Truncated 6-cubes

(Redirected from Bitruncated 6-cube)

6-cube

Truncated 6-cube

Bitruncated 6-cube

Tritruncated 6-cube

6-orthoplex

Truncated 6-orthoplex

Bitruncated 6-orthoplex
Orthogonal projections in B6 Coxeter plane

In six-dimensional geometry, a truncated 6-cube (or truncated hexeract) is a convex uniform 6-polytope, being a truncation of the regular 6-cube.

There are 5 truncations for the 6-cube. Vertices of the truncated 6-cube are located as pairs on the edge of the 6-cube. Vertices of the bitruncated 6-cube are located on the square faces of the 6-cube. Vertices of the tritruncated 6-cube are located inside the cubic cells of the 6-cube.

Truncated 6-cube

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Truncated 6-cube
Type uniform 6-polytope
Class B6 polytope
Schläfli symbol t{4,3,3,3,3}
Coxeter-Dynkin diagrams            
5-faces 76
4-faces 464
Cells 1120
Faces 1520
Edges 1152
Vertices 384
Vertex figure  
( )v{3,3,3}
Coxeter groups B6, [3,3,3,3,4]
Properties convex

Alternate names

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  • Truncated hexeract (Acronym: tox) (Jonathan Bowers)[1]

Construction and coordinates

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The truncated 6-cube may be constructed by truncating the vertices of the 6-cube at   of the edge length. A regular 5-simplex replaces each original vertex.

The Cartesian coordinates of the vertices of a truncated 6-cube having edge length 2 are the permutations of:

 

Images

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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]
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The truncated 6-cube, is fifth in a sequence of truncated hypercubes:

Truncated hypercubes
Image                     ...
Name Octagon Truncated cube Truncated tesseract Truncated 5-cube Truncated 6-cube Truncated 7-cube Truncated 8-cube
Coxeter diagram                                                                      
Vertex figure ( )v( )  
( )v{ }
 
( )v{3}
 
( )v{3,3}
( )v{3,3,3} ( )v{3,3,3,3} ( )v{3,3,3,3,3}

Bitruncated 6-cube

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

Alternate names

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  • Bitruncated hexeract (Acronym: botox) (Jonathan Bowers)[2]

Construction and coordinates

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The Cartesian coordinates of the vertices of a bitruncated 6-cube having edge length 2 are the permutations of:

 

Images

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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]
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The bitruncated 6-cube is fourth in a sequence of bitruncated hypercubes:

Bitruncated hypercubes
Image                   ...
Name Bitruncated cube Bitruncated tesseract Bitruncated 5-cube Bitruncated 6-cube Bitruncated 7-cube Bitruncated 8-cube
Coxeter                                                                  
Vertex figure  
( )v{ }
 
{ }v{ }
 
{ }v{3}
 
{ }v{3,3}
{ }v{3,3,3} { }v{3,3,3,3}

Tritruncated 6-cube

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

Alternate names

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  • Tritruncated hexeract (Acronym: xog) (Jonathan Bowers)[4]

Construction and coordinates

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The Cartesian coordinates of the vertices of a tritruncated 6-cube having edge length 2 are the permutations of:

 

Images

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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]
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2-isotopic hypercubes
Dim. 2 3 4 5 6 7 8 n
Name t{4} r{4,3} 2t{4,3,3} 2r{4,3,3,3} 3t{4,3,3,3,3} 3r{4,3,3,3,3,3} 4t{4,3,3,3,3,3,3} ...
Coxeter
diagram
                                      
Images                    
Facets {3}  
{4}  
t{3,3}  
t{3,4}  
r{3,3,3}  
r{3,3,4}  
2t{3,3,3,3}  
2t{3,3,3,4}  
2r{3,3,3,3,3}  
2r{3,3,3,3,4}  
3t{3,3,3,3,3,3}  
3t{3,3,3,3,3,4}  
Vertex
figure
( )v( )  
{ }×{ }
 
{ }v{ }
 
{3}×{4}
 
{3}v{4}
{3,3}×{3,4} {3,3}v{3,4}
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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

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  1. ^ Klitzing, (o3o3o3o3x4x - tox)
  2. ^ Klitzing, (o3o3o3x3x4o - botox)
  3. ^ "Squete".
  4. ^ Klitzing, (o3o3x3x3o4o - xog)

References

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  • 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)". o3o3o3o3x4x - tox, o3o3o3x3x4o - botox, o3o3x3x3o4o - xog
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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 polychoron Pentachoron 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