Cantellated 6-simplexes

  (Redirected from Cantellated 6-simplex)
6-simplex t0.svg
6-simplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
6-simplex t02.svg
Cantellated 6-simplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
6-simplex t13.svg
Bicantellated 6-simplex
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
6-simplex t2.svg
Birectified 6-simplex
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
6-simplex t012.svg
Cantitruncated 6-simplex
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
6-simplex t123.svg
Bicantitruncated 6-simplex
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
Orthogonal projections in A6 Coxeter plane

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

There are unique 4 degrees of cantellation for the 6-simplex, including truncations.

Cantellated 6-simplexEdit

Cantellated 6-simplex
Type uniform 6-polytope
Schläfli symbol rr{3,3,3,3,3}
or  
Coxeter-Dynkin diagrams            
         
5-faces 35
4-faces 210
Cells 560
Faces 805
Edges 525
Vertices 105
Vertex figure 5-cell prism
Coxeter group A6, [35], order 5040
Properties convex

Alternate namesEdit

  • Small rhombated heptapeton (Acronym: sril) (Jonathan Bowers)[1]

CoordinatesEdit

The vertices of the cantellated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,0,1,1,2). This construction is based on facets of the cantellated 7-orthoplex.

ImagesEdit

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

[2]

Bicantellated 6-simplexEdit

Bicantellated 6-simplex
Type uniform 6-polytope
Schläfli symbol 2rr{3,3,3,3,3}
or  
Coxeter-Dynkin diagrams            
       
5-faces 49
4-faces 329
Cells 980
Faces 1540
Edges 1050
Vertices 210
Vertex figure
Coxeter group A6, [35], order 5040
Properties convex

Alternate namesEdit

  • Small prismated heptapeton (Acronym: sabril) (Jonathan Bowers)[3]

CoordinatesEdit

The vertices of the bicantellated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,1,1,2,2). This construction is based on facets of the bicantellated 7-orthoplex.

ImagesEdit

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

Cantitruncated 6-simplexEdit

cantitruncated 6-simplex
Type uniform 6-polytope
Schläfli symbol tr{3,3,3,3,3}
or  
Coxeter-Dynkin diagrams            
         
5-faces 35
4-faces 210
Cells 560
Faces 805
Edges 630
Vertices 210
Vertex figure
Coxeter group A6, [35], order 5040
Properties convex

Alternate namesEdit

  • Great rhombated heptapeton (Acronym: gril) (Jonathan Bowers)[4]

CoordinatesEdit

The vertices of the cantitruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,0,1,2,3). This construction is based on facets of the cantitruncated 7-orthoplex.

ImagesEdit

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

Bicantitruncated 6-simplexEdit

bicantitruncated 6-simplex
Type uniform 6-polytope
Schläfli symbol 2tr{3,3,3,3,3}
or  
Coxeter-Dynkin diagrams            
       
5-faces 49
4-faces 329
Cells 980
Faces 1540
Edges 1260
Vertices 420
Vertex figure
Coxeter group A6, [35], order 5040
Properties convex

Alternate namesEdit

  • Great birhombated heptapeton (Acronym: gabril) (Jonathan Bowers)[5]

CoordinatesEdit

The vertices of the bicantitruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,1,2,3,3). This construction is based on facets of the bicantitruncated 7-orthoplex.

ImagesEdit

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

Related uniform 6-polytopesEdit

The truncated 6-simplex is one of 35 uniform 6-polytopes based on the [3,3,3,3,3] Coxeter group, all shown here in A6 Coxeter plane orthographic projections.

NotesEdit

  1. ^ Klitizing, (x3o3x3o3o3o - sril)
  2. ^ Klitzing, (x3o3x3o3o3o - sril)
  3. ^ Klitzing, (o3x3o3x3o3o - sabril)
  4. ^ Klitzing, (x3x3x3o3o3o - gril)
  5. ^ Klitzing, (o3x3x3x3o3o - gabril)

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)". x3o3x3o3o3o - sril, o3x3o3x3o3o - sabril, x3x3x3o3o3o - gril, o3x3x3x3o3o - gabril

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