- Pentellated 6-cube
-
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 BC6 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 polypeton 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 BC6, [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
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] Pentitruncated 6-cube
Pentitruncated 6-cube Type uniform polypeton 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 BC6, [4,3,3,3,3] Properties convex Alternate names
- Teritruncated hexeract (Acronym: tacog) (Jonathan Bowers)[2]
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] Penticantellated 6-cube
Penticantellated 6-cube Type uniform polypeton 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 BC6, [4,3,3,3,3] Properties convex Alternate names
- Terirhombated hexeract (Acronym: topag) (Jonathan Bowers)[3]
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] Penticantitruncated 6-cube
Penticantitruncated 6-cube Type uniform polypeton 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 BC6, [4,3,3,3,3] Properties convex Alternate names
- Terigreatorhombated hexeract (Acronym: togrix) (Jonathan Bowers)[4]
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] Pentiruncitruncated 6-cube
Pentiruncitruncated 6-cube Type uniform polypeton 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 BC6, [4,3,3,3,3] Properties convex Alternate names
- Tericellirhombated hexacontitetrapeton (Acronym: tocrag) (Jonathan Bowers)[5]
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] Pentiruncicantellated 6-cube
Pentiruncicantellated 6-cube Type uniform polypeton 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 BC6, [4,3,3,3,3] Properties convex Alternate names
- Teriprismatorhombi-hexeractihexacontitetrapeton (Acronym: tiprixog) (Jonathan Bowers)[6]
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] Pentiruncicantitruncated 6-cube
Pentiruncicantitruncated 6-cube Type uniform polypeton 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 BC6, [4,3,3,3,3] Properties convex Alternate names
- Terigreatoprismated hexeract (Acronym: tagpox) (Jonathan Bowers)[7]
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] Pentisteritruncated 6-cube
Pentisteritruncated 6-cube Type uniform polypeton 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 BC6, [4,3,3,3,3] Properties convex Alternate names
- Tericellitrunki-hexeractihexacontitetrapeton (Acronym: tactaxog) (Jonathan Bowers)[8]
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] Pentistericantitruncated 6-cube
Pentistericantitruncated 6-cube Type uniform polypeton 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 BC6, [4,3,3,3,3] Properties convex Alternate names
- Tericelligreatorhombated hexeract (Acronym: tocagrax) (Jonathan Bowers)[9]
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] 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 BC6, [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
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 polypeta generated from the B6 Coxeter plane, including the regular 6-cube or 6-orthoplex.
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, editied 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.
- Richard Klitzing, 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
- Glossary for hyperspace, George Olshevsky.
- Polytopes of Various Dimensions
- Multi-dimensional Glossary
Fundamental convex regular and uniform polytopes in dimensions 2–10 Family An BCn Dn E6 / E7 / E8 / F4 / G2 Hn Regular polygon Triangle Square Hexagon Pentagon Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron Uniform polychoron 5-cell 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 n-polytopes n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 pentagonal polytope Topics: Polytope families • Regular polytope • List of regular polytopes Categories:- 6-polytopes
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