Wikipedia

Cantellated 7-simplexes


7-simplex
Cantellated 7-simplex
Bicantellated 7-simplex
Tricantellated 7-simplex

Birectified 7-simplex
Cantitruncated 7-simplex
Bicantitruncated 7-simplex
Tricantitruncated 7-simplex
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-simplex edit

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 names edit

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

Coordinates edit

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.

Images edit

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-simplex edit

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 names edit

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

Coordinates edit

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.

Images edit

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-simplex edit

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 names edit

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

Coordinates edit

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.

Images edit

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-simplex edit

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 names edit

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

Coordinates edit

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.

Images edit

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-simplex edit

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 names edit

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

Coordinates edit

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.

Images edit

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-simplex edit

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 names edit

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

Coordinates edit

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.

Images edit

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 polytopes edit

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

A7 polytopes
 
t0 		 
t1 		 
t2 		 
t3 		 
t0,1 		 
t0,2 		 
t1,2 		 
t0,3
 
t1,3 		 
t2,3 		 
t0,4 		 
t1,4 		 
t2,4 		 
t0,5 		 
t1,5 		 
t0,6
 
t0,1,2 		 
t0,1,3 		 
t0,2,3 		 
t1,2,3 		 
t0,1,4 		 
t0,2,4 		 
t1,2,4 		 
t0,3,4
 
t1,3,4 		 
t2,3,4 		 
t0,1,5 		 
t0,2,5 		 
t1,2,5 		 
t0,3,5 		 
t1,3,5 		 
t0,4,5
 
t0,1,6 		 
t0,2,6 		 
t0,3,6 		 
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 		 
t1,2,3,5 		 
t0,1,4,5 		 
t0,2,4,5 		 
t1,2,4,5 		 
t0,3,4,5
 
t0,1,2,6 		 
t0,1,3,6 		 
t0,2,3,6 		 
t0,1,4,6 		 
t0,2,4,6 		 
t0,1,5,6 		 
t0,1,2,3,4 		 
t0,1,2,3,5
 
t0,1,2,4,5 		 
t0,1,3,4,5 		 
t0,2,3,4,5 		 
t1,2,3,4,5 		 
t0,1,2,3,6 		 
t0,1,2,4,6 		 
t0,1,3,4,6 		 
t0,2,3,4,6
 
t0,1,2,5,6 		 
t0,1,3,5,6 		 
t0,1,2,3,4,5 		 
t0,1,2,3,4,6 		 
t0,1,2,3,5,6 		 
t0,1,2,4,5,6 		 
t0,1,2,3,4,5,6

See also edit

Notes edit

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

References edit

  • 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 links edit

Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron
Uniform polychoron Pentachoron 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell
Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221
Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321
Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421
Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds
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