Langbahn Team – Weltmeisterschaft

Stericated 7-simplexes


7-simplex
Stericated 7-simplex
Bistericated 7-simplex

Steritruncated 7-simplex
Bisteritruncated 7-simplex
Stericantellated 7-simplex

Bistericantellated 7-simplex
Stericantitruncated 7-simplex
Bistericantitruncated 7-simplex

Steriruncinated 7-simplex
Steriruncitruncated 7-simplex
Steriruncicantellated 7-simplex

Bisteriruncitruncated 7-simplex
Steriruncicantitruncated 7-simplex
Bisteriruncicantitruncated 7-simplex

In seven-dimensional geometry, a stericated 7-simplex is a convex uniform 7-polytope with 4th order truncations (sterication) of the regular 7-simplex.

There are 14 unique sterication for the 7-simplex with permutations of truncations, cantellations, and runcinations.

Stericated 7-simplex

Stericated 7-simplex
Type uniform 7-polytope
Schläfli symbol t0,4{3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges 2240
Vertices 280
Vertex figure
Coxeter group A7, [36], order 40320
Properties convex

Alternate names

  • Small cellated octaexon (acronym: sco) (Jonathan Bowers)[1]

Coordinates

The vertices of the stericated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,1,1,1,2). This construction is based on facets of the stericated 8-orthoplex.

Images

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [5] [4] [3]

Bistericated 7-simplex

bistericated 7-simplex
Type uniform 7-polytope
Schläfli symbol t1,5{3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges 3360
Vertices 420
Vertex figure
Coxeter group A7×2, [[36]], order 80320
Properties convex

Alternate names

  • Small bicellated hexadecaexon (acronym: sabach) (Jonathan Bowers)[2]

Coordinates

The vertices of the bistericated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,1,1,1,2,2). This construction is based on facets of the bistericated 8-orthoplex.

Images

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph
Dihedral symmetry [8] [[7]] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [[5]] [4] [[3]]

Steritruncated 7-simplex

steritruncated 7-simplex
Type uniform 7-polytope
Schläfli symbol t0,1,4{3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges 7280
Vertices 1120
Vertex figure
Coxeter group A7, [36], order 40320
Properties convex

Alternate names

  • Cellitruncated octaexon (acronym: cato) (Jonathan Bowers)[3]

Coordinates

The vertices of the steritruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,1,1,2,3). This construction is based on facets of the steritruncated 8-orthoplex.

Images

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [5] [4] [3]

Bisteritruncated 7-simplex

bisteritruncated 7-simplex
Type uniform 7-polytope
Schläfli symbol t1,2,5{3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges 9240
Vertices 1680
Vertex figure
Coxeter group A7, [36], order 40320
Properties convex

Alternate names

  • Bicellitruncated octaexon (acronym: bacto) (Jonathan Bowers)[4]

Coordinates

The vertices of the bisteritruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,1,1,2,3,3). This construction is based on facets of the bisteritruncated 8-orthoplex.

Images

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [5] [4] [3]

Stericantellated 7-simplex

Stericantellated 7-simplex
Type uniform 7-polytope
Schläfli symbol t0,2,4{3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges 10080
Vertices 1680
Vertex figure
Coxeter group A7, [36], order 40320
Properties convex

Alternate names

  • Cellirhombated octaexon (acronym: caro) (Jonathan Bowers)[5]

Coordinates

The vertices of the stericantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,1,2,2,3). This construction is based on facets of the stericantellated 8-orthoplex.

Images

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [5] [4] [3]

Bistericantellated 7-simplex

Bistericantellated 7-simplex
Type uniform 7-polytope
Schläfli symbol t1,3,5{3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges 15120
Vertices 2520
Vertex figure
Coxeter group A7×2, [[36]], order 80320
Properties convex

Alternate names

  • Bicellirhombihexadecaexon (acronym: bacroh) (Jonathan Bowers)[6]

Coordinates

The vertices of the bistericantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,1,2,2,3,3). This construction is based on facets of the stericantellated 8-orthoplex.

Images

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [5] [4] [3]

Stericantitruncated 7-simplex

stericantitruncated 7-simplex
Type uniform 7-polytope
Schläfli symbol t0,1,2,4{3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges 16800
Vertices 3360
Vertex figure
Coxeter group A7, [36], order 40320
Properties convex

Alternate names

  • Celligreatorhombated octaexon (acronym: cagro) (Jonathan Bowers)[7]

Coordinates

The vertices of the stericantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,1,2,3,4). This construction is based on facets of the stericantitruncated 8-orthoplex.

Images

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [5] [4] [3]

Bistericantitruncated 7-simplex

bistericantitruncated 7-simplex
Type uniform 7-polytope
Schläfli symbol t1,2,3,5{3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges 22680
Vertices 5040
Vertex figure
Coxeter group A7, [36], order 40320
Properties convex

Alternate names

  • Bicelligreatorhombated octaexon (acronym: bacogro) (Jonathan Bowers)[8]

Coordinates

The vertices of the bistericantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,1,2,3,4,4). This construction is based on facets of the bistericantitruncated 8-orthoplex.

Images

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [5] [4] [3]

Steriruncinated 7-simplex

Steriruncinated 7-simplex
Type uniform 7-polytope
Schläfli symbol t0,3,4{3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges 5040
Vertices 1120
Vertex figure
Coxeter group A7, [36], order 40320
Properties convex

Alternate names

  • Celliprismated octaexon (acronym: cepo) (Jonathan Bowers)[9]

Coordinates

The vertices of the steriruncinated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,2,2,2,3). This construction is based on facets of the steriruncinated 8-orthoplex.

Images

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [5] [4] [3]

Steriruncitruncated 7-simplex

steriruncitruncated 7-simplex
Type uniform 7-polytope
Schläfli symbol t0,1,3,4{3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges 13440
Vertices 3360
Vertex figure
Coxeter group A7, [36], order 40320
Properties convex

Alternate names

  • Celliprismatotruncated octaexon (acronym: capto) (Jonathan Bowers)[10]

Coordinates

The vertices of the steriruncitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,2,2,3,4). This construction is based on facets of the steriruncitruncated 8-orthoplex.

Images

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [5] [4] [3]

Steriruncicantellated 7-simplex

steriruncicantellated 7-simplex
Type uniform 7-polytope
Schläfli symbol t0,2,3,4{3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges 13440
Vertices 3360
Vertex figure
Coxeter group A7, [36], order 40320
Properties convex

Alternate names

  • Celliprismatorhombated octaexon (acronym: capro) (Jonathan Bowers)[11]

Coordinates

The vertices of the steriruncicantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,2,3,3,4). This construction is based on facets of the steriruncicantellated 8-orthoplex.

Images

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [5] [4] [3]

Bisteriruncitruncated 7-simplex

bisteriruncitruncated 7-simplex
Type uniform 7-polytope
Schläfli symbol t1,2,4,5{3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges 20160
Vertices 5040
Vertex figure
Coxeter group A7×2, [[36]], order 80320
Properties convex

Alternate names

  • Bicelliprismatotruncated hexadecaexon (acronym: bicpath) (Jonathan Bowers)[12]

Coordinates

The vertices of the bisteriruncitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,2,2,3,4,4). This construction is based on facets of the bisteriruncitruncated 8-orthoplex.

Images

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph
Dihedral symmetry [8] [[7]] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [[5]] [4] [[3]]

Steriruncicantitruncated 7-simplex

steriruncicantitruncated 7-simplex
Type uniform 7-polytope
Schläfli symbol t0,1,2,3,4{3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges 23520
Vertices 6720
Vertex figure
Coxeter group A7, [36], order 40320
Properties convex

Alternate names

  • Great cellated octaexon (acronym: gecco) (Jonathan Bowers)[13]

Coordinates

The vertices of the steriruncicantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,2,3,4,5). This construction is based on facets of the steriruncicantitruncated 8-orthoplex.

Images

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [5] [4] [3]

Bisteriruncicantitruncated 7-simplex

bisteriruncicantitruncated 7-simplex
Type uniform 7-polytope
Schläfli symbol t1,2,3,4,5{3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges 35280
Vertices 10080
Vertex figure
Coxeter group A7×2, [[36]], order 80320
Properties convex

Alternate names

  • Great bicellated hexadecaexon (gabach) (Jonathan Bowers) [14]

Coordinates

The vertices of the bisteriruncicantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,1,2,3,4,5,5). This construction is based on facets of the bisteriruncicantitruncated 8-orthoplex.

Images

orthographic projections
Ak Coxeter plane A7 A6 A5
Graph
Dihedral symmetry [8] [[7]] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [[5]] [4] [[3]]

This polytope is one of 71 uniform 7-polytopes with A7 symmetry.

A7 polytopes

t0
t1
t2
t3
t0,1
t0,2
t1,2
t0,3

t1,3
t2,3
t0,4
t1,4
t2,4
t0,5
t1,5
t0,6

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

t1,3,4
t2,3,4
t0,1,5
t0,2,5
t1,2,5
t0,3,5
t1,3,5
t0,4,5

t0,1,6
t0,2,6
t0,3,6
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
t1,2,3,5
t0,1,4,5
t0,2,4,5
t1,2,4,5
t0,3,4,5

t0,1,2,6
t0,1,3,6
t0,2,3,6
t0,1,4,6
t0,2,4,6
t0,1,5,6
t0,1,2,3,4
t0,1,2,3,5

t0,1,2,4,5
t0,1,3,4,5
t0,2,3,4,5
t1,2,3,4,5
t0,1,2,3,6
t0,1,2,4,6
t0,1,3,4,6
t0,2,3,4,6

t0,1,2,5,6
t0,1,3,5,6
t0,1,2,3,4,5
t0,1,2,3,4,6
t0,1,2,3,5,6
t0,1,2,4,5,6
t0,1,2,3,4,5,6

Notes

  1. ^ Klitizing, (x3o3o3o3x3o3o - sco)
  2. ^ Klitizing, (o3x3o3o3o3x3o - sabach)
  3. ^ Klitizing, (x3x3o3o3x3o3o - cato)
  4. ^ Klitizing, (o3x3x3o3o3x3o - bacto)
  5. ^ Klitizing, (x3o3x3o3x3o3o - caro)
  6. ^ Klitizing, (o3x3o3x3o3x3o - bacroh)
  7. ^ Klitizing, (x3x3x3o3x3o3o - cagro)
  8. ^ Klitizing, (o3x3x3x3o3x3o - bacogro)
  9. ^ Klitizing, (x3o3o3x3x3o3o - cepo)
  10. ^ Klitizing, (x3x3x3o3x3o3o - capto)
  11. ^ Klitizing, (x3o3x3x3x3o3o - capro)
  12. ^ Klitizing, (o3x3x3o3x3x3o - bicpath)
  13. ^ Klitizing, (x3x3x3x3x3o3o - gecco)
  14. ^ Klitizing, (o3x3x3x3x3x3o - gabach)

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. "7D uniform polytopes (polyexa)". x3o3o3o3x3o3o - sco, o3x3o3o3o3x3o - sabach, x3x3o3o3x3o3o - cato, o3x3x3o3o3x3o - bacto, x3o3x3o3x3o3o - caro, o3x3o3x3o3x3o - bacroh, x3x3x3o3x3o3o - cagro, o3x3x3x3o3x3o - bacogro, x3o3o3x3x3o3o - cepo, x3x3x3o3x3o3o - capto, x3o3x3x3x3o3o - capro, o3x3x3o3x3x3o - bicpath, x3x3x3x3x3o3o - gecco, o3x3x3x3x3x3o - gabach
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
Quelle: https://en.wikipedia.org/wiki/Stericantellated_7-simplex