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.

Contents 1 Stericated 7-simplex 1.1 Alternate names 1.2 Coordinates 1.3 Images 2 Bistericated 7-simplex 2.1 Alternate names 2.2 Coordinates 2.3 Images 3 Steritruncated 7-simplex 3.1 Alternate names 3.2 Coordinates 3.3 Images 4 Bisteritruncated 7-simplex 4.1 Alternate names 4.2 Coordinates 4.3 Images 5 Stericantellated 7-simplex 5.1 Alternate names 5.2 Coordinates 5.3 Images 6 Bistericantellated 7-simplex 6.1 Alternate names 6.2 Coordinates 6.3 Images 7 Stericantitruncated 7-simplex 7.1 Alternate names 7.2 Coordinates 7.3 Images 8 Bistericantitruncated 7-simplex 8.1 Alternate names 8.2 Coordinates 8.3 Images 9 Steriruncinated 7-simplex 9.1 Alternate names 9.2 Coordinates 9.3 Images 10 Steriruncitruncated 7-simplex 10.1 Alternate names 10.2 Coordinates 10.3 Images 11 Steriruncicantellated 7-simplex 11.1 Alternate names 11.2 Coordinates 11.3 Images 12 Bisteriruncitruncated 7-simplex 12.1 Alternate names 12.2 Coordinates 12.3 Images 13 Steriruncicantitruncated 7-simplex 13.1 Alternate names 13.2 Coordinates 13.3 Images 14 Bisteriruncicantitruncated 7-simplex 14.1 Alternate names 14.2 Coordinates 14.3 Images 15 Related polytopes 16 Notes 17 References 18 External links

Stericated 7-simplex

Stericated 7-simplex
Type	uniform polyexon
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 polyexon
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, [[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 polyexon
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 polyexon
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 polyexon
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 polyexon
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, [[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 polyexon
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 polyexon
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 polyexon
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 polyexon
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 polyexon
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 polyexon
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, [36], order 40320
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 polyexon
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 80320
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 polyexon
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, [36], order 40320
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.

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

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, editied 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.
• Richard Klitzing, 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

External links

• Olshevsky, George, Cross polytope at Glossary for Hyperspace.
• Polytopes of Various Dimensions
• Multi-dimensional Glossary

Family	An	BCn	Dn	E6 / E7 / E8 / F4 / G2	Hn
Regular polygon	Triangle	Square		Hexagon	Pentagon
Uniform polyhedron	Tetrahedron	Octahedron • Cube	Demicube		Dodecahedron • Icosahedron
Uniform polychoron	5-cell	16-cell • Tesseract	Demitesseract	24-cell	120-cell • 600-cell
Uniform 5-polytope	5-simplex	5-orthoplex • 5-cube	5-demicube
Uniform 6-polytope	6-simplex	6-orthoplex • 6-cube	6-demicube	122 • 221
Uniform 7-polytope	7-simplex	7-orthoplex • 7-cube	7-demicube	132 • 231 • 321
Uniform 8-polytope	8-simplex	8-orthoplex • 8-cube	8-demicube	142 • 241 • 421
Uniform 9-polytope	9-simplex	9-orthoplex • 9-cube	9-demicube
Uniform 10-polytope	10-simplex	10-orthoplex • 10-cube	10-demicube
n-polytopes	n-simplex	n-orthoplex • n-cube	n-demicube	1k2 • 2k1 • k21	pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes