Stericated 7-simplexes

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


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

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

Stericated 7-simplex

Stericated 7-simplex
Typeuniform 7-polytope
Schläfli symbolt0,4{3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges2240
Vertices280
Vertex figure
Coxeter groupA7, [36], order 40320
Propertiesconvex

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
Typeuniform 7-polytope
Schläfli symbolt1,5{3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges3360
Vertices420
Vertex figure
Coxeter groupA7×2, [[36]], order 80320
Propertiesconvex

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
Typeuniform 7-polytope
Schläfli symbolt0,1,4{3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges7280
Vertices1120
Vertex figure
Coxeter groupA7, [36], order 40320
Propertiesconvex

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
Typeuniform 7-polytope
Schläfli symbolt1,2,5{3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges9240
Vertices1680
Vertex figure
Coxeter groupA7, [36], order 40320
Propertiesconvex

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
Typeuniform 7-polytope
Schläfli symbolt0,2,4{3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges10080
Vertices1680
Vertex figure
Coxeter groupA7, [36], order 40320
Propertiesconvex

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
Typeuniform 7-polytope
Schläfli symbolt1,3,5{3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges15120
Vertices2520
Vertex figure
Coxeter groupA7×2, [[36]], order 80320
Propertiesconvex

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
Typeuniform 7-polytope
Schläfli symbolt0,1,2,4{3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges16800
Vertices3360
Vertex figure
Coxeter groupA7, [36], order 40320
Propertiesconvex

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
Typeuniform 7-polytope
Schläfli symbolt1,2,3,5{3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges22680
Vertices5040
Vertex figure
Coxeter groupA7, [36], order 40320
Propertiesconvex

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
Typeuniform 7-polytope
Schläfli symbolt0,3,4{3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges5040
Vertices1120
Vertex figure
Coxeter groupA7, [36], order 40320
Propertiesconvex

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
Typeuniform 7-polytope
Schläfli symbolt0,1,3,4{3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges13440
Vertices3360
Vertex figure
Coxeter groupA7, [36], order 40320
Propertiesconvex

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
Typeuniform 7-polytope
Schläfli symbolt0,2,3,4{3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges13440
Vertices3360
Vertex figure
Coxeter groupA7, [36], order 40320
Propertiesconvex

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
Typeuniform 7-polytope
Schläfli symbolt1,2,4,5{3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges20160
Vertices5040
Vertex figure
Coxeter groupA7×2, [[36]], order 80320
Propertiesconvex

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
Typeuniform 7-polytope
Schläfli symbolt0,1,2,3,4{3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges23520
Vertices6720
Vertex figure
Coxeter groupA7, [36], order 40320
Propertiesconvex

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
Typeuniform 7-polytope
Schläfli symbolt1,2,3,4,5{3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges35280
Vertices10080
Vertex figure
Coxeter groupA7×2, [[36]], order 80320
Propertiesconvex

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
      • (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
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
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