Stericated 7-simplexes

(Redirected from Stericated 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

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]]

Related polytopes

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

Notes

1. ^ Klitizing, (x3o3o3o3x3o3o - sco)
2. ^ Klitizing, (x3o3x3o3x3o3o - 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, x3o3x3o3x3o3o - 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