". o3o3x3o3o4x - spox, o3x3o3o3x4o - sobpoxog, o3o3x3o3x4x - potax, o3x3o3x3x4o - boprag, o3o3x3x3o4x - prox, o3o3x3x3x4x - gippox, o3x3x3x3x4o - boprag
- Weisstein, Eric W. "Hypercube". MathWorld.
- Polytopes of Various Dimensions
- Multi-dimensional Glossary
More information Family, Regular polygon ...
Fundamental convex regular and uniform polytopes in dimensions 2–10
| ||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Family | An | Bn | I2(p) / Dn | E6 / E7 / E8 / F4 / G2 | Hn | |||||||
| Regular polygon | Triangle | Square | p-gon | Hexagon | Pentagon | |||||||
| Uniform polyhedron | Tetrahedron | Octahedron • Cube | Demicube | Dodecahedron • Icosahedron | ||||||||
| Uniform polychoron | Pentachoron | 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 | ||||||||||||
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