Runcinated 6-cubes
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![]() 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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In six-dimensional geometry, a runcinated 6-cube is a convex uniform 6-polytope with 3rd order truncations (runcination) of the regular 6-cube.
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
[edit]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
[edit]- Small prismated hexeract (spox) (Jonathan Bowers)[1]
Images
[edit]Coxeter plane | B6 | B5 | B4 |
---|---|---|---|
Graph | ![]() |
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Dihedral symmetry | [12] | [10] | [8] |
Coxeter plane | B3 | B2 | |
Graph | ![]() |
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Dihedral symmetry | [6] | [4] | |
Coxeter plane | A5 | A3 | |
Graph | ![]() |
![]() | |
Dihedral symmetry | [6] | [4] |
Biruncinated 6-cube
[edit]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
[edit]- Small biprismated hexeractihexacontatetrapeton (sobpoxog) (Jonathan Bowers)[2]
Images
[edit]Coxeter plane | B6 | B5 | B4 |
---|---|---|---|
Graph | ![]() |
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Dihedral symmetry | [12] | [10] | [8] |
Coxeter plane | B3 | B2 | |
Graph | ![]() |
![]() | |
Dihedral symmetry | [6] | [4] | |
Coxeter plane | A5 | A3 | |
Graph | ![]() |
![]() | |
Dihedral symmetry | [6] | [4] |
Runcitruncated 6-cube
[edit]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
[edit]- Prismatotruncated hexeract (potax) (Jonathan Bowers)[3]
Images
[edit]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] |
Biruncitruncated 6-cube
[edit]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
[edit]- Biprismatotruncated hexeract (boprag) (Jonathan Bowers)[4]
Images
[edit]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] |
Runcicantellated 6-cube
[edit]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
[edit]- Prismatorhombated hexeract (prox) (Jonathan Bowers)[5]
Images
[edit]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] |
Runcicantitruncated 6-cube
[edit]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
[edit]- Great prismated hexeract (gippox) (Jonathan Bowers)[6]
Images
[edit]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] |
Biruncitruncated 6-cube
[edit]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
[edit]- Biprismatotruncated hexeract (boprag) (Jonathan Bowers)[7]
Images
[edit]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] |
Related polytopes
[edit]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.
Notes
[edit]References
[edit]- 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. "6D uniform polytopes (polypeta)". o3o3x3o3o4x - spox, o3x3o3o3x4o - sobpoxog, o3o3x3o3x4x - potax, o3x3o3x3x4o - boprag, o3o3x3x3o4x - prox, o3o3x3x3x4x - gippox, o3x3x3x3x4o - boprag
External links
[edit]- Weisstein, Eric W. "Hypercube". MathWorld.
- Polytopes of Various Dimensions
- Multi-dimensional Glossary