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

Cantellated 7-simplex
Typeuniform 7-polytope
Schläfli symbolrr{3,3,3,3,3,3}
or
Coxeter-Dynkin diagram
or
6-faces
5-faces
4-faces
Cells
Faces
Edges1008
Vertices168
Vertex figure5-simplex prism
Coxeter groupsA7, [3,3,3,3,3,3]
Propertiesconvex

Alternate names

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

Coordinates

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

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

Bicantellated 7-simplex
Typeuniform 7-polytope
Schläfli symbolr2r{3,3,3,3,3,3}
or
Coxeter-Dynkin diagrams
or
6-faces
5-faces
4-faces
Cells
Faces
Edges2520
Vertices420
Vertex figure
Coxeter groupsA7, [3,3,3,3,3,3]
Propertiesconvex

Alternate names

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

Coordinates

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

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

Tricantellated 7-simplex
Typeuniform 7-polytope
Schläfli symbolr3r{3,3,3,3,3,3}
or
Coxeter-Dynkin diagrams
or
6-faces
5-faces
4-faces
Cells
Faces
Edges3360
Vertices560
Vertex figure
Coxeter groupsA7, [3,3,3,3,3,3]
Propertiesconvex

Alternate names

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

Coordinates

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

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

Cantitruncated 7-simplex
Typeuniform 7-polytope
Schläfli symboltr{3,3,3,3,3,3}
or
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges1176
Vertices336
Vertex figure
Coxeter groupsA7, [3,3,3,3,3,3]
Propertiesconvex

Alternate names

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

Coordinates

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

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

Bicantitruncated 7-simplex
Typeuniform 7-polytope
Schläfli symbolt2r{3,3,3,3,3,3}
or
Coxeter-Dynkin diagrams
or
6-faces
5-faces
4-faces
Cells
Faces
Edges2940
Vertices840
Vertex figure
Coxeter groupsA7, [3,3,3,3,3,3]
Propertiesconvex

Alternate names

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

Coordinates

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

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

Tricantitruncated 7-simplex
Typeuniform 7-polytope
Schläfli symbolt3r{3,3,3,3,3,3}
or
Coxeter-Dynkin diagrams
or
6-faces
5-faces
4-faces
Cells
Faces
Edges3920
Vertices1120
Vertex figure
Coxeter groupsA7, [3,3,3,3,3,3]
Propertiesconvex

Alternate names

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

Coordinates

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

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]]

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

Notes

  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

  • 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
      • (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
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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