Cantellated 6-cubes

  (Redirected from Bicantitruncated 6-cube)
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-cubeEdit

Cantellated 6-cube
Type uniform 6-polytope
Schläfli symbol rr{4,3,3,3,3}
or  
Coxeter-Dynkin diagrams            
         
5-faces
4-faces
Cells
Faces
Edges 4800
Vertices 960
Vertex figure
Coxeter groups B6, [3,3,3,3,4]
Properties convex

Alternate namesEdit

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

ImagesEdit

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]

Bicantellated 6-cubeEdit

Cantellated 6-cube
Type uniform 6-polytope
Schläfli symbol 2rr{4,3,3,3,3}
or  
Coxeter-Dynkin diagrams            
       
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups B6, [3,3,3,3,4]
Properties convex

Alternate namesEdit

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

ImagesEdit

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]

Cantitruncated 6-cubeEdit

Cantellated 6-cube
Type uniform 6-polytope
Schläfli symbol tr{4,3,3,3,3}
or  
Coxeter-Dynkin diagrams            
         
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups B6, [3,3,3,3,4]
Properties convex

Alternate namesEdit

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

ImagesEdit

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]

It is fourth in a series of cantitruncated hypercubes:

Petrie polygon projections
                 
Truncated cuboctahedron Cantitruncated tesseract Cantitruncated 5-cube Cantitruncated 6-cube Cantitruncated 7-cube Cantitruncated 8-cube
                                                                 

Bicantitruncated 6-cubeEdit

Cantellated 6-cube
Type uniform 6-polytope
Schläfli symbol 2tr{4,3,3,3,3}
or  
Coxeter-Dynkin diagrams            
       
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups B6, [3,3,3,3,4]
Properties convex

Alternate namesEdit

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

ImagesEdit

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]

Related polytopesEdit

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.

NotesEdit

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

ReferencesEdit

  • 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 linksEdit

Fundamental convex regular and uniform polytopes in dimensions 2–10
An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Triangle Square p-gon Hexagon Pentagon
Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
5-cell 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
5-simplex 5-orthoplex5-cube 5-demicube
6-simplex 6-orthoplex6-cube 6-demicube 122221
7-simplex 7-orthoplex7-cube 7-demicube 132231321
8-simplex 8-orthoplex8-cube 8-demicube 142241421
9-simplex 9-orthoplex9-cube 9-demicube
10-simplex 10-orthoplex10-cube 10-demicube
n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds