Stericated 6-simplexes

From Wikipedia, the free encyclopedia
  (Redirected from Stericated 6-simplex)
Jump to: navigation, search
6-simplex t0.svg
6-simplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
6-simplex t04.svg
Stericated 6-simplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
6-simplex t014.svg
Steritruncated 6-simplex
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
6-simplex t024.svg
Stericantellated 6-simplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
6-simplex t0124.svg
Stericantitruncated 6-simplex
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
6-simplex t034.svg
Steriruncinated 6-simplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
6-simplex t0134.svg
Steriruncitruncated 6-simplex
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
6-simplex t0234.svg
Steriruncicantellated 6-simplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
6-simplex t01234.svg
Steriruncicantitruncated 6-simplex
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
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[edit]

Stericated 6-simplex
Type uniform 6-polytope
Schläfli symbol t0,4{3,3,3,3,3}
Coxeter-Dynkin diagrams CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
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[edit]

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

Coordinates[edit]

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[edit]

orthographic projections
Ak Coxeter plane A6 A5 A4
Graph 6-simplex t04.svg 6-simplex t04 A5.svg 6-simplex t04 A4.svg
Dihedral symmetry [7] [6] [5]
Ak Coxeter plane A3 A2
Graph 6-simplex t04 A3.svg 6-simplex t04 A2.svg
Dihedral symmetry [4] [3]

Steritruncated 6-simplex[edit]

Steritruncated 6-simplex
Type uniform 6-polytope
Schläfli symbol t0,1,4{3,3,3,3,3}
Coxeter-Dynkin diagrams CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
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[edit]

  • Cellirhombated heptapeton (Acronym: catal) (Jonathan Bowers)[2]

Coordinates[edit]

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[edit]

orthographic projections
Ak Coxeter plane A6 A5 A4
Graph 6-simplex t014.svg 6-simplex t014 A5.svg 6-simplex t014 A4.svg
Dihedral symmetry [7] [6] [5]
Ak Coxeter plane A3 A2
Graph 6-simplex t014 A3.svg 6-simplex t014 A2.svg
Dihedral symmetry [4] [3]

Stericantellated 6-simplex[edit]

Stericantellated 6-simplex
Type uniform 6-polytope
Schläfli symbol t0,2,4{3,3,3,3,3}
Coxeter-Dynkin diagrams CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
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[edit]

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

Coordinates[edit]

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[edit]

orthographic projections
Ak Coxeter plane A6 A5 A4
Graph 6-simplex t024.svg 6-simplex t024 A5.svg 6-simplex t024 A4.svg
Dihedral symmetry [7] [6] [5]
Ak Coxeter plane A3 A2
Graph 6-simplex t024 A3.svg 6-simplex t024 A2.svg
Dihedral symmetry [4] [3]

Stericantitruncated 6-simplex[edit]

stericantitruncated 6-simplex
Type uniform 6-polytope
Schläfli symbol t0,1,2,4{3,3,3,3,3}
Coxeter-Dynkin diagrams CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
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[edit]

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

Coordinates[edit]

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[edit]

orthographic projections
Ak Coxeter plane A6 A5 A4
Graph 6-simplex t0124.svg 6-simplex t0124 A5.svg 6-simplex t0124 A4.svg
Dihedral symmetry [7] [6] [5]
Ak Coxeter plane A3 A2
Graph 6-simplex t0124 A3.svg 6-simplex t0124 A2.svg
Dihedral symmetry [4] [3]

Steriruncinated 6-simplex[edit]

steriruncinated 6-simplex
Type uniform 6-polytope
Schläfli symbol t0,3,4{3,3,3,3,3}
Coxeter-Dynkin diagrams CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
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[edit]

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

Coordinates[edit]

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[edit]

orthographic projections
Ak Coxeter plane A6 A5 A4
Graph 6-simplex t034.svg 6-simplex t034 A5.svg 6-simplex t034 A4.svg
Dihedral symmetry [7] [6] [5]
Ak Coxeter plane A3 A2
Graph 6-simplex t034 A3.svg 6-simplex t034 A2.svg
Dihedral symmetry [4] [3]

Steriruncitruncated 6-simplex[edit]

steriruncitruncated 6-simplex
Type uniform 6-polytope
Schläfli symbol t0,1,3,4{3,3,3,3,3}
Coxeter-Dynkin diagrams CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
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[edit]

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

Coordinates[edit]

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[edit]

orthographic projections
Ak Coxeter plane A6 A5 A4
Graph 6-simplex t0134.svg 6-simplex t0134 A5.svg 6-simplex t0134 A4.svg
Dihedral symmetry [7] [6] [5]
Ak Coxeter plane A3 A2
Graph 6-simplex t0134 A3.svg 6-simplex t0134 A2.svg
Dihedral symmetry [4] [3]

Steriruncicantellated 6-simplex[edit]

steriruncicantellated 6-simplex
Type uniform 6-polytope
Schläfli symbol t0,2,3,4{3,3,3,3,3}
Coxeter-Dynkin diagrams CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
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[edit]

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

Coordinates[edit]

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[edit]

orthographic projections
Ak Coxeter plane A6 A5 A4
Graph 6-simplex t0234.svg 6-simplex t0234 A5.svg 6-simplex t0234 A4.svg
Dihedral symmetry [7] [6] [5]
Ak Coxeter plane A3 A2
Graph 6-simplex t0234 A3.svg 6-simplex t0234 A2.svg
Dihedral symmetry [4] [3]

Steriruncicantitruncated 6-simplex[edit]

Steriuncicantitruncated 6-simplex
Type uniform 6-polytope
Schläfli symbol t0,1,2,3,4{3,3,3,3,3}
Coxeter-Dynkin diagrams CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
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[edit]

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

Coordinates[edit]

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[edit]

orthographic projections
Ak Coxeter plane A6 A5 A4
Graph 6-simplex t01234.svg 6-simplex t01234 A5.svg 6-simplex t01234 A4.svg
Dihedral symmetry [7] [6] [5]
Ak Coxeter plane A3 A2
Graph 6-simplex t01234 A3.svg 6-simplex t01234 A2.svg
Dihedral symmetry [4] [3]

Related uniform 6-polytopes[edit]

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[edit]

  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[edit]

  • 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[edit]

Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / E9 / E10 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform 4-polytope 5-cell 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds