Cantellated 6-simplexes

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


6-simplex
Cantellated 6-simplex
Bicantellated 6-simplex

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

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.

A6 polytopes

t0
t1
t2
t0,1
t0,2
t1,2
t0,3
t1,3
t2,3

t0,4
t1,4
t0,5
t0,1,2
t0,1,3
t0,2,3
t1,2,3
t0,1,4
t0,2,4

t1,2,4
t0,3,4
t0,1,5
t0,2,5
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
t0,1,4,5
t0,1,2,3,4
t0,1,2,3,5
t0,1,2,4,5
t0,1,2,3,4,5

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 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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