Pentellated 7-cubes

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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 of the 7-cube with permutations of truncations, cantellations, runcinations, and sterications ucted relative to the 7-orthoplex.

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7-cube
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Pentellated 7-cube
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Pentitruncated 7-cube
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Penticantellated 7-cube
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Penticantitruncated 7-cube
 120px
Pentiruncinated 7-cube
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Pentiruncitruncated 7-cube
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Pentiruncicantellated 7-cube
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Pentiruncicantitruncated 7-cube
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Pentistericated 7-cube
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Pentisteritruncated 7-cube
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Pentistericantellated 7-cube
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Pentistericantitruncated 7-cube
 120px
Pentisteriruncinated 7-cube
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Pentisteriruncitruncated 7-cube
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Pentisteriruncicantellated 7-cube
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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 hepteract (acronym: stesa) (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: tetsa) (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: tersa) (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: togresa) (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]

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: tapsa) (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: toptosa) (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: topresa) (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: togapsa) (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: tacosa) (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: tecatsa) (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: tecresa) (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: tecgresa) (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: tecpasa) (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: tecpetsa) (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: tocpresa) (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 hepteract (acronym: gotesa) (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]

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

Notes

  1. ↑ Klitzing, (x4o3o3o3o3x3o - stesa).
  2. ↑ Klitzing, (x4x3o3o3o3x3o - tetsa)
  3. ↑ Klitzing, (x4o3x3o3o3x3o - tersa)
  4. ↑ Klitzing, (x4x3x3o3o3x3o - togresa)
  5. ↑ Klitzing, (x4o3o3x3o3x3o - tapsa)
  6. ↑ Klitzing, (x4x3o3x3o3x3o - toptosa)
  7. ↑ Klitzing, (x4o3x3x3o3x3o - topresa)
  8. ↑ Klitzing, (x4x3x3x3o3x3o - togapsa)
  9. ↑ Klitzing, (x4o3o3o3x3x3o - tacosa)
  10. ↑ Klitzing, (x4x3o3o3x3x3o - tecatsa)
  11. ↑ Klitzing, (x4o3x3o3x3x3o - tecresa)
  12. ↑ Klitzing, (x4x3x3o3x3x3o - tecgresa)
  13. ↑ Klitzing, (x4o3o3x3x3x3o - tecpasa)
  14. ↑ Klitzing, (x4x3o3x3x3x3o - tecpetsa)
  15. ↑ Klitzing, (x4o3x3x3x3x3o - tocpresa)
  16. ↑ Klitzing, (x4x3x3x3x3x3o - gotesa)

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 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.
  • Klitzing, Richard. "7D uniform polytopes (polyexa) with acronyms". https://bendwavy.org/klitzing/dimensions/polyexa.htm. 
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 4-polytope 5-cell 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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