Cantellated 6-cubes

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6-cube t0.svg
6-cube
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
6-cube t02.svg
Cantellated 6-cube
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
6-cube t13.svg
Bicantellated 6-cube
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
6-cube t5.svg
6-orthoplex
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
6-cube t35.svg
Cantellated 6-orthoplex
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
6-cube t24.svg
Bicantellated 6-orthoplex
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
6-cube t012.svg
Cantitruncated 6-cube
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
6-cube t123.svg
Bicantitruncated 6-cube
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
6-cube t234.svg
Bicantitruncated 6-orthoplex
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
6-cube t345.svg
Cantitruncated 6-orthoplex
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Orthogonal projections in B6 Coxeter plane

In six-dimensional geometry, a cantellated 6-cube is a convex uniform 6-polytope, being a cantellation of the regular 6-cube.

There are 8 cantellations for the 6-cube, including truncations. Half of them are more easily constructed from the dual 5-orthoplex.

Cantellated 6-cube[edit]

Cantellated 6-cube
Type uniform 6-polytope
Schläfli symbol rr{4,3,3,3,3}
or
Coxeter-Dynkin diagrams CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
CDel node.pngCDel split1-43.pngCDel nodes 11.pngCDel 3b.pngCDel nodeb.pngCDel 3b.pngCDel nodeb.pngCDel 3b.pngCDel nodeb.png
5-faces
4-faces
Cells
Faces
Edges 4800
Vertices 960
Vertex figure
Coxeter groups B6, [3,3,3,3,4]
Properties convex

Alternate names[edit]

  • Cantellated hexeract
  • Small rhombated hexeract (acronym: srox) (Jonathan Bowers)[1]

Images[edit]

orthographic projections
Coxeter plane B6 B5 B4
Graph 6-cube t02.svg 6-cube t02 B5.svg 6-cube t02 B4.svg
Dihedral symmetry [12] [10] [8]
Coxeter plane B3 B2
Graph 6-cube t02 B3.svg 6-cube t02 B2.svg
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph 6-cube t02 A5.svg 6-cube t02 A3.svg
Dihedral symmetry [6] [4]

Bicantellated 6-cube[edit]

Cantellated 6-cube
Type uniform 6-polytope
Schläfli symbol 2rr{4,3,3,3,3}
or
Coxeter-Dynkin diagrams CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
CDel node.pngCDel split1.pngCDel nodes 11.pngCDel 4a3b.pngCDel nodes.pngCDel 3b.pngCDel nodeb.png
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups B6, [3,3,3,3,4]
Properties convex

Alternate names[edit]

  • Bicantellated hexeract
  • Small birhombated hexeract (acronym: saborx) (Jonathan Bowers)[2]

Images[edit]

orthographic projections
Coxeter plane B6 B5 B4
Graph 6-cube t13.svg 6-cube t13 B5.svg 6-cube t13 B4.svg
Dihedral symmetry [12] [10] [8]
Coxeter plane B3 B2
Graph 6-cube t13 B3.svg 6-cube t13 B2.svg
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph 6-cube t13 A5.svg 6-cube t13 A3.svg
Dihedral symmetry [6] [4]

Cantitruncated 6-cube[edit]

Cantellated 6-cube
Type uniform 6-polytope
Schläfli symbol tr{4,3,3,3,3}
or
Coxeter-Dynkin diagrams CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
CDel node 1.pngCDel split1-43.pngCDel nodes 11.pngCDel 3b.pngCDel nodeb.pngCDel 3b.pngCDel nodeb.pngCDel 3b.pngCDel nodeb.png
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups B6, [3,3,3,3,4]
Properties convex

Alternate names[edit]

  • Cantitruncated hexeract
  • Great rhombihexeract (acronym: grox) (Jonathan Bowers)[3]

Images[edit]

orthographic projections
Coxeter plane B6 B5 B4
Graph 6-cube t012.svg 6-cube t012 B5.svg 6-cube t012 B4.svg
Dihedral symmetry [12] [10] [8]
Coxeter plane B3 B2
Graph 6-cube t012 B3.svg 6-cube t012 B2.svg
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph 6-cube t012 A5.svg 6-cube t012 A3.svg
Dihedral symmetry [6] [4]

Bicantitruncated 6-cube[edit]

Cantellated 6-cube
Type uniform 6-polytope
Schläfli symbol 2tr{4,3,3,3,3}
or
Coxeter-Dynkin diagrams CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
CDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel 4a3b.pngCDel nodes.pngCDel 3b.pngCDel nodeb.png
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups B6, [3,3,3,3,4]
Properties convex

Alternate names[edit]

  • Bicantitruncated hexeract
  • Great birhombihexeract (acronym: gaborx) (Jonathan Bowers)[4]

Images[edit]

orthographic projections
Coxeter plane B6 B5 B4
Graph 6-cube t123.svg 6-cube t123 B5.svg 6-cube t123 B4.svg
Dihedral symmetry [12] [10] [8]
Coxeter plane B3 B2
Graph 6-cube t123 B3.svg 6-cube t123 B2.svg
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph 6-cube t123 A5.svg 6-cube t123 A3.svg
Dihedral symmetry [6] [4]

Related polytopes[edit]

These polytopes are part of a set of 63 uniform 6-polytopes generated from the B6 Coxeter plane, including the regular 6-cube or 6-orthoplex.

Notes[edit]

  1. ^ Klitzing, (o3o3o3x3o4x - srox)
  2. ^ Klitzing, (o3o3x3o3x4o - saborx)
  3. ^ Klitzing, (o3o3o3x3x4x - grox)
  4. ^ Klitzing, (o3o3x3x3x4o - gaborx)

References[edit]

  • 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)". o3o3o3x3o4x - srox, o3o3x3o3x4o - saborx, o3o3o3x3x4x - grox, o3o3x3x3x4o - gaborx

External links[edit]

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 OctahedronCube Demicube DodecahedronIcosahedron
Uniform 4-polytope 5-cell 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