Cantellated 7-cubes

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


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

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
Cantellated 7-cube
Typeuniform 7-polytope
Schläfli symbolrr{4,3,3,3,3,3}
Coxeter diagram
6-faces
5-faces
4-faces
Cells
Faces
Edges16128
Vertices2688
Vertex figure
Coxeter groupsB7, [4,3,3,3,3,3]
Propertiesconvex

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

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]

Bicantellated 7-cube
Bicantellated 7-cube
Typeuniform 7-polytope
Schläfli symbolr2r{4,3,3,3,3,3}
Coxeter diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges40320
Vertices6720
Vertex figure
Coxeter groupsB7, [4,3,3,3,3,3]
Propertiesconvex

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

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]

Tricantellated 7-cube
Tricantellated 7-cube
Typeuniform 7-polytope
Schläfli symbolr3r{4,3,3,3,3,3}
Coxeter diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges47040
Vertices6720
Vertex figure
Coxeter groupsB7, [4,3,3,3,3,3]
Propertiesconvex

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

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]

Cantitruncated 7-cube
Cantitruncated 7-cube
Typeuniform 7-polytope
Schläfli symboltr{4,3,3,3,3,3}
Coxeter diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges18816
Vertices5376
Vertex figure
Coxeter groupsB7, [4,3,3,3,3,3]
Propertiesconvex

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

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]

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
Bicantitruncated 7-cube
Typeuniform 7-polytope
Schläfli symbolr2r{4,3,3,3,3,3}
Coxeter diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges47040
Vertices13440
Vertex figure
Coxeter groupsB7, [4,3,3,3,3,3]
Propertiesconvex

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

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]

Tricantitruncated 7-cube
Tricantitruncated 7-cube
Typeuniform 7-polytope
Schläfli symbolt3r{4,3,3,3,3,3}
Coxeter diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges53760
Vertices13440
Vertex figure
Coxeter groupsB7, [4,3,3,3,3,3]
Propertiesconvex

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

Images
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]

These polytopes are from a family of 127 uniform 7-polytopes with B7 symmetry.

See also

Notes
  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
  • 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. "7D uniform polytopes (polyexa)". x3o3x3o3o3o4o- sersa, o3x3o3x3o3o4o - sibrosa, o3o3x3o3x3o4o - strasaz, x3x3x3o3o3o4o - gersa, o3x3x3x3o3o4o - gibrosa, o3o3x3x3x3o4o - gotrasaz
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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