Orthogonal projections in A7 Coxeter plane
7-simplexStericated 7-simplexBistericated 7-simplex
Steritruncated 7-simplexBisteritruncated 7-simplexStericantellated 7-simplex
Bistericantellated 7-simplexStericantitruncated 7-simplexBistericantitruncated 7-simplex
Steriruncinated 7-simplexSteriruncitruncated 7-simplexSteriruncicantellated 7-simplex
Bisteriruncitruncated 7-simplexSteriruncicantitruncated 7-simplexBisteriruncicantitruncated 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
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)

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 planeA7A6A5
Graph
Dihedral symmetry[8][7][6]
Ak Coxeter planeA4A3A2
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)

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 planeA7A6A5
Graph
Dihedral symmetry[8][[7]][6]
Ak Coxeter planeA4A3A2
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)

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 planeA7A6A5
Graph
Dihedral symmetry[8][7][6]
Ak Coxeter planeA4A3A2
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)

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 planeA7A6A5
Graph
Dihedral symmetry[8][7][6]
Ak Coxeter planeA4A3A2
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)

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 planeA7A6A5
Graph
Dihedral symmetry[8][7][6]
Ak Coxeter planeA4A3A2
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)

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 planeA7A6A5
Graph
Dihedral symmetry[8][7][6]
Ak Coxeter planeA4A3A2
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)

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 planeA7A6A5
Graph
Dihedral symmetry[8][7][6]
Ak Coxeter planeA4A3A2
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)

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 planeA7A6A5
Graph
Dihedral symmetry[8][7][6]
Ak Coxeter planeA4A3A2
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)

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 planeA7A6A5
Graph
Dihedral symmetry[8][7][6]
Ak Coxeter planeA4A3A2
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)

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 planeA7A6A5
Graph
Dihedral symmetry[8][7][6]
Ak Coxeter planeA4A3A2
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)

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 planeA7A6A5
Graph
Dihedral symmetry[8][7][6]
Ak Coxeter planeA4A3A2
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)

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 planeA7A6A5
Graph
Dihedral symmetry[8][[7]][6]
Ak Coxeter planeA4A3A2
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)

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 planeA7A6A5
Graph
Dihedral symmetry[8][7][6]
Ak Coxeter planeA4A3A2
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 (acronym: gabach) (Jonathan Bowers)

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 planeA7A6A5
Graph
Dihedral symmetry[8][[7]][6]
Ak Coxeter planeA4A3A2
Graph
Dihedral symmetry[[5]][4][[3]]

Related polytopes

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

A7 polytopes
t0t1t2t3t0,1t0,2t1,2t0,3
t1,3t2,3t0,4t1,4t2,4t0,5t1,5t0,6
t0,1,2t0,1,3t0,2,3t1,2,3t0,1,4t0,2,4t1,2,4t0,3,4
t1,3,4t2,3,4t0,1,5t0,2,5t1,2,5t0,3,5t1,3,5t0,4,5
t0,1,6t0,2,6t0,3,6t0,1,2,3t0,1,2,4t0,1,3,4t0,2,3,4t1,2,3,4
t0,1,2,5t0,1,3,5t0,2,3,5t1,2,3,5t0,1,4,5t0,2,4,5t1,2,4,5t0,3,4,5
t0,1,2,6t0,1,3,6t0,2,3,6t0,1,4,6t0,2,4,6t0,1,5,6t0,1,2,3,4t0,1,2,3,5
t0,1,2,4,5t0,1,3,4,5t0,2,3,4,5t1,2,3,4,5t0,1,2,3,6t0,1,2,4,6t0,1,3,4,6t0,2,3,4,6
t0,1,2,5,6t0,1,3,5,6t0,1,2,3,4,5t0,1,2,3,4,6t0,1,2,3,5,6t0,1,2,4,5,6t0,1,2,3,4,5,6

Notes

  • 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 Ivić 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. . 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

External links

vteFundamental convex regular and uniform polytopes in dimensions 2–10
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