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

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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

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

Coordinates

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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

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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

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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

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

Coordinates

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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

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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

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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

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

Coordinates

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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

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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

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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

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

Coordinates

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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

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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

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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

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

Coordinates

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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

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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

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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

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

Coordinates

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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

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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

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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

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

Coordinates

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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

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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

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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

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

Coordinates

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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

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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

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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

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

Coordinates

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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

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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

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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

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

Coordinates

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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

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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

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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

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

Coordinates

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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

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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

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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

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

Coordinates

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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

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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

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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

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

Coordinates

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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

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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

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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

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

Coordinates

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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

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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]]
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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

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  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

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  • 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
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Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform polychoron Pentachoron 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