- Hexicated 7-cube
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Orthogonal projections in BC4 Coxeter plane 
7-cube
Hexicated 7-cube
Hexitruncated 7-cube
Hexicantellated 7-cube
Hexiruncinated 7-cube
Hexicantitruncated 7-cube
Hexiruncitruncated 7-cube
Hexiruncicantellated 7-cube
Hexisteritruncated 7-cube
Hexistericantellated 7-cube
Hexipentitruncated 7-cube
Hexiruncicantitruncated 7-cube
Hexistericantitruncated 7-cube
Hexisteriruncitruncated 7-cube
Hexisteriruncicantellated 7-cube
Hexipenticantitruncated 7-cube
Hexipentiruncitruncated 7-cube
Hexisteriruncicantitruncated 7-cube
Hexipentiruncicantitruncated 7-cube
Hexipentistericantitruncated 7-cube
Hexipentisteriruncicantitruncated 7-cube
(Omnitruncated 7-cube)
In seven-dimensional geometry, a hexicated 7-cube is a convex uniform 7-polytope, including 6th-order truncations (hexication) from the regular 7-cube.
There are 32 hexications for the 7-cube, including all permutations of truncations, cantellations, runcinations, sterications, and pentellations. 20 are represented here, while 12 are more easily constructed from the 7-orthoplex.
The simple hexicated 7-cube is also called an expanded 7-cube, with only the first and last nodes ringed, is constructed by an expansion operation applied to the regular 7-cube. The highest form, the hexipentisteriruncicantitruncated 7-cube is more simply called a omnitruncated 7-cube with all of the nodes ringed.
These polytope are among a family of 127 uniform 7-polytopes with BC7 symmetry.
Hexicated 7-cube
Hexicated 7-cube Type uniform polyexon Schläfli symbol t0,6{4,35} Coxeter-Dynkin diagrams 6-faces 5-faces 4-faces Cells Faces Edges Vertices Vertex figure Coxeter groups BC7, [4,35] Properties convex In seven-dimensional geometry, a hexicated 7-cube is a convex uniform 7-polytope, a hexication (6th order truncation) of the regular 7-cube, or alternately can be seen as an expansion operation.
Alternate names
- Small petated hepteract (acronym: ) (Jonathan Bowers)
Images
orthographic projections Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4 Graph 
Dihedral symmetry [14] [12] [10] Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3 Graph 
Dihedral symmetry [8] [6] [4] Coxeter plane A5 A3 Graph 
Dihedral symmetry [6] [4] Hexitruncated 7-cube
hexitruncated 7-cube Type uniform polyexon Schläfli symbol t0,1,6{4,35} Coxeter-Dynkin diagrams 6-faces 5-faces 4-faces Cells Faces Edges Vertices Vertex figure Coxeter groups BC7, [4,35] Properties convex Alternate names
- Petitruncated hepteract (acronym: ) (Jonathan Bowers)
Images
orthographic projections Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4 Graph 
Dihedral symmetry [14] [12] [10] Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3 Graph 
Dihedral symmetry [8] [6] [4] Coxeter plane A5 A3 Graph 
Dihedral symmetry [6] [4] Hexicantellated 7-cube
Hexicantellated 7-cube Type uniform polyexon Schläfli symbol t0,2,6{4,35} Coxeter-Dynkin diagrams 6-faces 5-faces 4-faces Cells Faces Edges Vertices Vertex figure Coxeter groups BC7, [4,35] Properties convex Alternate names
- Petirhombated hepteract (acronym: ) (Jonathan Bowers)
Images
orthographic projections Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4 Graph 
Dihedral symmetry [14] [12] [10] Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3 Graph 
Dihedral symmetry [8] [6] [4] Coxeter plane A5 A3 Graph 
Dihedral symmetry [6] [4] Hexiruncinated 7-cube
Hexiruncinated 7-cube Type uniform polyexon Schläfli symbol t0,3,6{4,35} Coxeter-Dynkin diagrams 6-faces 5-faces 4-faces Cells Faces Edges Vertices Vertex figure Coxeter groups BC7, [4,35] Properties convex Alternate names
- Petiprismated hepteract (acronym: ) (Jonathan Bowers)
Images
orthographic projections Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4 Graph 150px 
Dihedral symmetry [14] [12] [10] Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3 Graph 
Dihedral symmetry [8] [6] [4] Coxeter plane A5 A3 Graph 
Dihedral symmetry [6] [4] Hexicantitruncated 7-cube
Hexicantitruncated 7-cube Type uniform polyexon Schläfli symbol t0,1,2,6{4,35} Coxeter-Dynkin diagrams 6-faces 5-faces 4-faces Cells Faces Edges Vertices Vertex figure Coxeter groups BC7, [4,35] Properties convex Alternate names
- Petigreatorhombated hepteract (acronym: ) (Jonathan Bowers)
Images
orthographic projections Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4 Graph 
Dihedral symmetry [14] [12] [10] Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3 Graph 
Dihedral symmetry [8] [6] [4] Coxeter plane A5 A3 Graph 
Dihedral symmetry [6] [4] Hexiruncitruncated 7-cube
Hexiruncitruncated 7-cube Type uniform polyexon Schläfli symbol t0,1,3,6{4,35} Coxeter-Dynkin diagrams 6-faces 5-faces 4-faces Cells Faces Edges Vertices Vertex figure Coxeter groups BC7, [4,35] Properties convex Alternate names
- Petiprismatotruncated hepteract (acronym: ) (Jonathan Bowers)
Images
orthographic projections Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4 Graph 
Dihedral symmetry [14] [12] [10] Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3 Graph 
Dihedral symmetry [8] [6] [4] Coxeter plane A5 A3 Graph 
Dihedral symmetry [6] [4] Hexiruncicantellated 7-cube
Hexiruncicantellated 7-cube Type uniform polyexon Schläfli symbol t0,2,3,6{4,35} Coxeter-Dynkin diagrams 6-faces 5-faces 4-faces Cells Faces Edges Vertices Vertex figure Coxeter groups BC7, [4,35] Properties convex In seven-dimensional geometry, a hexiruncicantellated 7-cube is a uniform 7-polytope.
Alternate names
- Petiprismatorhombated hepteract (acronym: ) (Jonathan Bowers)
Images
orthographic projections Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4 Graph 
Dihedral symmetry [14] [12] [10] Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3 Graph 
Dihedral symmetry [8] [6] [4] Coxeter plane A5 A3 Graph 
Dihedral symmetry [6] [4] Hexisteritruncated 7-cube
hexisteritruncated 7-cube Type uniform polyexon Schläfli symbol t0,1,4,6{4,35} Coxeter-Dynkin diagrams 6-faces 5-faces 4-faces Cells Faces Edges Vertices Vertex figure Coxeter groups BC7, [4,35] Properties convex Alternate names
- Peticellitruncated hepteract (acronym: ) (Jonathan Bowers)
Images
orthographic projections Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4 Graph 
Dihedral symmetry [14] [12] [10] Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3 Graph 
Dihedral symmetry [8] [6] [4] Coxeter plane A5 A3 Graph 
Dihedral symmetry [6] [4] Hexistericantellated 7-cube
hexistericantellated 7-cube Type uniform polyexon Schläfli symbol t0,2,4,6{4,35} Coxeter-Dynkin diagrams 6-faces 5-faces 4-faces Cells Faces Edges Vertices Vertex figure Coxeter groups BC7, [4,35] Properties convex Alternate names
- Peticellirhombihepteract (acronym: ) (Jonathan Bowers)
Images
orthographic projections Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4 Graph 
Dihedral symmetry [14] [12] [10] Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3 Graph 
Dihedral symmetry [8] [6] [4] Coxeter plane A5 A3 Graph 
Dihedral symmetry [6] [4] Hexipentitruncated 7-cube
Hexipentitruncated 7-cube Type uniform polyexon Schläfli symbol t0,1,5,6{4,35} Coxeter-Dynkin diagrams 6-faces 5-faces 4-faces Cells Faces Edges Vertices Vertex figure Coxeter groups BC7, [4,35] Properties convex Alternate names
- Petiteritruncated hepteract (acronym: ) (Jonathan Bowers)
Images
orthographic projections Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4 Graph 
Dihedral symmetry [14] [12] [10] Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3 Graph 
Dihedral symmetry [8] [6] [4] Coxeter plane A5 A3 Graph 
Dihedral symmetry [6] [4] Hexiruncicantitruncated 7-cube
Hexiruncicantitruncated 7-cube Type uniform polyexon Schläfli symbol t0,1,2,3,6{4,35} Coxeter-Dynkin diagrams 6-faces 5-faces 4-faces Cells Faces Edges Vertices Vertex figure Coxeter groups BC7, [4,35] Properties convex Alternate names
- Petigreatoprismated hepteract (acronym: ) (Jonathan Bowers)
Images
orthographic projections Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4 Graph 
Dihedral symmetry [14] [12] [10] Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3 Graph 
Dihedral symmetry [8] [6] [4] Coxeter plane A5 A3 Graph 150px 150px Dihedral symmetry [6] [4] Hexistericantitruncated 7-cube
Hexistericantitruncated 7-cube Type uniform polyexon Schläfli symbol t0,1,2,4,6{4,35} Coxeter-Dynkin diagrams 6-faces 5-faces 4-faces Cells Faces Edges Vertices Vertex figure Coxeter groups BC7, [4,35] Properties convex Alternate names
- Peticelligreatorhombated hepteract (acronym: ) (Jonathan Bowers)
Images
orthographic projections Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4 Graph 150px 
Dihedral symmetry [14] [12] [10] Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3 Graph 
Dihedral symmetry [8] [6] [4] Coxeter plane A5 A3 Graph 
Dihedral symmetry [6] [4] Hexisteriruncitruncated 7-cube
Hexisteriruncitruncated 7-cube Type uniform polyexon Schläfli symbol t0,1,3,4,6{4,35} Coxeter-Dynkin diagrams 6-faces 5-faces 4-faces Cells Faces Edges Vertices Vertex figure Coxeter groups BC7, [4,35] Properties convex Alternate names
- Peticelliprismatotruncated hepteract (acronym: ) (Jonathan Bowers)
Images
orthographic projections Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4 Graph 150px 
Dihedral symmetry [14] [12] [10] Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3 Graph 
Dihedral symmetry [8] [6] [4] Coxeter plane A5 A3 Graph 
Dihedral symmetry [6] [4] Hexisteriruncicantellated 7-cube
Hexisteriruncitruncated 7-cube Type uniform polyexon Schläfli symbol t0,2,3,4,6{4,35} Coxeter-Dynkin diagrams 6-faces 5-faces 4-faces Cells Faces Edges Vertices Vertex figure Coxeter groups BC7, [4,35] Properties convex Alternate names
- Peticelliprismatorhombihepteract (acronym: ) (Jonathan Bowers)
Images
orthographic projections Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4 Graph 150px 
Dihedral symmetry [14] [12] [10] Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3 Graph 
Dihedral symmetry [8] [6] [4] Coxeter plane A5 A3 Graph 
Dihedral symmetry [6] [4] Hexipenticantitruncated 7-cube
hexipenticantitruncated 7-cube Type uniform polyexon Schläfli symbol t0,1,2,5,6{4,35} Coxeter-Dynkin diagrams 6-faces 5-faces 4-faces Cells Faces Edges Vertices Vertex figure Coxeter groups BC7, [4,35] Properties convex Alternate names
- Petiterigreatorhombated hepteract (acronym: ) (Jonathan Bowers)
Images
orthographic projections Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4 Graph 
Dihedral symmetry [14] [12] [10] Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3 Graph 
Dihedral symmetry [8] [6] [4] Coxeter plane A5 A3 Graph 
Dihedral symmetry [6] [4] Hexipentiruncitruncated 7-cube
Hexisteriruncicantitruncated 7-cube Type uniform polyexon Schläfli symbol t0,1,2,3,4,6{4,35} Coxeter-Dynkin diagrams 6-faces 5-faces 4-faces Cells Faces Edges Vertices Vertex figure Coxeter groups BC7, [4,35] Properties convex Alternate names
- Great petacellated hepteract (acronym: ) (Jonathan Bowers)
Images
orthographic projections Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4 Graph 150px 
Dihedral symmetry [14] [12] [10] Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3 Graph 
Dihedral symmetry [8] [6] [4] Coxeter plane A5 A3 Graph 
Dihedral symmetry [6] [4] Hexisteriruncicantitruncated 7-cube
Hexisteriruncicantitruncated 7-cube Type uniform polyexon Schläfli symbol t0,1,2,3,4,6{4,35} Coxeter-Dynkin diagrams 6-faces 5-faces 4-faces Cells Faces Edges Vertices Vertex figure Coxeter groups BC7, [4,35] Properties convex Alternate names
- Great petacellated hepteract (acronym: ) (Jonathan Bowers)
Images
orthographic projections Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4 Graph 150px Dihedral symmetry [14] [12] [10] Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3 Graph Dihedral symmetry [8] [6] [4] Coxeter plane A5 A3 Graph Dihedral symmetry [6] [4] Hexipentiruncicantitruncated 7-cube
Hexipentiruncicantitruncated 7-cube Type uniform polyexon Schläfli symbol t0,1,2,3,5,6{4,35} Coxeter-Dynkin diagrams 6-faces 5-faces 4-faces Cells Faces Edges Vertices Vertex figure Coxeter groups BC7, [4,35] Properties convex Alternate names
- Petiterigreatoprismated hepteract (acronym: ) (Jonathan Bowers)
Images
orthographic projections Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4 Graph 150px 
Dihedral symmetry [14] [12] [10] Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3 Graph 
Dihedral symmetry [8] [6] [4] Coxeter plane A5 A3 Graph 
Dihedral symmetry [6] [4] Hexipentistericantitruncated 7-cube
Hexipentistericantitruncated 7-cube Type uniform polyexon Schläfli symbol t0,1,2,4,5,6{4,35} Coxeter-Dynkin diagrams 6-faces 5-faces 4-faces Cells Faces Edges Vertices Vertex figure Coxeter groups BC7, [4,35] Properties convex Alternate names
- Petitericelligreatorhombihepteract (acronym: putcagroh) (Jonathan Bowers)
Images
orthographic projections Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4 Graph 150px 
Dihedral symmetry [14] [12] [10] Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3 Graph 
Dihedral symmetry [8] [6] [4] Coxeter plane A5 A3 Graph 
Dihedral symmetry [6] [4] Omnitruncated 7-cube
Omnitruncated 7-cube Type uniform polyexon Schläfli symbol t0,1,2,3,4,5,6{36} Coxeter-Dynkin diagrams 6-faces 5-faces 4-faces Cells Faces Edges Vertices Vertex figure Coxeter groups BC7, [4,35] Properties convex The omnitruncated 7-cube is the largest uniform 7-polytope in the BC7 symmetry of the regular 7-cube. It can also be called the hexipentisteriruncicantitruncated 7-cube which is the long name for the omnitruncation for 7 dimensions, with all reflective mirrors active.
Alternate names
- Great petated hepteract (Acronym: ) (Jonathan Bowers)
Images
orthographic projections Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4 Graph 150px 
Dihedral symmetry [14] [12] [10] Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3 Graph 
Dihedral symmetry [8] [6] [4] Coxeter plane A5 A3 Graph 
Dihedral symmetry [6] [4] Notes
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, PhD (1966)
- Richard Klitzing, , 7D uniform polytopes (polyexa) x3o3o3o3o3o4x - , x3x3o3o3o3o3x- , x3o3o3x3o3o4x - , x3x3x3o3o3o4x - , x3x3o3x3o3o4x - , x3o3x3x3o3o4x - , x3o3x3o3o3x4x - , x3o3x3o3x3o4x - , x3x3o3o3o3x4x - , x3x3x3x3o3o4x - , x3x3x3o3x3o4x - , x3x3o3x3x3o4x - , x3o3x3x3x3o4x - , x3x3x3oxo3x4x - , x3x3x3x3x3o4x - , x3x3x3o3x3x4x - , x3x3o3x3x3x4x - , x3x3x3x3x3x4x -
External links
- Olshevsky, George, Cross polytope at Glossary for Hyperspace.
- 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:- 7-polytopes
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