Pentellated 6-cubes

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6-cube
6-orthoplex
Pentellated 6-cube

Pentitruncated 6-cube
Penticantellated 6-cube
Penticantitruncated 6-cube

Pentiruncitruncated 6-cube
Pentiruncicantellated 6-cube
Pentiruncicantitruncated 6-cube

Pentisteritruncated 6-cube
Pentistericantitruncated 6-cube
Omnitruncated 6-cube
Orthogonal projections in B6 Coxeter plane

In six-dimensional geometry, a pentellated 6-cube is a convex uniform 6-polytope with 5th order truncations of the regular 6-cube.

There are unique 16 degrees of pentellations of the 6-cube with permutations of truncations, cantellations, runcinations, and sterications. The simple pentellated 6-cube is also called an expanded 6-cube, constructed by an expansion operation applied to the regular 6-cube. The highest form, the pentisteriruncicantitruncated 6-cube, is called an omnitruncated 6-cube with all of the nodes ringed. Six of them are better constructed from the 6-orthoplex given at pentellated 6-orthoplex.

Pentellated 6-cube

Pentellated 6-cube
Type Uniform 6-polytope
Schläfli symbol t0,5{4,3,3,3,3}
Coxeter-Dynkin diagram
5-faces
4-faces
Cells
Faces
Edges 1920
Vertices 384
Vertex figure 5-cell antiprism
Coxeter group B6, [4,3,3,3,3]
Properties convex

Alternate names

  • Pentellated 6-orthoplex
  • Expanded 6-cube, expanded 6-orthoplex
  • Small teri-hexeractihexacontitetrapeton (Acronym: stoxog) (Jonathan Bowers)[1]

Images

Pentitruncated 6-cube

Pentitruncated 6-cube
Type uniform 6-polytope
Schläfli symbol t0,1,5{4,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces
4-faces
Cells
Faces
Edges 8640
Vertices 1920
Vertex figure
Coxeter groups B6, [4,3,3,3,3]
Properties convex

Alternate names

  • Teritruncated hexeract (Acronym: tacog) (Jonathan Bowers)[2]

Images

orthographic projections
Coxeter plane B6 B5 B4
Graph
Dihedral symmetry [12] [10] [8]
Coxeter plane B3 B2
Graph
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]

Penticantellated 6-cube

Penticantellated 6-cube
Type uniform 6-polytope
Schläfli symbol t0,2,5{4,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces
4-faces
Cells
Faces
Edges 21120
Vertices 3840
Vertex figure
Coxeter groups B6, [4,3,3,3,3]
Properties convex

Alternate names

  • Terirhombated hexeract (Acronym: topag) (Jonathan Bowers)[3]

Images

orthographic projections
Coxeter plane B6 B5 B4
Graph
Dihedral symmetry [12] [10] [8]
Coxeter plane B3 B2
Graph
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]

Penticantitruncated 6-cube

Penticantitruncated 6-cube
Type uniform 6-polytope
Schläfli symbol t0,1,2,5{4,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces
4-faces
Cells
Faces
Edges 30720
Vertices 7680
Vertex figure
Coxeter groups B6, [4,3,3,3,3]
Properties convex

Alternate names

  • Terigreatorhombated hexeract (Acronym: togrix) (Jonathan Bowers)[4]

Images

orthographic projections
Coxeter plane B6 B5 B4
Graph
Dihedral symmetry [12] [10] [8]
Coxeter plane B3 B2
Graph
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]

Pentiruncitruncated 6-cube

Pentiruncitruncated 6-cube
Type uniform 6-polytope
Schläfli symbol t0,1,3,5{4,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces
4-faces
Cells
Faces
Edges 151840
Vertices 11520
Vertex figure
Coxeter groups B6, [4,3,3,3,3]
Properties convex

Alternate names

  • Tericellirhombated hexacontitetrapeton (Acronym: tocrag) (Jonathan Bowers)[5]

Images

orthographic projections
Coxeter plane B6 B5 B4
Graph
Dihedral symmetry [12] [10] [8]
Coxeter plane B3 B2
Graph
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]

Pentiruncicantellated 6-cube

Pentiruncicantellated 6-cube
Type uniform 6-polytope
Schläfli symbol t0,2,3,5{4,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces
4-faces
Cells
Faces
Edges 46080
Vertices 11520
Vertex figure
Coxeter groups B6, [4,3,3,3,3]
Properties convex

Alternate names

  • Teriprismatorhombi-hexeractihexacontitetrapeton (Acronym: tiprixog) (Jonathan Bowers)[6]

Images

Pentiruncicantitruncated 6-cube

Pentiruncicantitruncated 6-cube
Type uniform 6-polytope
Schläfli symbol t0,1,2,3,5{4,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces
4-faces
Cells
Faces
Edges 80640
Vertices 23040
Vertex figure
Coxeter groups B6, [4,3,3,3,3]
Properties convex

Alternate names

  • Terigreatoprismated hexeract (Acronym: tagpox) (Jonathan Bowers)[7]

Images

orthographic projections
Coxeter plane B6 B5 B4
Graph
Dihedral symmetry [12] [10] [8]
Coxeter plane B3 B2
Graph
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]

Pentisteritruncated 6-cube

Pentisteritruncated 6-cube
Type uniform 6-polytope
Schläfli symbol t0,1,4,5{4,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces
4-faces
Cells
Faces
Edges 30720
Vertices 7680
Vertex figure
Coxeter groups B6, [4,3,3,3,3]
Properties convex

Alternate names

  • Tericellitrunki-hexeractihexacontitetrapeton (Acronym: tactaxog) (Jonathan Bowers)[8]

Images

Pentistericantitruncated 6-cube

Pentistericantitruncated 6-cube
Type uniform 6-polytope
Schläfli symbol t0,1,2,4,5{4,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces
4-faces
Cells
Faces
Edges 80640
Vertices 23040
Vertex figure
Coxeter groups B6, [4,3,3,3,3]
Properties convex

Alternate names

  • Tericelligreatorhombated hexeract (Acronym: tocagrax) (Jonathan Bowers)[9]

Images

orthographic projections
Coxeter plane B6 B5 B4
Graph
Dihedral symmetry [12] [10] [8]
Coxeter plane B3 B2
Graph
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]

Omnitruncated 6-cube

Omnitruncated 6-cube
Type Uniform 6-polytope
Schläfli symbol t0,1,2,3,4,5{35}
Coxeter-Dynkin diagrams
5-faces 728:
12 t0,1,2,3,4{3,3,3,4}
60 {}×t0,1,2,3{3,3,4} 40px×40px
160 {6}×t0,1,2{3,4} 40px×40px
240 {8}×t0,1,2{3,3} 40px×40px
192 {}×t0,1,2,3{33} 40px×40px
64 t0,1,2,3,4{34}
4-faces 14168
Cells 72960
Faces 151680
Edges 138240
Vertices 46080
Vertex figure irregular 5-simplex
Coxeter group B6, [4,3,3,3,3]
Properties convex, isogonal

The omnitruncated 6-cube has 5040 vertices, 15120 edges, 16800 faces (4200 hexagons and 1260 squares), 8400 cells, 1806 4-faces, and 126 5-faces. With 5040 vertices, it is the largest of 35 uniform 6-polytopes generated from the regular 6-cube.

Alternate names

  • Pentisteriruncicantituncated 6-cube or 6-orthoplex (omnitruncation for 6-polytopes)
  • Omnitruncated hexeract
  • Great teri-hexeractihexacontitetrapeton (Acronym: gotaxog) (Jonathan Bowers)[10]

Images

Full snub 6-cube

The full snub 6-cube or omnisnub 6-cube, defined as an alternation of the omnitruncated 6-cube is not uniform, but it can be given Coxeter diagram and symmetry [4,3,3,3,3]+, and constructed from 12 snub 5-cubes, 64 snub 5-simplexes, 60 snub tesseract antiprisms, 192 snub 5-cell antiprisms, 160 3-sr{4,3} duoantiprisms, 240 4-s{3,4} duoantiprisms, and 23040 irregular 5-simplexes filling the gaps at the deleted vertices.

These polytopes are from a set of 63 uniform 6-polytopes generated from the B6 Coxeter plane, including the regular 6-cube or 6-orthoplex.


Notes

  1. ↑ Klitzing, (x4o3o3o3o3x - stoxog)
  2. ↑ Klitzing, (x4x3o3o3o3x - tacog)
  3. ↑ Klitzing, (x4o3x3o3o3x - topag)
  4. ↑ Klitzing, (x4x3x3o3o3x - togrix)
  5. ↑ Klitzing, (x4x3o3x3o3x - tocrag)
  6. ↑ Klitzing, (x4o3x3x3o3x - tiprixog)
  7. ↑ Klitzing, (x4x3x3o3x3x - tagpox)
  8. ↑ Klitzing, (x4x3o3o3x3x - tactaxog)
  9. ↑ Klitzing, (x4x3x3o3x3x - tocagrax)
  10. ↑ Klitzing, (x4x3x3x3x3x - gotaxog)

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 [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.
  • Klitzing, Richard. "6D uniform polytopes (polypeta)". https://bendwavy.org/klitzing/dimensions/polypeta.htm.  x4o3o3o3o3x - stoxog, x4x3o3o3o3x - tacog, x4o3x3o3o3x - topag, x4x3x3o3o3x - togrix, x4x3o3x3o3x - tocrag, x4o3x3x3o3x - tiprixog, x4x3x3o3x3x - tagpox, x4x3o3o3x3x - tactaxog, x4x3x3o3x3x - tocagrax, x4x3x3x3x3x - gotaxog
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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