6-simplex

Rectified 6-simplex

Birectified 6-simplex
Orthogonal projections in A6 Coxeter plane

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

There are three unique degrees of rectifications, including the zeroth, the 6-simplex itself. Vertices of the rectified 6-simplex are located at the edge-centers of the 6-simplex. Vertices of the birectified 6-simplex are located in the triangular face centers of the 6-simplex.

Rectified 6-simplex

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Rectified 6-simplex
Type uniform polypeton
Schläfli symbol t1{35}
r{35} = {34,1}
or  
Coxeter diagrams            
         
Elements

f5 = 14, f4 = 63, C = 140, F = 175, E = 105, V = 21
(χ=0)

Coxeter group A6, [35], order 5040
Bowers name
and (acronym)
Rectified heptapeton
(ril)
Vertex figure 5-cell prism
Circumradius 0.845154
Properties convex, isogonal

E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S1
6
. It is also called 04,1 for its branching Coxeter-Dynkin diagram, shown as          .

Alternate names

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  • Rectified heptapeton (Acronym: ril) (Jonathan Bowers)

Coordinates

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The vertices of the rectified 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,0,0,1,1). This construction is based on facets of the rectified 7-orthoplex.

Images

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orthographic projections
Ak Coxeter plane A6 A5 A4
Graph      
Dihedral symmetry [7] [6] [5]
Ak Coxeter plane A3 A2
Graph    
Dihedral symmetry [4] [3]

Birectified 6-simplex

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Birectified 6-simplex
Type uniform 6-polytope
Class A6 polytope
Schläfli symbol t2{3,3,3,3,3}
2r{35} = {33,2}
or  
Coxeter symbol 032
Coxeter diagrams            
       
5-faces 14 total:
7 t1{3,3,3,3}
7 t2{3,3,3,3}
4-faces 84
Cells 245
Faces 350
Edges 210
Vertices 35
Vertex figure {3}x{3,3}
Petrie polygon Heptagon
Coxeter groups A6, [3,3,3,3,3]
Properties convex

E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S2
6
. It is also called 03,2 for its branching Coxeter-Dynkin diagram, shown as        .

Alternate names

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  • Birectified heptapeton (Acronym: bril) (Jonathan Bowers)

Coordinates

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The vertices of the birectified 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,0,1,1,1). This construction is based on facets of the birectified 7-orthoplex.

Images

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orthographic projections
Ak Coxeter plane A6 A5 A4
Graph      
Dihedral symmetry [7] [6] [5]
Ak Coxeter plane A3 A2
Graph    
Dihedral symmetry [4] [3]
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The rectified 6-simplex polytope is the vertex figure of the 7-demicube, and the edge figure of the uniform 241 polytope.

These polytopes are a part of 35 uniform 6-polytopes based on the [3,3,3,3,3] Coxeter group, all shown here in A6 Coxeter plane orthographic projections.

A6 polytopes
 
t0
 
t1
 
t2
 
t0,1
 
t0,2
 
t1,2
 
t0,3
 
t1,3
 
t2,3
 
t0,4
 
t1,4
 
t0,5
 
t0,1,2
 
t0,1,3
 
t0,2,3
 
t1,2,3
 
t0,1,4
 
t0,2,4
 
t1,2,4
 
t0,3,4
 
t0,1,5
 
t0,2,5
 
t0,1,2,3
 
t0,1,2,4
 
t0,1,3,4
 
t0,2,3,4
 
t1,2,3,4
 
t0,1,2,5
 
t0,1,3,5
 
t0,2,3,5
 
t0,1,4,5
 
t0,1,2,3,4
 
t0,1,2,3,5
 
t0,1,2,4,5
 
t0,1,2,3,4,5

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)". o3x3o3o3o3o - ril, o3x3o3o3o3o - bril
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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