Stericated 6-simplexes

In six-dimensional geometry, a stericated 6-simplex is a convex uniform 6-polytope with 4th order truncations (sterication) of the regular 6-simplex.


6-simplex
Stericated 6-simplex
Steritruncated 6-simplex

Stericantellated 6-simplex
Stericantitruncated 6-simplex
Steriruncinated 6-simplex

Steriruncitruncated 6-simplex
Steriruncicantellated 6-simplex
Steriruncicantitruncated 6-simplex
Orthogonal projections in A6 Coxeter plane

There are 8 unique sterications for the 6-simplex with permutations of truncations, cantellations, and runcinations.

Stericated 6-simplex
Stericated 6-simplex
Typeuniform 6-polytope
Schläfli symbolt0,4{3,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces105
4-faces700
Cells1470
Faces1400
Edges630
Vertices105
Vertex figure
Coxeter groupA6, [35], order 5040
Propertiesconvex

Alternate names
  • Small cellated heptapeton (Acronym: scal) (Jonathan Bowers)[1]

Coordinates

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

Steritruncated 6-simplex
Steritruncated 6-simplex
Typeuniform 6-polytope
Schläfli symbolt0,1,4{3,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces105
4-faces945
Cells2940
Faces3780
Edges2100
Vertices420
Vertex figure
Coxeter groupA6, [35], order 5040
Propertiesconvex

Alternate names
  • Cellitruncated heptapeton (Acronym: catal) (Jonathan Bowers)[2]

Coordinates

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

Stericantellated 6-simplex
Stericantellated 6-simplex
Typeuniform 6-polytope
Schläfli symbolt0,2,4{3,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces105
4-faces1050
Cells3465
Faces5040
Edges3150
Vertices630
Vertex figure
Coxeter groupA6, [35], order 5040
Propertiesconvex

Alternate names
  • Cellirhombated heptapeton (Acronym: cral) (Jonathan Bowers)[3]

Coordinates

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

Stericantitruncated 6-simplex
stericantitruncated 6-simplex
Typeuniform 6-polytope
Schläfli symbolt0,1,2,4{3,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces105
4-faces1155
Cells4410
Faces7140
Edges5040
Vertices1260
Vertex figure
Coxeter groupA6, [35], order 5040
Propertiesconvex

Alternate names
  • Celligreatorhombated heptapeton (Acronym: cagral) (Jonathan Bowers)[4]

Coordinates

The vertices of the stericanttruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,1,2,3,4). This construction is based on facets of the stericantitruncated 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]

Steriruncinated 6-simplex
steriruncinated 6-simplex
Typeuniform 6-polytope
Schläfli symbolt0,3,4{3,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces105
4-faces700
Cells1995
Faces2660
Edges1680
Vertices420
Vertex figure
Coxeter groupA6, [35], order 5040
Propertiesconvex

Alternate names
  • Celliprismated heptapeton (Acronym: copal) (Jonathan Bowers)[5]

Coordinates

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

Steriruncitruncated 6-simplex
steriruncitruncated 6-simplex
Typeuniform 6-polytope
Schläfli symbolt0,1,3,4{3,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces105
4-faces945
Cells3360
Faces5670
Edges4410
Vertices1260
Vertex figure
Coxeter groupA6, [35], order 5040
Propertiesconvex

Alternate names
  • Celliprismatotruncated heptapeton (Acronym: captal) (Jonathan Bowers)[6]

Coordinates

The vertices of the steriruncittruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,1,2,3,4). This construction is based on facets of the steriruncitruncated 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]

Steriruncicantellated 6-simplex
steriruncicantellated 6-simplex
Typeuniform 6-polytope
Schläfli symbolt0,2,3,4{3,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces105
4-faces1050
Cells3675
Faces5880
Edges4410
Vertices1260
Vertex figure
Coxeter groupA6, [35], order 5040
Propertiesconvex

Alternate names
  • Bistericantitruncated 6-simplex as t1,2,3,5{3,3,3,3,3}
  • Celliprismatorhombated heptapeton (Acronym: copril) (Jonathan Bowers)[7]

Coordinates

The vertices of the steriruncitcantellated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,1,2,3,4). This construction is based on facets of the steriruncicantellated 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]

Steriruncicantitruncated 6-simplex
Steriuncicantitruncated 6-simplex
Typeuniform 6-polytope
Schläfli symbolt0,1,2,3,4{3,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces105
4-faces1155
Cells4620
Faces8610
Edges7560
Vertices2520
Vertex figure
Coxeter groupA6, [35], order 5040
Propertiesconvex

Alternate names
  • Great cellated heptapeton (Acronym: gacal) (Jonathan Bowers)[8]

Coordinates

The vertices of the steriruncicantittruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,1,2,3,4,5). This construction is based on facets of the steriruncicantitruncated 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. Klitzing, (x3o3o3o3x3o - scal)
  2. Klitzing, (x3x3o3o3x3o - catal)
  3. Klitzing, (x3o3x3o3x3o - cral)
  4. Klitzing, (x3x3x3o3x3o - cagral)
  5. Klitzing, (x3o3o3x3x3o - copal)
  6. Klitzing, (x3x3o3x3x3o - captal)
  7. Klitzing, ( x3o3x3x3x3o - copril)
  8. Klitzing, (x3x3x3x3x3o - gacal)

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)".
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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