Cantellated 7-simplexes

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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-simplex[edit]

Cantellated 7-simplex
Type uniform 7-polytope
Schläfli symbol rr{3,3,3,3,3,3}
or
Coxeter-Dynkin diagram 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
or CDel node.pngCDel split1.pngCDel nodes 11.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
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 7-simplex t02.svg 7-simplex t02 A6.svg 7-simplex t02 A5.svg
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph 7-simplex t02 A4.svg 7-simplex t02 A3.svg 7-simplex t02 A2.svg
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 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
or CDel node.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
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 7-simplex t13.svg 7-simplex t13 A6.svg 7-simplex t13 A5.svg
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph 7-simplex t13 A4.svg 7-simplex t13 A3.svg 7-simplex t13 A2.svg
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 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
or CDel node.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.png
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 7-simplex t24.svg 7-simplex t24 A6.svg 7-simplex t24 A5.svg
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph 7-simplex t24 A4.svg 7-simplex t24 A3.svg 7-simplex t24 A2.svg
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 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
CDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
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 7-simplex t012.svg 7-simplex t012 A6.svg 7-simplex t012 A5.svg
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph 7-simplex t012 A4.svg 7-simplex t012 A3.svg 7-simplex t012 A2.svg
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 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
or CDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
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 7-simplex t123.svg 7-simplex t123 A6.svg 7-simplex t123 A5.svg
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph 7-simplex t123 A4.svg 7-simplex t123 A3.svg 7-simplex t123 A2.svg
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 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
or CDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.png
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 7-simplex t234.svg 7-simplex t234 A6.svg 7-simplex t234 A5.svg
Dihedral symmetry [8] [[7]] [6]
Ak Coxeter plane A4 A3 A2
Graph 7-simplex t234 A4.svg 7-simplex t234 A3.svg 7-simplex t234 A2.svg
Dihedral symmetry [[5]] [4] [[3]]

Related polytopes[edit]

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

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 / E9 / E10 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform 4-polytope 5-cell 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