Runcinated 6-orthoplexes
In six-dimensional geometry, a runcinated 6-orthplex is a convex uniform 6-polytope with 3rd order truncations (runcination) of the regular 6-orthoplex.
|  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            |  Runcicantitruncated 6-cube            |  Biruncicantitruncated 6-cube            |  Runcicantitruncated 6-orthoplex            | |
| Orthogonal projections in BC6 Coxeter plane | ||||
|---|---|---|---|---|
There are 12 unique runcinations of the 6-orthoplex with permutations of truncations, and cantellations. Half are expressed relative to the dual 6-cube.
Runcinated 6-orthoplex
    
    Alternate names
    
- Small prismatohexacontatetrapeton (spog) (Jonathan Bowers)[1]
Images
    
| 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-orthoplex
    
    Alternate names
    
- Prismatorhombated hexacontatetrapeton (prog) (Jonathan Bowers)[2]
Images
    
| 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-orthoplex
    
    Alternate names
    
- Prismatotruncated hexacontatetrapeton (potag) (Jonathan Bowers)[3]
Images
    
| 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] | 
Biruncicantellated 6-cube
    
    Alternate names
    
- Great biprismated hexeractihexacontatetrapeton (gobpoxog) (Jonathan Bowers)[4]
Images
    
| 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 or 6-orthoplex.
Notes
    
- Klitzing, (x3o3o3x3o4o - spog)
- Klitzing, (x3o3x3x3o4o - prog)
- Klitzing, (x3x3o3x3o4o - potag)
- Klitzing, (o3x3x3x3x4o - gobpoxog)
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)". x3o3o3x3o4o - spog, x3o3x3x3o4o - prog, x3x3o3x3o4o - potag, o3x3x3x3x4o - gobpoxog
External links
    
- Weisstein, Eric W. "Hypercube". MathWorld.
- Polytopes of Various Dimensions
- Multi-dimensional Glossary
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