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
Orthogonal projections in A7Coxeter plane

In seven-dimensionalgeometry, astericated 7-simplex is a convexuniform 7-polytope with 4th ordertruncations (sterication) of the regular7-simplex.

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

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

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

The vertices of thestericated 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 onfacets of thestericated 8-orthoplex.

orthographic projections

AkCoxeter 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
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

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

The vertices of thebistericated 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 onfacets of thebistericated 8-orthoplex.

orthographic projections

AkCoxeter 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
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

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

The vertices of thesteritruncated 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 onfacets of thesteritruncated 8-orthoplex.

orthographic projections

AkCoxeter 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
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

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

The vertices of thebisteritruncated 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 onfacets of thebisteritruncated 8-orthoplex.

orthographic projections

AkCoxeter 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
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

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

The vertices of thestericantellated 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 onfacets of thestericantellated 8-orthoplex.

orthographic projections

AkCoxeter 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
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

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

The vertices of thebistericantellated 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 onfacets of thestericantellated 8-orthoplex.

orthographic projections

AkCoxeter 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
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

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

The vertices of thestericantitruncated 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 onfacets of thestericantitruncated 8-orthoplex.

orthographic projections

AkCoxeter 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
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

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

The vertices of thebistericantitruncated 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 onfacets of thebistericantitruncated 8-orthoplex.

orthographic projections

AkCoxeter 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
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

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

The vertices of thesteriruncinated 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 onfacets of thesteriruncinated 8-orthoplex.

orthographic projections

AkCoxeter 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
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

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

The vertices of thesteriruncitruncated 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 onfacets of thesteriruncitruncated 8-orthoplex.

orthographic projections

AkCoxeter 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
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

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

The vertices of thesteriruncicantellated 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 onfacets of thesteriruncicantellated 8-orthoplex.

orthographic projections

AkCoxeter 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
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

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

The vertices of thebisteriruncitruncated 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 onfacets of thebisteriruncitruncated 8-orthoplex.

orthographic projections

AkCoxeter 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
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

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

The vertices of thesteriruncicantitruncated 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 onfacets of thesteriruncicantitruncated 8-orthoplex.

orthographic projections

AkCoxeter 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
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

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

The vertices of thebisteriruncicantitruncated 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 onfacets of thebisteriruncicantitruncated 8-orthoplex.

orthographic projections

AkCoxeter 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 71uniform 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

• 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,wiley.com,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 JohnsonUniform 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

• Polytopes of Various Dimensions
• Multi-dimensional Glossary

• 7-polytopes