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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 B7 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

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Runcinated 7-cube
Type uniform 7-polytope
Schläfli symbol t0,3{4,35}
Coxeter-Dynkin diagrams              
6-faces
5-faces
4-faces
Cells
Faces
Edges 33600
Vertices 4480
Vertex figure
Coxeter groups B7, [4,35]
Properties convex

Alternate names

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  • Small prismated hepteract (acronym: spesa) (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]

Biruncinated 7-cube

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Biruncinated 7-cube
Type uniform 7-polytope
Schläfli symbol t1,4{4,35}
Coxeter-Dynkin diagrams              
6-faces
5-faces
4-faces
Cells
Faces
Edges 67200
Vertices 8960
Vertex figure
Coxeter groups B7, [4,35]
Properties convex

Alternate names

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  • Small biprismated hepteract (Acronym sibposa) (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]

Runcitruncated 7-cube

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Runcitruncated 7-cube
Type uniform 7-polytope
Schläfli symbol t0,1,3{4,35}
Coxeter-Dynkin diagrams              
6-faces
5-faces
4-faces
Cells
Faces
Edges 73920
Vertices 13440
Vertex figure
Coxeter groups B7, [4,35]
Properties convex

Alternate names

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  • Prismatotruncated hepteract (acronym: petsa) (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]

Biruncitruncated 7-cube

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Biruncitruncated 7-cube
Type uniform 7-polytope
Schläfli symbol t1,2,4{4,35}
Coxeter-Dynkin diagrams              
6-faces
5-faces
4-faces
Cells
Faces
Edges 134400
Vertices 26880
Vertex figure
Coxeter groups B7, [4,35]
Properties convex

Alternate names

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  • Biprismatotruncated hepteract (acronym: biptesa) (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]

Runcicantellated 7-cube

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Runcicantellated 7-cube
Type uniform 7-polytope
Schläfli symbol t0,2,3{4,35}
Coxeter-Dynkin diagrams              
6-faces
5-faces
4-faces
Cells
Faces
Edges 53760
Vertices 13440
Vertex figure
Coxeter groups B7, [4,35]
Properties convex

Alternate names

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  • Prismatorhombated hepteract (acronym: parsa) (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]

Biruncicantellated 7-cube

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biruncicantellated 7-cube
Type uniform 7-polytope
Schläfli symbol t1,3,4{4,35}
Coxeter-Dynkin diagrams              
6-faces
5-faces
4-faces
Cells
Faces
Edges 120960
Vertices 26880
Vertex figure
Coxeter groups B7, [4,35]
Properties convex

Alternate names

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

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]

Runcicantitruncated 7-cube

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Runcicantitruncated 7-cube
Type uniform 7-polytope
Schläfli symbol t0,1,2,3{4,35}
Coxeter-Dynkin diagrams              
6-faces
5-faces
4-faces
Cells
Faces
Edges 94080
Vertices 26880
Vertex figure
Coxeter groups B7, [4,35]
Properties convex

Alternate names

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  • 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

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biruncicantitruncated 7-cube
Type uniform 7-polytope
Schläfli symbol t1,2,3,4{4,35}
Coxeter-Dynkin diagrams              
6-faces
5-faces
4-faces
Cells
Faces
Edges 188160
Vertices 53760
Vertex figure
Coxeter groups B7, [4,35]
Properties convex

Alternate names

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  • 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

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  1. ^ Klitzing, (x3o3o3x3o3o4o - spesa)
  2. ^ Klitzing, (o3x3o3o3x3o4o - sibposa)
  3. ^ Klitzing, (x3x3o3x3o3o4o - petsa)
  4. ^ Klitzing, (o3x3x3o3x3o4o - biptesa)
  5. ^ Klitzing, (x3o3x3x3o3o4o - parsa)
  6. ^ Klitzing, (o3o3x3x3o3x4o - bopresa)
  7. ^ Klitzing, (x3x3x3x3o3o4o - gapsa)
  8. ^ Klitzing, (o3x3x3x3x3o3o - gibposa)

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. (1966)
  • Klitzing, Richard. "7D uniform polytopes (polyexa)". x3o3o3x3o3o4o - spo, o3x3o3o3x3o4o - sibpo, x3x3o3x3o3o4o - patto, o3x3x3o3x3o4o - bipto, x3o3x3x3o3o4o - paro, x3x3x3x3o3o4o - gapo, o3x3x3x3x3o3o- gibpo
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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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