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Runcinated 7-cubes

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7-cube
Runcinated 7-cube
Biruncinated 7-cube

Runcitruncated 7-cube
Biruncitruncated 7-cube
Runcicantellated 7-cube

Biruncicantellated 7-cube
Runcicantitruncated 7-cube
Biruncicantitruncated 7-cube
Orthogonal projections in B6 Coxeter plane

In seven-dimensional geometry, a runcinated 7-cube is a convex uniform 7-polytope with 3rd order truncations (runcination) of the regular 7-cube.

There are 16 unique runcinations of the 7-cube with permutations of truncations, and cantellations. 8 are more simply constructed from the 7-orthoplex.

These polytopes are among 127 uniform 7-polytopes with B7 symmetry.

Runcinated 7-cube

[edit]
Runcinated 7-cube
Typeuniform 7-polytope
Schläfli symbolt0,3{4,35}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges33600
Vertices4480
Vertex figure
Coxeter groupsB7, [4,35]
Propertiesconvex

Alternate names

[edit]
  • Small prismated hepteract (acronym: spesa) (Jonathan Bowers)[1]

Images

[edit]
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]

Biruncinated 7-cube

[edit]
Biruncinated 7-cube
Typeuniform 7-polytope
Schläfli symbolt1,4{4,35}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges67200
Vertices8960
Vertex figure
Coxeter groupsB7, [4,35]
Propertiesconvex

Alternate names

[edit]
  • Small biprismated hepteract (Acronym sibposa) (Jonathan Bowers)[2]

Images

[edit]
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]

Runcitruncated 7-cube

[edit]
Runcitruncated 7-cube
Typeuniform 7-polytope
Schläfli symbolt0,1,3{4,35}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges73920
Vertices13440
Vertex figure
Coxeter groupsB7, [4,35]
Propertiesconvex

Alternate names

[edit]
  • Prismatotruncated hepteract (acronym: petsa) (Jonathan Bowers)[3]

Images

[edit]
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]

Biruncitruncated 7-cube

[edit]
Biruncitruncated 7-cube
Typeuniform 7-polytope
Schläfli symbolt1,2,4{4,35}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges134400
Vertices26880
Vertex figure
Coxeter groupsB7, [4,35]
Propertiesconvex

Alternate names

[edit]
  • Biprismatotruncated hepteract (acronym: biptesa) (Jonathan Bowers)[4]

Images

[edit]
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]

Runcicantellated 7-cube

[edit]
Runcicantellated 7-cube
Typeuniform 7-polytope
Schläfli symbolt0,2,3{4,35}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges53760
Vertices13440
Vertex figure
Coxeter groupsB7, [4,35]
Propertiesconvex

Alternate names

[edit]
  • Prismatorhombated hepteract (acronym: parsa) (Jonathan Bowers)[5]

Images

[edit]
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]

Biruncicantellated 7-cube

[edit]
Biruncicantellated 7-cube
Typeuniform 7-polytope
Schläfli symbolt1,3,4{4,35}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges120960
Vertices26880
Vertex figure
Coxeter groupsB7, [4,35]
Propertiesconvex

Alternate names

[edit]
  • Biprismatorhombated hepteract (acronym: bopresa) (Jonathan Bowers)[6]

Images

[edit]
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]

Runcicantitruncated 7-cube

[edit]
Runcicantitruncated 7-cube
Typeuniform 7-polytope
Schläfli symbolt0,1,2,3{4,35}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges94080
Vertices26880
Vertex figure
Coxeter groupsB7, [4,35]
Propertiesconvex

Alternate names

[edit]
  • Great prismated hepteract (acronym: gapsa) (Jonathan Bowers)[7]

Images

[edit]
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]

Biruncicantitruncated 7-cube

[edit]
Biruncicantitruncated 7-cube
Typeuniform 7-polytope
Schläfli symbolt1,2,3,4{4,35}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges188160
Vertices53760
Vertex figure
Coxeter groupsB7, [4,35]
Propertiesconvex

Alternate names

[edit]
  • Great biprismated hepteract (acronym: gibposa) (Jonathan Bowers)[8]

Images

[edit]
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]

Notes

[edit]
  1. ↑ Klitzing, (x4o3o3x3o3o3o - spesa)
  2. ↑ Klitzing, (o4x3o3o3x3o3o - sibposa)
  3. ↑ Klitzing, (x4x3o3x3o3o3o - petsa)
  4. ↑ Klitzing, (o4x3x3o3x3o3o - biptesa)
  5. ↑ Klitzing, (x4o3x3x3o3o3o - parsa)
  6. ↑ Klitzing, (o4x3o3x3x3o3o - bopresa)
  7. ↑ Klitzing, (x4x3x3x3o3o3o - gapsa)
  8. ↑ Klitzing, (o4x3x3x3x3o3o - gibposa)

References

[edit]
  • 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 Ivić Weiss, Wiley-Interscience Publication, 1995, wiley.com, 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. (1966)
  • Klitzing, Richard. "7D uniform polytopes (polyexa) with acronyms". x4o3o3x3o3o3o - spesa, o4x3o3o3x3o3o - sibposa, x4x3o3x3o3o3o - petsa, o4x3x3o3x3o3o - biptesa, x4o3x3x3o3o3o - parsa, o4x3o3x3x3o3o - bopresa, x4x3x3x3o3o3o - gapsa, o4x3x3x3x3o3o - gibposa
[edit]
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 • Polytope operations
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