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. 5 are expressed relative to the dual 6-orthoplex.
Runcinated 6-cube
| Runcinated 6-cube |
| Type | Uniform 6-polytope |
| Schläfli symbol | t0,3{4,3,3,3,3} |
| Coxeter-Dynkin diagram | |
| 5-faces | |
| 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)
Images
Orthographic projections| 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] |
Biruncinated 6-cube
| Biruncinated 6-cube |
| Type | Uniform 6-polytope |
| Schläfli symbol | t1,4{4,3,3,3,3} |
| Coxeter-Dynkin diagram | |
| 5-faces | |
| 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)
Images
Orthographic projections| 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] |
Runcitruncated 6-cube
| Runcitruncated 6-cube |
| Type | Uniform 6-polytope |
| Schläfli symbol | t0,1,3{4,3,3,3,3} |
| Coxeter-Dynkin diagram | |
| 5-faces | |
| 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)
Images
Orthographic projections| 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
| Biruncitruncated 6-cube |
| Type | Uniform 6-polytope |
| Schläfli symbol | t1,2,4{4,3,3,3,3} |
| Coxeter-Dynkin diagram | |
| 5-faces | |
| 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)
Images
Orthographic projections| 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
| Runcicantellated 6-cube |
| Type | Uniform 6-polytope |
| Schläfli symbol | t0,2,3{4,3,3,3,3} |
| Coxeter-Dynkin diagram | |
| 5-faces | |
| 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)
Images
Orthographic projections| 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
| Runcicantitruncated 6-cube |
| Type | Uniform 6-polytope |
| Schläfli symbol | t0,1,2,3{4,3,3,3,3} |
| Coxeter-Dynkin diagram | |
| 5-faces | |
| 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)
Images
Orthographic projections| 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] |
Biruncicantitruncated 6-cube
| Biruncicantitruncated 6-cube |
| Type | Uniform 6-polytope |
| Schläfli symbol | t1,2,3,4{4,3,3,3,3} |
| Coxeter-Dynkin diagram | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 23040 |
| Vertices | 5760 |
| Vertex figure | |
| Coxeter group | B6 [4,3,3,3,3] |
| Properties | convex |
Alternate names
- Great biprismated hexeractihexacontatetrapeton (gobpoxog) (Jonathan Bowers)
Images
Orthographic projections| 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
These polytopes are from a set of 63 uniform 6-polytopes generated from the B6 Coxeter plane, including the regular 6-cube and 6-orthoplex.
| B6 polytopes |
|---|
| β6 | t1β6 | t2β6 | t2γ6 | t1γ6 | γ6 | t0,1β6 | t0,2β6 |
| t1,2β6 | t0,3β6 | t1,3β6 | t2,3γ6 | t0,4β6 | t1,4γ6 | t1,3γ6 | t1,2γ6 |
| t0,5γ6 | t0,4γ6 | t0,3γ6 | t0,2γ6 | t0,1γ6 | t0,1,2β6 | t0,1,3β6 | t0,2,3β6 |
| t1,2,3β6 | t0,1,4β6 | t0,2,4β6 | t1,2,4β6 | t0,3,4β6 | t1,2,4γ6 | t1,2,3γ6 | t0,1,5β6 |
| t0,2,5β6 | t0,3,4γ6 | t0,2,5γ6 | t0,2,4γ6 | t0,2,3γ6 | t0,1,5γ6 | t0,1,4γ6 | t0,1,3γ6 |
| t0,1,2γ6 | t0,1,2,3β6 | t0,1,2,4β6 | t0,1,3,4β6 | t0,2,3,4β6 | t1,2,3,4γ6 | t0,1,2,5β6 | t0,1,3,5β6 |
| t0,2,3,5γ6 | t0,2,3,4γ6 | t0,1,4,5γ6 | t0,1,3,5γ6 | t0,1,3,4γ6 | t0,1,2,5γ6 | t0,1,2,4γ6 | t0,1,2,3γ6 |
| t0,1,2,3,4β6 | t0,1,2,3,5β6 | t0,1,2,4,5β6 | t0,1,2,4,5γ6 | t0,1,2,3,5γ6 | t0,1,2,3,4γ6 | t0,1,2,3,4,5γ6 |
Notes
- 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 Ivić 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. . o3o3x3o3o4x - spox, o3x3o3o3x4o - sobpoxog, o3o3x3o3x4x - potax, o3x3o3x3x4o - boprag, o3o3x3x3o4x - prox, o3o3x3x3x4x - gippox, o3x3x3x3x4o - gobpoxog
External links