Cantellated 6-simplexes


6-simplex
Cantellated 6-simplex
Bicantellated 6-simplex

Birectified 6-simplex
Cantitruncated 6-simplex
Bicantitruncated 6-simplex
Orthogonal projections in A6 Coxeter plane

In six-dimensional geometry, a cantellated 6-simplex is a convex uniform 6-polytope, being a cantellation of the regular 6-simplex.

There are unique 4 degrees of cantellation for the 6-simplex, including truncations.

Cantellated 6-simplex

Cantellated 6-simplex
Typeuniform 6-polytope
Schläfli symbolrr{3,3,3,3,3}
or
Coxeter-Dynkin diagrams
5-faces35
4-faces210
Cells560
Faces805
Edges525
Vertices105
Vertex figure5-cell prism
Coxeter groupA6, [35], order 5040
Propertiesconvex

Alternate names

  • Small rhombated heptapeton (Acronym: sril) (Jonathan Bowers)[1]

Coordinates

The vertices of the cantellated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,0,1,1,2). This construction is based on facets of the cantellated 7-orthoplex.

Images

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

[2]

Bicantellated 6-simplex

Bicantellated 6-simplex
Typeuniform 6-polytope
Schläfli symbol2rr{3,3,3,3,3}
or
Coxeter-Dynkin diagrams
5-faces49
4-faces329
Cells980
Faces1540
Edges1050
Vertices210
Vertex figure
Coxeter groupA6, [35], order 5040
Propertiesconvex

Alternate names

  • Small prismated heptapeton (Acronym: sabril) (Jonathan Bowers)[3]

Coordinates

The vertices of the bicantellated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,1,1,2,2). This construction is based on facets of the bicantellated 7-orthoplex.

Images

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

Cantitruncated 6-simplex

cantitruncated 6-simplex
Typeuniform 6-polytope
Schläfli symboltr{3,3,3,3,3}
or
Coxeter-Dynkin diagrams
5-faces35
4-faces210
Cells560
Faces805
Edges630
Vertices210
Vertex figure
Coxeter groupA6, [35], order 5040
Propertiesconvex

Alternate names

  • Great rhombated heptapeton (Acronym: gril) (Jonathan Bowers)[4]

Coordinates

The vertices of the cantitruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,0,1,2,3). This construction is based on facets of the cantitruncated 7-orthoplex.

Images

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

Bicantitruncated 6-simplex

bicantitruncated 6-simplex
Typeuniform 6-polytope
Schläfli symbol2tr{3,3,3,3,3}
or
Coxeter-Dynkin diagrams
5-faces49
4-faces329
Cells980
Faces1540
Edges1260
Vertices420
Vertex figure
Coxeter groupA6, [35], order 5040
Propertiesconvex

Alternate names

  • Great birhombated heptapeton (Acronym: gabril) (Jonathan Bowers)[5]

Coordinates

The vertices of the bicantitruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,1,2,3,3). This construction is based on facets of the bicantitruncated 7-orthoplex.

Images

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

The truncated 6-simplex is one of 35 uniform 6-polytopes based on the [3,3,3,3,3] Coxeter group, all shown here in A6 Coxeter plane orthographic projections.

Notes

  1. ↑ Klitizing, (x3o3x3o3o3o - sril)
  2. ↑ Klitzing, (x3o3x3o3o3o - sril)
  3. ↑ Klitzing, (o3x3o3x3o3o - sabril)
  4. ↑ Klitzing, (x3x3x3o3o3o - gril)
  5. ↑ Klitzing, (o3x3x3x3o3o - gabril)

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. "6D uniform polytopes (polypeta)". x3o3x3o3o3o - sril, o3x3o3x3o3o - sabril, x3x3x3o3o3o - gril, o3x3x3x3o3o - gabril
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 4-polytope 5-cell 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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