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Cantellated 7-cubes

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7-cube
Cantellated 7-cube
Bicantellated 7-cube
Tricantellated 7-cube

Birectified 7-cube
Cantitruncated 7-cube
Bicantitruncated 7-cube
Tricantitruncated 7-cube

Cantellated 7-orthoplex
Bicantellated 7-orthoplex
Cantitruncated 7-orthoplex
Bicantitruncated 7-orthoplex
Orthogonal projections in B6 Coxeter plane

In seven-dimensional geometry, a cantellated 7-cube is a convex uniform 7-polytope, being a cantellation of the regular 7-cube.

There are 10 degrees of cantellation for the 7-cube, including truncations. 4 are most simply constructible from the dual 7-orthoplex.

Cantellated 7-cube

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Cantellated 7-cube
Type uniform 7-polytope
Schläfli symbol rr{4,3,3,3,3,3}
Coxeter diagram              
6-faces
5-faces
4-faces
Cells
Faces
Edges 16128
Vertices 2688
Vertex figure
Coxeter groups B7, [4,3,3,3,3,3]
Properties convex

Alternate names

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  • Small rhombated hepteract (acronym: sersa) (Jonathan Bowers)[1]

Images

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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]

Bicantellated 7-cube

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Bicantellated 7-cube
Type uniform 7-polytope
Schläfli symbol r2r{4,3,3,3,3,3}
Coxeter diagrams              
           
6-faces
5-faces
4-faces
Cells
Faces
Edges 40320
Vertices 6720
Vertex figure
Coxeter groups B7, [4,3,3,3,3,3]
Properties convex

Alternate names

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  • Small birhombated hepteract (acronym: sibrosa) (Jonathan Bowers)[2]

Images

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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]

Tricantellated 7-cube

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Tricantellated 7-cube
Type uniform 7-polytope
Schläfli symbol r3r{4,3,3,3,3,3}
Coxeter diagrams              
           
6-faces
5-faces
4-faces
Cells
Faces
Edges 47040
Vertices 6720
Vertex figure
Coxeter groups B7, [4,3,3,3,3,3]
Properties convex

Alternate names

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  • Small trirhombihepteractihecatonicosoctaexon (acronym: strasaz) (Jonathan Bowers)[3]

Images

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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]

Cantitruncated 7-cube

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Cantitruncated 7-cube
Type uniform 7-polytope
Schläfli symbol tr{4,3,3,3,3,3}
Coxeter diagrams              
6-faces
5-faces
4-faces
Cells
Faces
Edges 18816
Vertices 5376
Vertex figure
Coxeter groups B7, [4,3,3,3,3,3]
Properties convex

Alternate names

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  • Great rhombated hepteract (acronym: gersa) (Jonathan Bowers)[4]

Images

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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]

It is fifth in a series of cantitruncated hypercubes:

Petrie polygon projections
                 
Truncated cuboctahedron Cantitruncated tesseract Cantitruncated 5-cube Cantitruncated 6-cube Cantitruncated 7-cube Cantitruncated 8-cube
                                                                 

Bicantitruncated 7-cube

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Bicantitruncated 7-cube
Type uniform 7-polytope
Schläfli symbol r2r{4,3,3,3,3,3}
Coxeter diagrams              
           
6-faces
5-faces
4-faces
Cells
Faces
Edges 47040
Vertices 13440
Vertex figure
Coxeter groups B7, [4,3,3,3,3,3]
Properties convex

Alternate names

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  • Great birhombated hepteract (acronym: gibrosa) (Jonathan Bowers)[5]

Images

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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]

Tricantitruncated 7-cube

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Tricantitruncated 7-cube
Type uniform 7-polytope
Schläfli symbol t3r{4,3,3,3,3,3}
Coxeter diagrams              
           
6-faces
5-faces
4-faces
Cells
Faces
Edges 53760
Vertices 13440
Vertex figure
Coxeter groups B7, [4,3,3,3,3,3]
Properties convex

Alternate names

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  • Great trirhombihepteractihecatonicosoctaexon (acronym: gotrasaz) (Jonathan Bowers)[6]

Images

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orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph too complex    
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]
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These polytopes are from a family of 127 uniform 7-polytopes with B7 symmetry.

See also

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Notes

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  1. ^ Klitizing, (x3o3x3o3o3o4o - sersa)
  2. ^ Klitizing, (o3x3o3x3o3o4o - sibrosa)
  3. ^ Klitizing, (o3o3x3o3x3o4o - strasaz)
  4. ^ Klitizing, (x3x3x3o3o3o4o - gersa)
  5. ^ Klitizing, (o3x3x3x3o3o4o - gibrosa)
  6. ^ Klitizing, (o3o3x3x3x3o4o - gotrasaz)

References

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  • 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 [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.
  • Klitzing, Richard. "7D uniform polytopes (polyexa)". x3o3x3o3o3o4o- sersa, o3x3o3x3o3o4o - sibrosa, o3o3x3o3x3o4o - strasaz, x3x3x3o3o3o4o - gersa, o3x3x3x3o3o4o - gibrosa, o3o3x3x3x3o4o - gotrasaz
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Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron
Uniform polychoron Pentachoron 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
Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds
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