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Stericated 6-cubes


6-cube

Stericated 6-cube

Steritruncated 6-cube

Stericantellated 6-cube

Stericantitruncated 6-cube

Steriruncinated 6-cube

Steriruncitruncated 6-cube

Steriruncicantellated 6-cube

Steriruncicantitruncated 6-cube
Orthogonal projections in B6 Coxeter plane

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

There are 8 unique sterications for the 6-cube with permutations of truncations, cantellations, and runcinations.

Stericated 6-cube

Stericated 6-cube
Typeuniform 6-polytope
Schläfli symbol2r2r{4,3,3,3,3}
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 hexeract (Acronym: scox) (Jonathan Bowers)[1]

Images

orthographic projections
Coxeter planeB6B5B4
Graph
Dihedral symmetry[12][10][8]
Coxeter planeB3B2
Graph
Dihedral symmetry[6][4]
Coxeter planeA5A3
Graph
Dihedral symmetry[6][4]

Steritruncated 6-cube

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

Alternate names

  • Cellirhombated hexeract (Acronym: catax) (Jonathan Bowers)[2]

Images

orthographic projections
Coxeter planeB6B5B4
Graph
Dihedral symmetry[12][10][8]
Coxeter planeB3B2
Graph
Dihedral symmetry[6][4]
Coxeter planeA5A3
Graph
Dihedral symmetry[6][4]

Stericantellated 6-cube

Stericantellated 6-cube
Typeuniform 6-polytope
Schläfli symbol2r2r{4,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces
4-faces
Cells
Faces
Edges28800
Vertices5760
Vertex figure
Coxeter groupsB6, [4,3,3,3,3]
Propertiesconvex

Alternate names

  • Cellirhombated hexeract (Acronym: crax) (Jonathan Bowers)[3]

Images

orthographic projections
Coxeter planeB6B5B4
Graph
Dihedral symmetry[12][10][8]
Coxeter planeB3B2
Graph
Dihedral symmetry[6][4]
Coxeter planeA5A3
Graph
Dihedral symmetry[6][4]

Stericantitruncated 6-cube

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

Alternate names

  • Celligreatorhombated hexeract (Acronym: cagorx) (Jonathan Bowers)[4]

Images

orthographic projections
Coxeter planeB6B5B4
Graph
Dihedral symmetry[12][10][8]
Coxeter planeB3B2
Graph
Dihedral symmetry[6][4]
Coxeter planeA5A3
Graph
Dihedral symmetry[6][4]

Steriruncinated 6-cube

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

Alternate names

  • Celliprismated hexeract (Acronym: copox) (Jonathan Bowers)[5]

Images

orthographic projections
Coxeter planeB6B5B4
Graph
Dihedral symmetry[12][10][8]
Coxeter planeB3B2
Graph
Dihedral symmetry[6][4]
Coxeter planeA5A3
Graph
Dihedral symmetry[6][4]

Steriruncitruncated 6-cube

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

Alternate names

  • Celliprismatotruncated hexeract (Acronym: captix) (Jonathan Bowers)[6]

Images

orthographic projections
Coxeter planeB6B5B4
Graph
Dihedral symmetry[12][10][8]
Coxeter planeB3B2
Graph
Dihedral symmetry[6][4]
Coxeter planeA5A3
Graph
Dihedral symmetry[6][4]

Steriruncicantellated 6-cube

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

Alternate names

  • Celliprismatorhombated hexeract (Acronym: coprix) (Jonathan Bowers)[7]

Images

orthographic projections
Coxeter planeB6B5B4
Graph
Dihedral symmetry[12][10][8]
Coxeter planeB3B2
Graph
Dihedral symmetry[6][4]
Coxeter planeA5A3
Graph
Dihedral symmetry[6][4]

Steriruncicantitruncated 6-cube

Steriuncicantitruncated 6-cube
Typeuniform 6-polytope
Schläfli symboltr2r{4,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces
4-faces
Cells
Faces
Edges69120
Vertices23040
Vertex figure
Coxeter groupsB6, [4,3,3,3,3]
Propertiesconvex

Alternate names

  • Great cellated hexeract (Acronym: gocax) (Jonathan Bowers)[8]

Images

orthographic projections
Coxeter planeB6B5B4
Graph
Dihedral symmetry[12][10][8]
Coxeter planeB3B2
Graph
Dihedral symmetry[6][4]
Coxeter planeA5A3
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

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)".
Fundamental convex regular and uniform polytopes in dimensions 2–10
FamilyAnBnI2(p) / DnE6 / E7 / E8 / F4 / G2Hn
Regular polygonTriangleSquarep-gonHexagonPentagon
Uniform polyhedronTetrahedronOctahedron • CubeDemicubeDodecahedron • Icosahedron
Uniform polychoronPentachoron16-cell • TesseractDemitesseract24-cell120-cell • 600-cell
Uniform 5-polytope5-simplex5-orthoplex • 5-cube5-demicube
Uniform 6-polytope6-simplex6-orthoplex • 6-cube6-demicube122 • 221
Uniform 7-polytope7-simplex7-orthoplex • 7-cube7-demicube132 • 231 • 321
Uniform 8-polytope8-simplex8-orthoplex • 8-cube8-demicube142 • 241 • 421
Uniform 9-polytope9-simplex9-orthoplex • 9-cube9-demicube
Uniform 10-polytope10-simplex10-orthoplex • 10-cube10-demicube
Uniform n-polytopen-simplexn-orthoplex • n-cuben-demicube1k2 • 2k1 • k21n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds
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