Rectified 6-orthoplexes

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6-orthoplex

Rectified 6-orthoplex

Birectified 6-orthoplex

Birectified 6-cube

Rectified 6-cube

6-cube
Orthogonal projections in B6 Coxeter plane

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

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

Rectified 6-orthoplex

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Rectified hexacross
Type uniform 6-polytope
Schläfli symbols t1{34,4} or r{34,4}
 
r{3,3,3,31,1}
Coxeter-Dynkin diagrams             =          
          =      
5-faces 76 total:
64 rectified 5-simplex
12 5-orthoplex
4-faces 576 total:
192 rectified 5-cell
384 5-cell
Cells 1200 total:
240 octahedron
960 tetrahedron
Faces 1120 total:
160 and 960 triangles
Edges 480
Vertices 60
Vertex figure 16-cell prism
Petrie polygon Dodecagon
Coxeter groups B6, [3,3,3,3,4]
D6, [33,1,1]
Properties convex

The rectified 6-orthoplex is the vertex figure for the demihexeractic honeycomb.

          or            

Alternate names

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  • rectified hexacross
  • rectified hexacontitetrapeton (acronym: rag) (Jonathan Bowers)

Construction

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There are two Coxeter groups associated with the rectified hexacross, one with the C6 or [4,3,3,3,3] Coxeter group, and a lower symmetry with two copies of pentacross facets, alternating, with the D6 or [33,1,1] Coxeter group.

Cartesian coordinates

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Cartesian coordinates for the vertices of a rectified hexacross, centered at the origin, edge length   are all permutations of:

(±1,±1,0,0,0,0)

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]

Root vectors

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The 60 vertices represent the root vectors of the simple Lie group D6. The vertices can be seen in 3 hyperplanes, with the 15 vertices rectified 5-simplices cells on opposite sides, and 30 vertices of an expanded 5-simplex passing through the center. When combined with the 12 vertices of the 6-orthoplex, these vertices represent the 72 root vectors of the B6 and C6 simple Lie groups.

The 60 roots of D6 can be geometrically folded into H3 (Icosahedral symmetry), as       to      , creating 2 copies of 30-vertex icosidodecahedra, with the Golden ratio between their radii:[1]

Rectified 6-orthoplex 2 icosidodecahedra
3D (H3 projection) A4/B5/D6 Coxeter plane H2 Coxeter plane
     

Birectified 6-orthoplex

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Birectified 6-orthoplex
Type uniform 6-polytope
Schläfli symbols t2{34,4} or 2r{34,4}
 
t2{3,3,3,31,1}
Coxeter-Dynkin diagrams             =        
          =      
5-faces 76
4-faces 636
Cells 2160
Faces 2880
Edges 1440
Vertices 160
Vertex figure {3}×{3,4} duoprism
Petrie polygon Dodecagon
Coxeter groups B6, [3,3,3,3,4]
D6, [33,1,1]
Properties convex

The birectified 6-orthoplex can tessellation space in the trirectified 6-cubic honeycomb.

Alternate names

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  • birectified hexacross
  • birectified hexacontitetrapeton (acronym: brag) (Jonathan Bowers)

Cartesian coordinates

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Cartesian coordinates for the vertices of a rectified hexacross, centered at the origin, edge length   are all permutations of:

(±1,±1,±1,0,0,0)

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]

It can also be projected into 3D-dimensions as           , a dodecahedron envelope.

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These polytopes are a part a family 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. ^ Icosidodecahedron from D6 John Baez, January 1, 2015

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 - rag, o3o3x3o3o4o - brag
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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