Runcinated 6-cubes

  (Redirected from Biruncitruncated 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 t03.svg
Runcinated 6-cube
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
6-cube t14.svg
Biruncinated 6-cube
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
6-cube t25.svg
Runcinated 6-orthoplex
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.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 t013.svg
Runcitruncated 6-cube
CDel node 1.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 t124.svg
Biruncitruncated 6-cube
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
6-cube t235.svg
Runcicantellated 6-orthoplex
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
6-cube t023.svg
Runcicantellated 6-cube
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
6-cube t134.svg
Biruncitruncated 6-orthoplex
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
6-cube t245.svg
Runcitruncated 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 1.png
6-cube t0123.svg
Runcicanti-truncated 6-cube
CDel node 1.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 t1234.svg
Biruncicanti-truncated 6-cube
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
6-cube t2345.svg
Runcicanti-truncated 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 1.png
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-cubeEdit

Runcinated 6-cube
Type Uniform 6-polytope
Schläfli symbol t0,3{4,3,3,3,3}
Coxeter-Dynkin diagram            
4-faces
Cells
Faces
Edges 7680
Vertices 1280
Vertex figure
Coxeter group B6 [4,3,3,3,3]
Properties convex

Alternate namesEdit

  • Small prismated hexeract (spox) (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]

Biruncinated 6-cubeEdit

Biruncinated 6-cube
Type Uniform 6-polytope
Schläfli symbol t1,4{4,3,3,3,3}
Coxeter-Dynkin diagram            
4-faces
Cells
Faces
Edges 11520
Vertices 1920
Vertex figure
Coxeter group B6 [4,3,3,3,3]
Properties convex

Alternate namesEdit

  • Small biprismated hexeractihexacontatetrapeton (sobpoxog) (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]

Runcitruncated 6-cubeEdit

Runcitruncated 6-cube
Type Uniform 6-polytope
Schläfli symbol t0,1,3{4,3,3,3,3}
Coxeter-Dynkin diagram            
4-faces
Cells
Faces
Edges 17280
Vertices 3840
Vertex figure
Coxeter group B6 [4,3,3,3,3]
Properties convex

Alternate namesEdit

  • Prismatotruncated hexeract (potax) (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]

Biruncitruncated 6-cubeEdit

Biruncitruncated 6-cube
Type Uniform 6-polytope
Schläfli symbol t1,2,4{4,3,3,3,3}
Coxeter-Dynkin diagram            
4-faces
Cells
Faces
Edges 23040
Vertices 5760
Vertex figure
Coxeter group B6 [4,3,3,3,3]
Properties convex

Alternate namesEdit

  • Biprismatotruncated hexeract (boprag) (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]

Runcicantellated 6-cubeEdit

Runcicantellated 6-cube
Type Uniform 6-polytope
Schläfli symbol t0,2,3{4,3,3,3,3}
Coxeter-Dynkin diagram            
4-faces
Cells
Faces
Edges 13440
Vertices 3840
Vertex figure
Coxeter group B6 [4,3,3,3,3]
Properties convex

Alternate namesEdit

  • Prismatorhombated hexeract (prox) (Jonathan Bowers)[5]

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]

Runcicantitruncated 6-cubeEdit

Runcicantitruncated 6-cube
Type Uniform 6-polytope
Schläfli symbol t0,1,2,3{4,3,3,3,3}
Coxeter-Dynkin diagram            
4-faces
Cells
Faces
Edges 23040
Vertices 7680
Vertex figure
Coxeter group B6 [4,3,3,3,3]
Properties convex

Alternate namesEdit

  • Great prismated hexeract (gippox) (Jonathan Bowers)[6]

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]

Biruncitruncated 6-cubeEdit

Biruncitruncated 6-cube
Type Uniform 6-polytope
Schläfli symbol t1,2,3,4{4,3,3,3,3}
Coxeter-Dynkin diagram            
4-faces
Cells
Faces
Edges 23040
Vertices 5760
Vertex figure
Coxeter group B6 [4,3,3,3,3]
Properties convex

Alternate namesEdit

  • Biprismatotruncated hexeract (boprag) (Jonathan Bowers)[7]

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

NotesEdit

  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)

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

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