Cantellated 7-simplexes

  (Redirected from Cantitruncated 7-simplex)
7-simplex t0.svg
7-simplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
7-simplex t02.svg
Cantellated 7-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.pngCDel 3.pngCDel node.png
7-simplex t13.svg
Bicantellated 7-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.pngCDel 3.pngCDel node.png
7-simplex t24.svg
Tricantellated 7-simplex
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
7-simplex t2.svg
Birectified 7-simplex
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
7-simplex t012.svg
Cantitruncated 7-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.pngCDel 3.pngCDel node.png
7-simplex t123.svg
Bicantitruncated 7-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.pngCDel 3.pngCDel node.png
7-simplex t234.svg
Tricantitruncated 7-simplex
CDel node.pngCDel 3.pngCDel 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 A7 Coxeter plane

In seven-dimensional geometry, a cantellated 7-simplex is a convex uniform 7-polytope, being a cantellation of the regular 7-simplex.

There are unique 6 degrees of cantellation for the 7-simplex, including truncations.

Cantellated 7-simplexEdit

Cantellated 7-simplex
Type uniform 7-polytope
Schläfli symbol rr{3,3,3,3,3,3}
or  
Coxeter-Dynkin diagram              
or            
6-faces
5-faces
4-faces
Cells
Faces
Edges 1008
Vertices 168
Vertex figure 5-simplex prism
Coxeter groups A7, [3,3,3,3,3,3]
Properties convex

Alternate namesEdit

  • Small rhombated octaexon (acronym: saro) (Jonathan Bowers)[1]

CoordinatesEdit

The vertices of the cantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,0,1,1,2). This construction is based on facets of the cantellated 8-orthoplex.

ImagesEdit

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph      
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph      
Dihedral symmetry [5] [4] [3]

Bicantellated 7-simplexEdit

Bicantellated 7-simplex
Type uniform 7-polytope
Schläfli symbol r2r{3,3,3,3,3,3}
or  
Coxeter-Dynkin diagrams              
or          
6-faces
5-faces
4-faces
Cells
Faces
Edges 2520
Vertices 420
Vertex figure
Coxeter groups A7, [3,3,3,3,3,3]
Properties convex

Alternate namesEdit

  • Small birhombated octaexon (acronym: sabro) (Jonathan Bowers)[2]

CoordinatesEdit

The vertices of the bicantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,1,1,2,2). This construction is based on facets of the bicantellated 8-orthoplex.

ImagesEdit

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph      
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph      
Dihedral symmetry [5] [4] [3]

Tricantellated 7-simplexEdit

Tricantellated 7-simplex
Type uniform 7-polytope
Schläfli symbol r3r{3,3,3,3,3,3}
or  
Coxeter-Dynkin diagrams              
or        
6-faces
5-faces
4-faces
Cells
Faces
Edges 3360
Vertices 560
Vertex figure
Coxeter groups A7, [3,3,3,3,3,3]
Properties convex

Alternate namesEdit

  • Small trirhombihexadecaexon (stiroh) (Jonathan Bowers)[3]

CoordinatesEdit

The vertices of the tricantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,1,2,2,2). This construction is based on facets of the tricantellated 8-orthoplex.

ImagesEdit

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph      
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph      
Dihedral symmetry [5] [4] [3]

Cantitruncated 7-simplexEdit

Cantitruncated 7-simplex
Type uniform 7-polytope
Schläfli symbol tr{3,3,3,3,3,3}
or  
Coxeter-Dynkin diagrams              
           
6-faces
5-faces
4-faces
Cells
Faces
Edges 1176
Vertices 336
Vertex figure
Coxeter groups A7, [3,3,3,3,3,3]
Properties convex

Alternate namesEdit

  • Great rhombated octaexon (acronym: garo) (Jonathan Bowers)[4]

CoordinatesEdit

The vertices of the cantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,0,1,2,3). This construction is based on facets of the cantitruncated 8-orthoplex.

ImagesEdit

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph      
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph      
Dihedral symmetry [5] [4] [3]

Bicantitruncated 7-simplexEdit

Bicantitruncated 7-simplex
Type uniform 7-polytope
Schläfli symbol t2r{3,3,3,3,3,3}
or  
Coxeter-Dynkin diagrams              
or          
6-faces
5-faces
4-faces
Cells
Faces
Edges 2940
Vertices 840
Vertex figure
Coxeter groups A7, [3,3,3,3,3,3]
Properties convex

Alternate namesEdit

  • Great birhombated octaexon (acronym: gabro) (Jonathan Bowers)[5]

CoordinatesEdit

The vertices of the bicantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,1,2,3,3). This construction is based on facets of the bicantitruncated 8-orthoplex.

ImagesEdit

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph      
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph      
Dihedral symmetry [5] [4] [3]

Tricantitruncated 7-simplexEdit

Tricantitruncated 7-simplex
Type uniform 7-polytope
Schläfli symbol t3r{3,3,3,3,3,3}
or  
Coxeter-Dynkin diagrams              
or        
6-faces
5-faces
4-faces
Cells
Faces
Edges 3920
Vertices 1120
Vertex figure
Coxeter groups A7, [3,3,3,3,3,3]
Properties convex

Alternate namesEdit

  • Great trirhombihexadecaexon (acronym: gatroh) (Jonathan Bowers)[6]

CoordinatesEdit

The vertices of the tricantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,2,3,4,4). This construction is based on facets of the tricantitruncated 8-orthoplex.

ImagesEdit

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph      
Dihedral symmetry [8] [[7]] [6]
Ak Coxeter plane A4 A3 A2
Graph      
Dihedral symmetry [[5]] [4] [[3]]

Related polytopesEdit

This polytope is one of 71 uniform 7-polytopes with A7 symmetry.

See alsoEdit

NotesEdit

  1. ^ Klitizing, (x3o3x3o3o3o3o - saro)
  2. ^ Klitizing, (o3x3o3x3o3o3o - sabro)
  3. ^ Klitizing, (o3o3x3o3x3o3o - stiroh)
  4. ^ Klitizing, (x3x3x3o3o3o3o - garo)
  5. ^ Klitizing, (o3x3x3x3o3o3o - gabro)
  6. ^ Klitizing, (o3o3x3x3x3o3o - gatroh)

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. "7D uniform polytopes (polyexa)". x3o3x3o3o3o3o - saro, o3x3o3x3o3o3o - sabro, o3o3x3o3x3o3o - stiroh, x3x3x3o3o3o3o - garo, o3x3x3x3o3o3o - gabro, o3o3x3x3x3o3o - gatroh

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