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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

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]

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

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]

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

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]

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} ×
160 {6}×t0,1,2{3,4} ×
240 {8}×t0,1,2{3,3} ×
192 {}×t0,1,2,3{33} ×
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

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]

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.

B6 polytopes

β6
t1β6
t2β6
t2γ6
t1γ6
γ6
t0,1β6
t0,2β6

t1,2β6
t0,3β6
t1,3β6
t2,3γ6
t0,4β6
t1,4γ6
t1,3γ6
t1,2γ6

t0,5γ6
t0,4γ6
t0,3γ6
t0,2γ6
t0,1γ6
t0,1,2β6
t0,1,3β6
t0,2,3β6

t1,2,3β6
t0,1,4β6
t0,2,4β6
t1,2,4β6
t0,3,4β6
t1,2,4γ6
t1,2,3γ6
t0,1,5β6

t0,2,5β6
t0,3,4γ6
t0,2,5γ6
t0,2,4γ6
t0,2,3γ6
t0,1,5γ6
t0,1,4γ6
t0,1,3γ6

t0,1,2γ6
t0,1,2,3β6
t0,1,2,4β6
t0,1,3,4β6
t0,2,3,4β6
t1,2,3,4γ6
t0,1,2,5β6
t0,1,3,5β6

t0,2,3,5γ6
t0,2,3,4γ6
t0,1,4,5γ6
t0,1,3,5γ6
t0,1,3,4γ6
t0,1,2,5γ6
t0,1,2,4γ6
t0,1,2,3γ6

t0,1,2,3,4β6
t0,1,2,3,5β6
t0,1,2,4,5β6
t0,1,2,4,5γ6
t0,1,2,3,5γ6
t0,1,2,3,4γ6
t0,1,2,3,4,5γ6

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)". 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 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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