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Pentellated 7-cubes

From Wikipedia, the free encyclopedia

In seven-dimensional geometry, a pentellated 7-cube is a convex uniform 7-polytope with 5th order truncations (pentellation) of the regular 7-cube. There are 32 unique pentellations of the 7-cube with permutations of truncations, cantellations, runcinations, and sterications. 16 are more simply constructed relative to the 7-orthoplex.


7-cube

Pentellated 7-cube

Pentitruncated 7-cube

Penticantellated 7-cube

Penticantitruncated 7-cube
Pentiruncinated 7-cube

Pentiruncitruncated 7-cube

Pentiruncicantellated 7-cube

Pentiruncicantitruncated 7-cube

Pentistericated 7-cube

Pentisteritruncated 7-cube

Pentistericantellated 7-cube

Pentistericantitruncated 7-cube

Pentisteriruncinated 7-cube

Pentisteriruncitruncated 7-cube

Pentisteriruncicantellated 7-cube

Pentisteriruncicantitruncated 7-cube

Pentellated 7-cube

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

Alternate names

  • Small terated hepteract (acronym:) (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]

Pentitruncated 7-cube

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

Alternate names

  • Teritruncated hepteract (acronym: ) (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]

Penticantellated 7-cube

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

Alternate names

  • Terirhombated hepteract (acronym: ) (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]

Penticantitruncated 7-cube

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

Alternate names

  • Terigreatorhombated hepteract (acronym: ) (Jonathan Bowers)[4]


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]

Pentiruncinated 7-cube

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

Alternate names

  • Teriprismated hepteract (acronym: ) (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]

Pentiruncitruncated 7-cube

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

Alternate names

  • Teriprismatotruncated hepteract (acronym: ) (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]

Pentiruncicantellated 7-cube

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

Alternate names

  • Teriprismatorhombated hepteract (acronym: ) (Jonathan Bowers)[7]

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]

Pentiruncicantitruncated 7-cube

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

Alternate names

  • Terigreatoprismated hepteract (acronym: ) (Jonathan Bowers)[8]

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 too complex too complex
Dihedral symmetry [6] [4]

Pentistericated 7-cube

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

Alternate names

  • Tericellated hepteract (acronym: ) (Jonathan Bowers)[9]

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]

Pentisteritruncated 7-cube

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

Alternate names

  • Tericellitruncated hepteract (acronym: ) (Jonathan Bowers)[10]

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]

Pentistericantellated 7-cube

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

Alternate names

  • Tericellirhombated hepteract (acronym: ) (Jonathan Bowers)[11]

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]

Pentistericantitruncated 7-cube

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

Alternate names

  • Tericelligreatorhombated hepteract (acronym: ) (Jonathan Bowers)[12]

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]

Pentisteriruncinated 7-cube

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

Alternate names

  • Bipenticantitruncated 7-cube as t1,2,3,6{4,35}
  • Tericelliprismated hepteract (acronym: ) (Jonathan Bowers)[13]

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]

Pentisteriruncitruncated 7-cube

pentisteriruncitruncated 7-cube
Type uniform 7-polytope
Schläfli symbol t0,1,3,4,5{4,35}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges 40320
Vertices 10080
Vertex figure
Coxeter groups B7, [4,35]
Properties convex

Alternate names

  • Tericelliprismatotruncated hepteract (acronym: ) (Jonathan Bowers)[14]

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]

Pentisteriruncicantellated 7-cube

pentisteriruncicantellated 7-cube
Type uniform 7-polytope
Schläfli symbol t0,2,3,4,5{4,35}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges 40320
Vertices 10080
Vertex figure
Coxeter groups B7, [4,35]
Properties convex

Alternate names

  • Bipentiruncicantitruncated 7-cube as t1,2,3,4,6{4,35}
  • Tericelliprismatorhombated hepteract (acronym: ) (Jonathan Bowers)[15]

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]

Pentisteriruncicantitruncated 7-cube

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

Alternate names

  • Great terated hepteract (acronym:) (Jonathan Bowers)[16]

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]

Related polytopes

These polytopes are a part of a set of 127 uniform 7-polytopes with B7 symmetry.

Notes

  1. ^ Klitzing, (x3o3o3o3o3x4o - )
  2. ^ Klitzing, (x3x3o3o3o3x4o - )
  3. ^ Klitzing, (x3o3x3o3o3x4o - )
  4. ^ Klitzing, (x3x3x3oxo3x4o - )
  5. ^ Klitzing, (x3o3o3x3o3x4o - )
  6. ^ Klitzing, (x3x3o3x3o3x4o - )
  7. ^ Klitzing, (x3o3x3x3o3x4o - )
  8. ^ Klitzing, (x3x3x3x3o3x4o - )
  9. ^ Klitzing, (x3o3o3o3x3x4o - )
  10. ^ Klitzing, (x3x3o3o3x3x4o - )
  11. ^ Klitzing, (x3o3x3o3x3x4o - )
  12. ^ Klitzing, (x3x3x3o3x3x4o - )
  13. ^ Klitzing, (x3o3o3x3x3x4o - )
  14. ^ Klitzing, (x3x3o3x3x3x4o - )
  15. ^ Klitzing, (x3o3x3x3x3x4o - )
  16. ^ Klitzing, (x3x3x3x3x3x4o - )

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.
  • Klitzing, Richard. "7D uniform polytopes (polyexa)". x3o3o3o3o3x4o, x3x3o3o3o3x4o, x3o3x3o3o3x4o, x3x3x3oxo3x4o, x3o3o3x3o3x4o, x3x3o3x3o3x4o, x3o3x3x3o3x4o, x3x3x3x3o3x4o, x3o3o3o3x3x4o, x3x3o3o3x3x4o, x3o3x3o3x3x4o, x3x3x3o3x3x4o, x3o3o3x3x3x4o, x3x3o3x3x3x4o, x3o3x3x3x3x4o, x3x3x3x3x3x3o

External links

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
This page was last edited on 7 August 2021, at 18:51
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