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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

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

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

Stericated 6-simplex

Stericated 6-simplex
Type uniform 6-polytope
Schläfli symbol t0,4{3,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces 105
4-faces 700
Cells 1470
Faces 1400
Edges 630
Vertices 105
Vertex figure
Coxeter group A6, [35], order 5040
Properties convex

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
Type uniform 6-polytope
Schläfli symbol t0,1,4{3,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces 105
4-faces 945
Cells 2940
Faces 3780
Edges 2100
Vertices 420
Vertex figure
Coxeter group A6, [35], order 5040
Properties convex

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
Type uniform 6-polytope
Schläfli symbol t0,2,4{3,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces 105
4-faces 1050
Cells 3465
Faces 5040
Edges 3150
Vertices 630
Vertex figure
Coxeter group A6, [35], order 5040
Properties convex

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
Type uniform 6-polytope
Schläfli symbol t0,1,2,4{3,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces 105
4-faces 1155
Cells 4410
Faces 7140
Edges 5040
Vertices 1260
Vertex figure
Coxeter group A6, [35], order 5040
Properties convex

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
Type uniform 6-polytope
Schläfli symbol t0,3,4{3,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces 105
4-faces 700
Cells 1995
Faces 2660
Edges 1680
Vertices 420
Vertex figure
Coxeter group A6, [35], order 5040
Properties convex

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
Type uniform 6-polytope
Schläfli symbol t0,1,3,4{3,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces 105
4-faces 945
Cells 3360
Faces 5670
Edges 4410
Vertices 1260
Vertex figure
Coxeter group A6, [35], order 5040
Properties convex

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
Type uniform 6-polytope
Schläfli symbol t0,2,3,4{3,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces 105
4-faces 1050
Cells 3675
Faces 5880
Edges 4410
Vertices 1260
Vertex figure
Coxeter group A6, [35], order 5040
Properties convex

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
Type uniform 6-polytope
Schläfli symbol t0,1,2,3,4{3,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces 105
4-faces 1155
Cells 4620
Faces 8610
Edges 7560
Vertices 2520
Vertex figure
Coxeter group A6, [35], order 5040
Properties convex

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]

Related uniform 6-polytopes

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 [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. "6D uniform polytopes (polypeta)".

External links

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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