Stericated 6-orthoplexes

In six-dimensional geometry, a stericated 6-orthoplex is a convex uniform 6-polytope, constructed as a sterication (4th order truncation) of the regular 6-orthoplex.


6-orthoplex

Stericated 6-orthoplex

Steritruncated 6-orthoplex

Stericantellated 6-orthoplex

Stericantitruncated 6-orthoplex

Steriruncinated 6-orthoplex

Steriruncitruncated 6-orthoplex

Steriruncicantellated 6-orthoplex

Steriruncicantitruncated 6-orthoplex
Orthogonal projections in B6 Coxeter plane

There are 16 unique sterications for the 6-orthoplex with permutations of truncations, cantellations, and runcinations. Eight are better represented from the stericated 6-cube.

Stericated 6-orthoplex

Stericated 6-orthoplex
Typeuniform 6-polytope
Schläfli symbol2r2r{3,3,3,3,4}
Coxeter-Dynkin diagrams
5-faces
4-faces
Cells
Faces
Edges5760
Vertices960
Vertex figure
Coxeter groupsB6, [4,3,3,3,3]
Propertiesconvex

Alternate names

  • Small cellated hexacontatetrapeton (Acronym: scag) (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]

Steritruncated 6-orthoplex

Steritruncated 6-orthoplex
Typeuniform 6-polytope
Schläfli symbolt0,1,4{3,3,3,3,4}
Coxeter-Dynkin diagrams
5-faces
4-faces
Cells
Faces
Edges19200
Vertices3840
Vertex figure
Coxeter groupsB6, [4,3,3,3,3]
Propertiesconvex

Alternate names

  • Cellitruncated hexacontatetrapeton (Acronym: catog) (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]

Stericantellated 6-orthoplex

Stericantellated 6-orthoplex
Typeuniform 6-polytope
Schläfli symbolst0,2,4{34,4}
rr2r{3,3,3,3,4}
Coxeter-Dynkin diagrams
5-faces
4-faces
Cells
Faces
Edges28800
Vertices5760
Vertex figure
Coxeter groupsB6, [4,3,3,3,3]
Propertiesconvex

Alternate names

  • Cellirhombated hexacontatetrapeton (Acronym: crag) (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]

Stericantitruncated 6-orthoplex

Stericantitruncated 6-orthoplex
Typeuniform 6-polytope
Schläfli symbolt0,1,2,4{3,3,3,3,4}
Coxeter-Dynkin diagrams
5-faces
4-faces
Cells
Faces
Edges46080
Vertices11520
Vertex figure
Coxeter groupsB6, [4,3,3,3,3]
Propertiesconvex

Alternate names

  • Celligreatorhombated hexacontatetrapeton (Acronym: cagorg) (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]

Steriruncinated 6-orthoplex

Steriruncinated 6-orthoplex
Typeuniform 6-polytope
Schläfli symbolt0,3,4{3,3,3,3,4}
Coxeter-Dynkin diagrams
5-faces
4-faces
Cells
Faces
Edges15360
Vertices3840
Vertex figure
Coxeter groupsB6, [4,3,3,3,3]
Propertiesconvex

Alternate names

  • Celliprismated hexacontatetrapeton (Acronym: copog) (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]

Steriruncitruncated 6-orthoplex

Steriruncitruncated 6-orthoplex
Typeuniform 6-polytope
Schläfli symbol2t2r{3,3,3,3,4}
Coxeter-Dynkin diagrams
5-faces
4-faces
Cells
Faces
Edges40320
Vertices11520
Vertex figure
Coxeter groupsB6, [4,3,3,3,3]
Propertiesconvex

Alternate names

  • Celliprismatotruncated hexacontatetrapeton (Acronym: captog) (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]

Steriruncicantellated 6-orthoplex

Steriruncicantellated 6-orthoplex
Typeuniform 6-polytope
Schläfli symbolt0,2,3,4{3,3,3,3,4}
Coxeter-Dynkin diagrams
5-faces
4-faces
Cells
Faces
Edges40320
Vertices11520
Vertex figure
Coxeter groupsB6, [4,3,3,3,3]
Propertiesconvex

Alternate names

  • Celliprismatorhombated hexacontatetrapeton (Acronym: coprag) (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]

Steriruncicantitruncated 6-orthoplex

Steriuncicantitruncated 6-orthoplex
Typeuniform 6-polytope
Schläfli symbolst0,1,2,3,4{34,4}
tr2r{3,3,3,3,4}
Coxeter-Dynkin diagrams
5-faces536:
12 t0,1,2,3{3,3,3,4}
60 {}×t0,1,2{3,3,4} ×
160 {6}×t0,1,2{3,3} ×
240 {4}×t0,1,2{3,3} ×
64 t0,1,2,3,4{34}
4-faces8216
Cells38400
Faces76800
Edges69120
Vertices23040
Vertex figureirregular 5-simplex
Coxeter groupsB6, [4,3,3,3,3]
Propertiesconvex

Alternate names

  • Great cellated hexacontatetrapeton (Acronym: gocog) (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]

Snub 6-demicube

The snub 6-demicube defined as an alternation of the omnitruncated 6-demicube is not uniform, but it can be given Coxeter diagram or and symmetry [32,1,1,1]+ or [4,(3,3,3,3)+], and constructed from 12 snub 5-demicubes, 64 snub 5-simplexes, 60 snub 24-cell antiprisms, 160 3-s{3,4} duoantiprisms, 240 2-sr{3,3} duoantiprisms, and 11520 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-orthoplex 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, (x3o3o3o3x4o - scag)
  2. Klitzing, (x3x3o3o3x4o - catog)
  3. Klitzing, (x3o3x3o3x4o - crag)
  4. Klitzing, (x3x3x3o3x4o - cagorg)
  5. Klitzing, (x3o3o3x3x4o - copog)
  6. Klitzing, (x3x3o3x3x4o - captog)
  7. Klitzing, (x3o3x3x3x4o - coprag)
  8. Klitzing, (x3x3x3x3x4o - gocog)

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)".
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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