Pentellated 6-orthoplexes

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

Orthogonal projections in B6 Coxeter plane

6-orthoplex

Pentellated 6-orthoplex
Pentellated 6-cube

6-cube

Pentitruncated 6-orthoplex

Penticantellated 6-orthoplex

Penticantitruncated 6-orthoplex

Pentiruncitruncated 6-orthoplex

Pentiruncicantellated 6-cube

Pentiruncicantitruncated 6-orthoplex

Pentisteritruncated 6-cube

Pentistericantitruncated 6-orthoplex

Pentisteriruncicantitruncated 6-orthoplex
(Omnitruncated 6-cube)

There are unique 16 degrees of pentellations of the 6-orthoplex with permutations of truncations, cantellations, runcinations, and sterications. Ten are shown, with the other 6 more easily constructed as a pentellated 6-cube. The simple pentellated 6-orthoplex (Same as pentellated 5-cube) is also called an expanded 6-orthoplex, constructed by an expansion operation applied to the regular 6-orthoplex. The highest form, the pentisteriruncicantitruncated 6-orthoplex, is called an omnitruncated 6-orthoplex with all of the nodes ringed.

Pentitruncated 6-orthoplex

Pentitruncated 6-orthoplex
Typeuniform 6-polytope
Schläfli symbolt0,1,5{3,3,3,3,4}
Coxeter-Dynkin diagrams
5-faces
4-faces
Cells
Faces
Edges8640
Vertices1920
Vertex figure
Coxeter groupsB6, [4,3,3,3,3]
Propertiesconvex

Alternate names

  • Teritruncated hexacontatetrapeton (Acronym: tacox) (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]

Penticantellated 6-orthoplex

Penticantellated 6-orthoplex
Typeuniform 6-polytope
Schläfli symbolt0,2,5{3,3,3,3,4}
Coxeter-Dynkin diagrams
5-faces
4-faces
Cells
Faces
Edges21120
Vertices3840
Vertex figure
Coxeter groupsB6, [4,3,3,3,3]
Propertiesconvex

Alternate names

  • Terirhombated hexacontitetrapeton (Acronym: tapox) (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]

Penticantitruncated 6-orthoplex

Penticantitruncated 6-orthoplex
Typeuniform 6-polytope
Schläfli symbolt0,1,2,5{3,3,3,3,4}
Coxeter-Dynkin diagrams
5-faces
4-faces
Cells
Faces
Edges30720
Vertices7680
Vertex figure
Coxeter groupsB6, [4,3,3,3,3]
Propertiesconvex

Alternate names

  • Terigreatorhombated hexacontitetrapeton (Acronym: togrig) (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]

Pentiruncitruncated 6-orthoplex

Pentiruncitruncated 6-orthoplex
Typeuniform 6-polytope
Schläfli symbolt0,1,3,5{3,3,3,3,4}
Coxeter-Dynkin diagrams
5-faces
4-faces
Cells
Faces
Edges51840
Vertices11520
Vertex figure
Coxeter groupsB6, [4,3,3,3,3]
Propertiesconvex

Alternate names

  • Teriprismatotruncated hexacontitetrapeton (Acronym: tocrax) (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]

Pentiruncicantitruncated 6-orthoplex

Pentiruncicantitruncated 6-orthoplex
Typeuniform 6-polytope
Schläfli symbolt0,1,2,3,5{3,3,3,3,4}
Coxeter-Dynkin diagrams
5-faces
4-faces
Cells
Faces
Edges80640
Vertices23040
Vertex figure
Coxeter groupsB6, [4,3,3,3,3]
Propertiesconvex

Alternate names

  • Terigreatoprismated hexacontitetrapeton (Acronym: tagpog) (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]

Pentistericantitruncated 6-orthoplex

Pentistericantitruncated 6-orthoplex
Typeuniform 6-polytope
Schläfli symbolt0,1,2,4,5{3,3,3,3,4}
Coxeter-Dynkin diagrams
5-faces
4-faces
Cells
Faces
Edges80640
Vertices23040
Vertex figure
Coxeter groupsB6, [4,3,3,3,3]
Propertiesconvex

Alternate names

  • Tericelligreatorhombated hexacontitetrapeton (Acronym: tecagorg) (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]


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, (x4o3o3o3x3x - tacox)
  2. Klitzing, (x4o3o3x3o3x - tapox)
  3. Klitzing, (x4o3o3x3x3x - togrig)
  4. Klitzing, (x4o3x3o3x3x - tocrax)
  5. Klitzing, (x4x3o3x3x3x - tagpog)
  6. Klitzing, (x4x3o3x3x3x - tecagorg)

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, [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)". x4o3o3o3x3x - tacox, x4o3o3x3o3x - tapox, x4o3o3x3x3x - togrig, x4o3x3o3x3x - tocrax, x4x3o3x3x3x - tagpog, x4x3o3x3x3x - tecagorg
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform polychoron Pentachoron 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds
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