Orthogonal projections in B6 Coxeter plane

7-orthoplex
Runcinated 7-orthoplex
Biruncinated 7-orthoplex

Runcitruncated 7-orthoplex
Biruncitruncated 7-orthoplex
Runcicantellated 7-orthoplex

Biruncicantellated 7-orthoplex
Runcicantitruncated 7-orthoplex
Biruncicantitruncated 7-orthoplex

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

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

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

Runcinated 7-orthoplex

Runcinated 7-orthoplex
Typeuniform 7-polytope
Schläfli symbolt0,3{35,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges23520
Vertices2240
Vertex figure
Coxeter groupsB7, [4,35]
Propertiesconvex

Alternate names

  • Small prismated hecatonicosoctaexon (acronym: spaz) (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]

Biruncinated 7-orthoplex

Biruncinated 7-orthoplex
Typeuniform 7-polytope
Schläfli symbolt1,4{35,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges60480
Vertices6720
Vertex figure
Coxeter groupsB7, [4,35]
Propertiesconvex

Alternate names

  • Small biprismated hecatonicosoctaexon (Acronym sibpaz) (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]

Runcitruncated 7-orthoplex

Runcitruncated 7-orthoplex
Typeuniform 7-polytope
Schläfli symbolt0,1,3{35,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges50400
Vertices6720
Vertex figure
Coxeter groupsB7, [4,35]
Propertiesconvex

Alternate names

  • Prismatotruncated hecatonicosoctaexon (acronym: potaz) (Jonathan Bowers)[3]

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]

Biruncitruncated 7-orthoplex

Biruncitruncated 7-orthoplex
Typeuniform 7-polytope
Schläfli symbolt1,2,4{35,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges120960
Vertices20160
Vertex figure
Coxeter groupsB7, [4,35]
Propertiesconvex

Alternate names

  • Biprismatotruncated hecatonicosoctaexon (acronym: baptize) (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]

Runcicantellated 7-orthoplex

Runcicantellated 7-orthoplex
Typeuniform 7-polytope
Schläfli symbolt0,2,3{35,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges33600
Vertices6720
Vertex figure
Coxeter groupsB7, [4,35]
Propertiesconvex

Alternate names

  • Prismatorhombated hecatonicosoctaexon (acronym: parz) (Jonathan Bowers)[5]

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]

Biruncicantellated 7-orthoplex

biruncicantellated 7-orthoplex
Typeuniform 7-polytope
Schläfli symbolt1,3,4{35,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges100800
Vertices20160
Vertex figure
Coxeter groupsB7, [4,35]
Propertiesconvex

Alternate names

  • Biprismatorhombated hecatonicosoctaexon (acronym: boparz) (Jonathan Bowers)[6]

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]

Runcicantitruncated 7-orthoplex

Runcicantitruncated 7-orthoplex
Typeuniform 7-polytope
Schläfli symbolt0,1,2,3{35,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges60480
Vertices13440
Vertex figure
Coxeter groupsB7, [4,35]
Propertiesconvex

Alternate names

  • Great prismated hecatonicosoctaexon (acronym: gopaz) (Jonathan Bowers)[7]

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]

Biruncicantitruncated 7-orthoplex

biruncicantitruncated 7-orthoplex
Typeuniform 7-polytope
Schläfli symbolt1,2,3,4{35,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges161280
Vertices40320
Vertex figure
Coxeter groupsB7, [4,35]
Propertiesconvex

Alternate names

  • Great biprismated hecatonicosoctaexon (acronym: gibpaz) (Jonathan Bowers)[8]

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]

Notes

  1. ↑ Klitzing, (o3o3o3x3o3o4x - spaz)
  2. ↑ Klitzing, (o3x3o3o3x3o4o - sibpaz)
  3. ↑ Klitzing, (o3o3o3x3x3o4x - potaz)
  4. ↑ Klitzing, (o3o3x3o3x3x4o - baptize)
  5. ↑ Klitzing, (o3o3o3x3x3o4x - parz)
  6. ↑ Klitzing, (o3x3o3x3x3o4o - boparz)
  7. ↑ Klitzing, (o3o3o3x3x3x4x - gopaz)
  8. ↑ Klitzing, (o3o3x3x3x3x3o - gibpaz)

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 Wiley: Kaleidoscopes: Selected Writings of H.S.M. Coxeter
      • (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)". o3o3o3x3o3o4x - spaz, o3x3o3o3x3o4o - sibpaz, o3o3o3x3x3o4x - potaz, o3o3x3o3x3x4o - baptize, o3o3o3x3x3o4x - parz, o3x3o3x3x3o4o - boparz, o3o3o3x3x3x4x - gopaz, o3o3x3x3x3x3o - gibpaz
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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