Pentellated 6-cubes
 6-cube
           |
 6-orthoplex
           |
 Pentellated 6-cube
           |
 Pentitruncated 6-cube
           |
 Penticantellated 6-cube
          |
 Penticantitruncated 6-cube
          |
 Pentiruncitruncated 6-cube
          |
 Pentiruncicantellated 6-cube
          |
 Pentiruncicantitruncated 6-cube
          |
 Pentisteritruncated 6-cube
           |
 Pentistericantitruncated 6-cube
          |
 Omnitruncated 6-cube
           |
Orthogonal projections in B6 Coxeter plane |
In six-dimensional geometry, a pentellated 6-cube is a convex uniform 6-polytope with 5th order truncations of the regular 6-cube.
There are unique 16 degrees of pentellations of the 6-cube with permutations of truncations, cantellations, runcinations, and sterications. The simple pentellated 6-cube is also called an expanded 6-cube, constructed by an expansion operation applied to the regular 6-cube. The highest form, the pentisteriruncicantitruncated 6-cube, is called an omnitruncated 6-cube with all of the nodes ringed. Six of them are better constructed from the 6-orthoplex given at pentellated 6-orthoplex.
Pentellated 6-cube
Pentellated 6-cube |
Type | Uniform 6-polytope |
Schläfli symbol | t0,5{4,3,3,3,3} |
Coxeter-Dynkin diagram |            |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | 1920 |
Vertices | 384 |
Vertex figure | 5-cell antiprism |
Coxeter group | B6, [4,3,3,3,3] |
Properties | convex |
Alternate names
- Pentellated 6-orthoplex
- Expanded 6-cube, expanded 6-orthoplex
- Small teri-hexeractihexacontitetrapeton (Acronym: stoxog) (Jonathan Bowers)[1]
Images
Pentitruncated 6-cube
Pentitruncated 6-cube |
Type | uniform 6-polytope |
Schläfli symbol | t0,1,5{4,3,3,3,3} |
Coxeter-Dynkin diagrams |            |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | 8640 |
Vertices | 1920 |
Vertex figure | |
Coxeter groups | B6, [4,3,3,3,3] |
Properties | convex |
Alternate names
- Teritruncated hexeract (Acronym: tacog) (Jonathan Bowers)[2]
Images
Penticantellated 6-cube
Penticantellated 6-cube |
Type | uniform 6-polytope |
Schläfli symbol | t0,2,5{4,3,3,3,3} |
Coxeter-Dynkin diagrams |            |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | 21120 |
Vertices | 3840 |
Vertex figure | |
Coxeter groups | B6, [4,3,3,3,3] |
Properties | convex |
Alternate names
- Terirhombated hexeract (Acronym: topag) (Jonathan Bowers)[3]
Images
Penticantitruncated 6-cube
Penticantitruncated 6-cube |
Type | uniform 6-polytope |
Schläfli symbol | t0,1,2,5{4,3,3,3,3} |
Coxeter-Dynkin diagrams |            |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | 30720 |
Vertices | 7680 |
Vertex figure | |
Coxeter groups | B6, [4,3,3,3,3] |
Properties | convex |
Alternate names
- Terigreatorhombated hexeract (Acronym: togrix) (Jonathan Bowers)[4]
Images
Pentiruncitruncated 6-cube
Pentiruncitruncated 6-cube |
Type | uniform 6-polytope |
Schläfli symbol | t0,1,3,5{4,3,3,3,3} |
Coxeter-Dynkin diagrams |            |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | 151840 |
Vertices | 11520 |
Vertex figure | |
Coxeter groups | B6, [4,3,3,3,3] |
Properties | convex |
Alternate names
- Tericellirhombated hexacontitetrapeton (Acronym: tocrag) (Jonathan Bowers)[5]
Images
Pentiruncicantellated 6-cube
Pentiruncicantellated 6-cube |
Type | uniform 6-polytope |
Schläfli symbol | t0,2,3,5{4,3,3,3,3} |
Coxeter-Dynkin diagrams |            |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | 46080 |
Vertices | 11520 |
Vertex figure | |
Coxeter groups | B6, [4,3,3,3,3] |
Properties | convex |
Alternate names
- Teriprismatorhombi-hexeractihexacontitetrapeton (Acronym: tiprixog) (Jonathan Bowers)[6]
Images
Pentiruncicantitruncated 6-cube
Pentiruncicantitruncated 6-cube |
Type | uniform 6-polytope |
Schläfli symbol | t0,1,2,3,5{4,3,3,3,3} |
Coxeter-Dynkin diagrams |            |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | 80640 |
Vertices | 23040 |
Vertex figure | |
Coxeter groups | B6, [4,3,3,3,3] |
Properties | convex |
Alternate names
- Terigreatoprismated hexeract (Acronym: tagpox) (Jonathan Bowers)[7]
Images
Pentisteritruncated 6-cube
Pentisteritruncated 6-cube |
Type | uniform 6-polytope |
Schläfli symbol | t0,1,4,5{4,3,3,3,3} |
Coxeter-Dynkin diagrams |            |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | 30720 |
Vertices | 7680 |
Vertex figure | |
Coxeter groups | B6, [4,3,3,3,3] |
Properties | convex |
Alternate names
- Tericellitrunki-hexeractihexacontitetrapeton (Acronym: tactaxog) (Jonathan Bowers)[8]
Images
Pentistericantitruncated 6-cube
Pentistericantitruncated 6-cube |
Type | uniform 6-polytope |
Schläfli symbol | t0,1,2,4,5{4,3,3,3,3} |
Coxeter-Dynkin diagrams |            |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | 80640 |
Vertices | 23040 |
Vertex figure | |
Coxeter groups | B6, [4,3,3,3,3] |
Properties | convex |
Alternate names
- Tericelligreatorhombated hexeract (Acronym: tocagrax) (Jonathan Bowers)[9]
Images
Omnitruncated 6-cube
Omnitruncated 6-cube |
Type |
Uniform 6-polytope |
Schläfli symbol | t0,1,2,3,4,5{35} |
Coxeter-Dynkin diagrams |            |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | 138240 |
Vertices | 46080 |
Vertex figure | irregular 5-simplex |
Coxeter group | B6, [4,3,3,3,3] |
Properties | convex, isogonal |
The omnitruncated 6-cube has 5040 vertices, 15120 edges, 16800 faces (4200 hexagons and 1260 squares), 8400 cells, 1806 4-faces, and 126 5-faces. With 5040 vertices, it is the largest of 35 uniform 6-polytopes generated from the regular 6-cube.
Alternate names
- Pentisteriruncicantituncated 6-cube or 6-orthoplex (omnitruncation for 6-polytopes)
- Omnitruncated hexeract
- Great teri-hexeractihexacontitetrapeton (Acronym: gotaxog) (Jonathan Bowers)[10]
Images
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.
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
- ↑ Klitzing, (x4o3o3o3o3x - stoxog)
- ↑ Klitzing, (x4x3o3o3o3x - tacog)
- ↑ Klitzing, (x4o3x3o3o3x - topag)
- ↑ Klitzing, (x4x3x3o3o3x - togrix)
- ↑ Klitzing, (x4x3o3x3o3x - tocrag)
- ↑ Klitzing, (x4o3x3x3o3x - tiprixog)
- ↑ Klitzing, (x4x3x3o3x3x - tagpox)
- ↑ Klitzing, (x4x3o3o3x3x - tactaxog)
- ↑ Klitzing, (x4x3x3o3x3x - tocagrax)
- ↑ Klitzing, (x4x3x3x3x3x - gotaxog)
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)". x4o3o3o3o3x - stoxog, x4x3o3o3o3x - tacog, x4o3x3o3o3x - topag, x4x3x3o3o3x - togrix, x4x3o3x3o3x - tocrag, x4o3x3x3o3x - tiprixog, x4x3x3o3x3x - tagpox, x4x3o3o3x3x - tactaxog, x4x3x3o3x3x - tocagrax, x4x3x3x3x3x - gotaxog
External links