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Cantellated 7-simplexes

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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]]
[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
[edit]
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