Knowledge (XXG)

Pentellated 7-orthoplexes

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Orthogonal projections in B6 Coxeter plane

7-orthoplex
Pentellated 7-orthoplex
Pentitruncated 7-orthoplex
Penticantellated 7-orthoplex

Penticantitruncated 7-orthoplex
Pentiruncinated 7-orthoplex
Pentiruncitruncated 7-orthoplex
Pentiruncicantellated 7-orthoplex

Pentiruncicantitruncated 7-orthoplex
Pentistericated 7-orthoplex
Pentisteritruncated 7-orthoplex
Pentistericantellated 7-orthoplex

Pentistericantitruncated 7-orthoplex
Pentisteriruncinated 7-orthoplex
Pentisteriruncitruncated 7-orthoplex
Pentisteriruncicantellated 7-orthoplex

Pentisteriruncicantitruncated 7-orthoplex

In seven-dimensional geometry, a pentellated 7-orthoplex is a convex uniform 7-polytope with 5th order truncations (pentellation) of the regular 7-orthoplex.

There are 32 unique pentellations of the 7-orthoplex with permutations of truncations, cantellations, runcinations, and sterications. 16 are more simply constructed relative to the 7-cube.

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

Pentellated 7-orthoplex

Pentellated 7-orthoplex
Type uniform 7-polytope
Schläfli symbol t0,5{3,4}
Coxeter diagram
6-faces
5-faces
4-faces
Cells
Faces
Edges 20160
Vertices 2688
Vertex figure
Coxeter groups B7,
Properties convex

Alternate names

  • Small terated hecatonicosoctaexon (acronym: Staz) (Jonathan Bowers)

Coordinates

Coordinates are permutations of (0,1,1,1,1,1,2)√2

Images

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph
Dihedral symmetry
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph
Dihedral symmetry
Coxeter plane A5 A3
Graph
Dihedral symmetry

Pentitruncated 7-orthoplex

pentitruncated 7-orthoplex
Type uniform 7-polytope
Schläfli symbol t0,1,5{3,4}
Coxeter diagram
6-faces
5-faces
4-faces
Cells
Faces
Edges 87360
Vertices 13440
Vertex figure
Coxeter groups B7,
Properties convex

Alternate names

  • Teritruncated hecatonicosoctaexon (acronym: Tetaz) (Jonathan Bowers)

Images

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph
Dihedral symmetry
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph
Dihedral symmetry
Coxeter plane A5 A3
Graph
Dihedral symmetry

Coordinates

Coordinates are permutations of (0,1,1,1,1,2,3).

Penticantellated 7-orthoplex

Penticantellated 7-orthoplex
Type uniform 7-polytope
Schläfli symbol t0,2,5{3,4}
Coxeter diagram
6-faces
5-faces
4-faces
Cells
Faces
Edges 188160
Vertices 26880
Vertex figure
Coxeter groups B7,
Properties convex

Alternate names

  • Terirhombated hecatonicosoctaexon (acronym: Teroz) (Jonathan Bowers)

Coordinates

Coordinates are permutations of (0,1,1,1,2,2,3)√2.

Images

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph
Dihedral symmetry
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph
Dihedral symmetry
Coxeter plane A5 A3
Graph
Dihedral symmetry

Penticantitruncated 7-orthoplex

penticantitruncated 7-orthoplex
Type uniform 7-polytope
Schläfli symbol t0,1,2,5{3,4}
Coxeter diagram
6-faces
5-faces
4-faces
Cells
Faces
Edges 295680
Vertices 53760
Vertex figure
Coxeter groups B7,
Properties convex

Alternate names

  • Terigreatorhombated hecatonicosoctaexon (acronym: Tograz) (Jonathan Bowers)

Coordinates

Coordinates are permutations of (0,1,1,1,2,3,4)√2.

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph
Dihedral symmetry
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph
Dihedral symmetry
Coxeter plane A5 A3
Graph
Dihedral symmetry

Pentiruncinated 7-orthoplex

pentiruncinated 7-orthoplex
Type uniform 7-polytope
Schläfli symbol t0,3,5{3,4}
Coxeter diagram
6-faces
5-faces
4-faces
Cells
Faces
Edges 174720
Vertices 26880
Vertex figure
Coxeter groups B7,
Properties convex

Alternate names

  • Teriprismated hecatonicosoctaexon (acronym: Topaz) (Jonathan Bowers)

Coordinates

The coordinates are permutations of (0,1,1,2,2,2,3)√2.

Images

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph
Dihedral symmetry
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph
Dihedral symmetry
Coxeter plane A5 A3
Graph
Dihedral symmetry

Pentiruncitruncated 7-orthoplex

pentiruncitruncated 7-orthoplex
Type uniform 7-polytope
Schläfli symbol t0,1,3,5{3,4}
Coxeter diagram
6-faces
5-faces
4-faces
Cells
Faces
Edges 443520
Vertices 80640
Vertex figure
Coxeter groups B7,
Properties convex

Alternate names

  • Teriprismatotruncated hecatonicosoctaexon (acronym: Toptaz) (Jonathan Bowers)

Coordinates

Coordinates are permutations of (0,1,1,2,2,3,4)√2.

Images

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph
Dihedral symmetry
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph
Dihedral symmetry
Coxeter plane A5 A3
Graph
Dihedral symmetry

Pentiruncicantellated 7-orthoplex

pentiruncicantellated 7-orthoplex
Type uniform 7-polytope
Schläfli symbol t0,2,3,5{3,4}
Coxeter diagram
6-faces
5-faces
4-faces
Cells
Faces
Edges 403200
Vertices 80640
Vertex figure
Coxeter groups B7,
Properties convex

Alternate names

  • Teriprismatorhombated hecatonicosoctaexon (acronym: Toparz) (Jonathan Bowers)

Coordinates

Coordinates are permutations of (0,1,1,2,3,3,4)√2.

Images

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph
Dihedral symmetry
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph
Dihedral symmetry
Coxeter plane A5 A3
Graph
Dihedral symmetry

Pentiruncicantitruncated 7-orthoplex

pentiruncicantitruncated 7-orthoplex
Type uniform 7-polytope
Schläfli symbol t0,1,2,3,5{3,4}
Coxeter diagram
6-faces
5-faces
4-faces
Cells
Faces
Edges 725760
Vertices 161280
Vertex figure
Coxeter groups B7,
Properties convex

Alternate names

  • Terigreatoprismated hecatonicosoctaexon (acronym: Tegopaz) (Jonathan Bowers)

Coordinates

Coordinates are permutations of (0,1,1,2,3,4,5)√2.

Images

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph too complex
Dihedral symmetry
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph
Dihedral symmetry
Coxeter plane A5 A3
Graph
Dihedral symmetry

Pentistericated 7-orthoplex

pentistericated 7-orthoplex
Type uniform 7-polytope
Schläfli symbol t0,4,5{3,4}
Coxeter diagram
6-faces
5-faces
4-faces
Cells
Faces
Edges 67200
Vertices 13440
Vertex figure
Coxeter groups B7,
Properties convex

Alternate names

  • Tericellated hecatonicosoctaexon (acronym: Tocaz) (Jonathan Bowers)

Images

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph
Dihedral symmetry
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph
Dihedral symmetry
Coxeter plane A5 A3
Graph
Dihedral symmetry

Coordinates

Coordinates are permutations of (0,1,2,2,2,2,3)√2.

Pentisteritruncated 7-orthoplex

pentisteritruncated 7-orthoplex
Type uniform 7-polytope
Schläfli symbol t0,1,4,5{3,4}
Coxeter diagram
6-faces
5-faces
4-faces
Cells
Faces
Edges 241920
Vertices 53760
Vertex figure
Coxeter groups B7,
Properties convex

Alternate names

  • Tericellitruncated hecatonicosoctaexon (acronym: Tacotaz) (Jonathan Bowers)

Coordinates

Coordinates are permutations of (0,1,2,2,2,3,4)√2.

Images

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph
Dihedral symmetry
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph
Dihedral symmetry
Coxeter plane A5 A3
Graph
Dihedral symmetry

Pentistericantellated 7-orthoplex

pentistericantellated 7-orthoplex
Type uniform 7-polytope
Schläfli symbol t0,2,4,5{3,4}
Coxeter diagram
6-faces
5-faces
4-faces
Cells
Faces
Edges 403200
Vertices 80640
Vertex figure
Coxeter groups B7,
Properties convex

Alternate names

  • Tericellirhombated hecatonicosoctaexon (acronym: Tocarz) (Jonathan Bowers)

Coordinates

Coordinates are permutations of (0,1,2,2,3,3,4)√2.

Images

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph
Dihedral symmetry
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph
Dihedral symmetry
Coxeter plane A5 A3
Graph
Dihedral symmetry

Pentistericantitruncated 7-orthoplex

pentistericantitruncated 7-orthoplex
Type uniform 7-polytope
Schläfli symbol t0,1,2,4,5{3,4}
Coxeter diagram
6-faces
5-faces
4-faces
Cells
Faces
Edges 645120
Vertices 161280
Vertex figure
Coxeter groups B7,
Properties convex

Alternate names

  • Tericelligreatorhombated hecatonicosoctaexon (acronym: Tecagraz) (Jonathan Bowers)

Coordinates

Coordinates are permutations of (0,1,2,2,3,4,5)√2.

Images

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph too complex
Dihedral symmetry
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph
Dihedral symmetry
Coxeter plane A5 A3
Graph
Dihedral symmetry

Pentisteriruncinated 7-orthoplex

Pentisteriruncinated 7-orthoplex
Type uniform 7-polytope
Schläfli symbol t0,3,4,5{3,4}
Coxeter diagram
6-faces
5-faces
4-faces
Cells
Faces
Edges 241920
Vertices 53760
Vertex figure
Coxeter groups B7,
Properties convex

Alternate names

  • Bipenticantitruncated 7-orthoplex as t1,2,3,6{3,4}
  • Tericelliprismated hecatonicosoctaexon (acronym: Tecpaz) (Jonathan Bowers)

Coordinates

Coordinates are permutations of (0,1,2,3,3,3,4)√2.

Images

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph
Dihedral symmetry
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph
Dihedral symmetry
Coxeter plane A5 A3
Graph
Dihedral symmetry

Pentisteriruncitruncated 7-orthoplex

pentisteriruncitruncated 7-orthoplex
Type uniform 7-polytope
Schläfli symbol t0,1,3,4,5{3,4}
Coxeter diagram
6-faces
5-faces
4-faces
Cells
Faces
Edges 645120
Vertices 161280
Vertex figure
Coxeter groups B7,
Properties convex

Alternate names

  • Tericelliprismatotruncated hecatonicosoctaexon (acronym: Tecpotaz) (Jonathan Bowers)

Coordinates

Coordinates are permutations of (0,1,2,3,3,4,5)√2.

Images

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph too complex
Dihedral symmetry
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph
Dihedral symmetry
Coxeter plane A5 A3
Graph
Dihedral symmetry

Pentisteriruncicantellated 7-orthoplex

pentisteriruncicantellated 7-orthoplex
Type uniform 7-polytope
Schläfli symbol t0,2,3,4,5{3,4}
Coxeter diagram
6-faces
5-faces
4-faces
Cells
Faces
Edges 645120
Vertices 161280
Vertex figure
Coxeter groups B7,
Properties convex

Alternate names

  • Bipentiruncicantitruncated 7-orthoplex as t1,2,3,4,6{3,4}
  • Tericelliprismatorhombated hecatonicosoctaexon (acronym: Tacparez) (Jonathan Bowers)

Coordinates

Coordinates are permutations of (0,1,2,3,4,4,5)√2.

Images

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph too complex
Dihedral symmetry
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph
Dihedral symmetry
Coxeter plane A5 A3
Graph
Dihedral symmetry

Pentisteriruncicantitruncated 7-orthoplex

pentisteriruncicantitruncated 7-orthoplex
Type uniform 7-polytope
Schläfli symbol t0,1,2,3,4,5{3,4}
Coxeter diagram
6-faces
5-faces
4-faces
Cells
Faces
Edges 1128960
Vertices 322560
Vertex figure
Coxeter groups B7,
Properties convex

Alternate names

  • Great terated hecatonicosoctaexon (acronym: Gotaz) (Jonathan Bowers)

Coordinates

Coordinates are permutations of (0,1,2,3,4,5,6)√2.

Images

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph too complex
Dihedral symmetry
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph
Dihedral symmetry
Coxeter plane A5 A3
Graph
Dihedral 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
      • (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I,
      • (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II,
      • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III,
  • Norman Johnson Uniform Polytopes, Manuscript (1991)
    • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D.
  • Klitzing, Richard. "7D uniform polytopes (polyexa)".

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

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