6-cube
Runcinated 6-cube
Biruncinated 6-cube
Runcinated 6-orthoplex
6-orthoplex

Runcitruncated 6-cube
Biruncitruncated 6-cube
Runcicantellated 6-orthoplex
Runcicantellated 6-cube
Biruncitruncated 6-orthoplex

Runcitruncated 6-orthoplex
Runcicanti-truncated 6-cube
Biruncicanti-truncated 6-cube
Runcicanti-truncated 6-orthoplex
Orthogonal projections in B6 Coxeter plane

In six-dimensional geometry, a runcinated 6-cube is a convex uniform 6-polytope with 3rd order truncations (runcination) of the regular 6-cube.

There are 12 unique runcinations of the 6-cube with permutations of truncations, and cantellations. Half are expressed relative to the dual 6-orthoplex.

Runcinated 6-cube

Runcinated 6-cube
TypeUniform 6-polytope
Schläfli symbolt0,3{4,3,3,3,3}
Coxeter-Dynkin diagram
4-faces
Cells
Faces
Edges7680
Vertices1280
Vertex figure
Coxeter groupB6 [4,3,3,3,3]
Propertiesconvex

Alternate names

  • Small prismated hexeract (spox) (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]

Biruncinated 6-cube

Biruncinated 6-cube
TypeUniform 6-polytope
Schläfli symbolt1,4{4,3,3,3,3}
Coxeter-Dynkin diagram
4-faces
Cells
Faces
Edges11520
Vertices1920
Vertex figure
Coxeter groupB6 [4,3,3,3,3]
Propertiesconvex

Alternate names

  • Small biprismated hexeractihexacontatetrapeton (sobpoxog) (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]

Runcitruncated 6-cube

Runcitruncated 6-cube
TypeUniform 6-polytope
Schläfli symbolt0,1,3{4,3,3,3,3}
Coxeter-Dynkin diagram
4-faces
Cells
Faces
Edges17280
Vertices3840
Vertex figure
Coxeter groupB6 [4,3,3,3,3]
Propertiesconvex

Alternate names

  • Prismatotruncated hexeract (potax) (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]

Biruncitruncated 6-cube

Biruncitruncated 6-cube
TypeUniform 6-polytope
Schläfli symbolt1,2,4{4,3,3,3,3}
Coxeter-Dynkin diagram
4-faces
Cells
Faces
Edges23040
Vertices5760
Vertex figure
Coxeter groupB6 [4,3,3,3,3]
Propertiesconvex

Alternate names

  • Biprismatotruncated hexeract (boprag) (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]

Runcicantellated 6-cube

Runcicantellated 6-cube
TypeUniform 6-polytope
Schläfli symbolt0,2,3{4,3,3,3,3}
Coxeter-Dynkin diagram
4-faces
Cells
Faces
Edges13440
Vertices3840
Vertex figure
Coxeter groupB6 [4,3,3,3,3]
Propertiesconvex

Alternate names

  • Prismatorhombated hexeract (prox) (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]

Runcicantitruncated 6-cube

Runcicantitruncated 6-cube
TypeUniform 6-polytope
Schläfli symbolt0,1,2,3{4,3,3,3,3}
Coxeter-Dynkin diagram
4-faces
Cells
Faces
Edges23040
Vertices7680
Vertex figure
Coxeter groupB6 [4,3,3,3,3]
Propertiesconvex

Alternate names

  • Great prismated hexeract (gippox) (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]

Biruncitruncated 6-cube

Biruncitruncated 6-cube
TypeUniform 6-polytope
Schläfli symbolt1,2,3,4{4,3,3,3,3}
Coxeter-Dynkin diagram
4-faces
Cells
Faces
Edges23040
Vertices5760
Vertex figure
Coxeter groupB6 [4,3,3,3,3]
Propertiesconvex

Alternate names

  • Biprismatotruncated hexeract (boprag) (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]

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, (o3o3x3o3o4x - spox)
  2. ↑ Klitzing, (o3x3o3o3x4o - sobpoxog)
  3. ↑ Klitzing, (o3o3x3o3x4x - potax)
  4. ↑ Klitzing, (o3x3o3x3x4o - boprag)
  5. ↑ Klitzing, (o3o3x3x3o4x - prox)
  6. ↑ Klitzing, (o3o3x3x3x4x - gippox)
  7. ↑ Klitzing, (o3x3x3x3x4o - boprag)

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)". o3o3x3o3o4x - spox, o3x3o3o3x4o - sobpoxog, o3o3x3o3x4x - potax, o3x3o3x3x4o - boprag, o3o3x3x3o4x - prox, o3o3x3x3x4x - gippox, o3x3x3x3x4o - boprag
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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