Rectified 6-cubes

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6-cube

Rectified 6-cube

Birectified 6-cube

Birectified 6-orthoplex

Rectified 6-orthoplex

6-orthoplex
Orthogonal projections in A6 Coxeter plane

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

There are unique 6 degrees of rectifications, the zeroth being the 6-cube, and the 6th and last being the 6-orthoplex. Vertices of the rectified 6-cube are located at the edge-centers of the 6-cube. Vertices of the birectified 6-cube are located in the square face centers of the 6-cube.

Rectified 6-cube

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Rectified 6-cube
Type uniform 6-polytope
Schläfli symbol t1{4,34} or r{4,34}
 
Coxeter-Dynkin diagrams             =          
         
5-faces 76
4-faces 444
Cells 1120
Faces 1520
Edges 960
Vertices 192
Vertex figure 5-cell prism
Petrie polygon Dodecagon
Coxeter groups B6, [3,3,3,3,4]
D6, [33,1,1]
Properties convex

Alternate names

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  • Rectified hexeract (acronym: rax) (Jonathan Bowers)

Construction

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The rectified 6-cube may be constructed from the 6-cube by truncating its vertices at the midpoints of its edges.

Coordinates

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The Cartesian coordinates of the vertices of the rectified 6-cube with edge length 2 are all 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]

Birectified 6-cube

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Birectified 6-cube
Type uniform 6-polytope
Coxeter symbol 0311
Schläfli symbol t2{4,34} or 2r{4,34}
 
Coxeter-Dynkin diagrams             =        
          =      
5-faces 76
4-faces 636
Cells 2080
Faces 3200
Edges 1920
Vertices 240
Vertex figure {4}x{3,3} duoprism
Coxeter groups B6, [3,3,3,3,4]
D6, [33,1,1]
Properties convex

Alternate names

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  • Birectified hexeract (acronym: brox) (Jonathan Bowers)
  • Rectified 6-demicube

Construction

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The birectified 6-cube may be constructed from the 6-cube by truncating its vertices at the midpoints of its edges.

Coordinates

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The Cartesian coordinates of the vertices of the rectified 6-cube with edge length 2 are all 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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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.

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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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)". o3x3o3o3o4o - rax, o3o3x3o3o4o - brox,
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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