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
Type	uniform 7-polytope
Schläfli symbol	t_0,4{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
A_k Coxeter plane	A₇	A₆	A₅
Graph
Dihedral symmetry	[8]	[7]	[6]
A_k Coxeter plane	A₄	A₃	A₂
Graph
Dihedral symmetry	[5]	[4]	[3]

Bistericated 7-simplex

bistericated 7-simplex
Type	uniform 7-polytope
Schläfli symbol	t_1,5{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
A_k Coxeter plane	A₇	A₆	A₅
Graph
Dihedral symmetry	[8]	[[7]]	[6]
A_k Coxeter plane	A₄	A₃	A₂
Graph
Dihedral symmetry	[[5]]	[4]	[[3]]

Steritruncated 7-simplex

steritruncated 7-simplex
Type	uniform 7-polytope
Schläfli symbol	t_0,1,4{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
A_k Coxeter plane	A₇	A₆	A₅
Graph
Dihedral symmetry	[8]	[7]	[6]
A_k Coxeter plane	A₄	A₃	A₂
Graph
Dihedral symmetry	[5]	[4]	[3]

Bisteritruncated 7-simplex

bisteritruncated 7-simplex
Type	uniform 7-polytope
Schläfli symbol	t_1,2,5{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
A_k Coxeter plane	A₇	A₆	A₅
Graph
Dihedral symmetry	[8]	[7]	[6]
A_k Coxeter plane	A₄	A₃	A₂
Graph
Dihedral symmetry	[5]	[4]	[3]

Stericantellated 7-simplex

Stericantellated 7-simplex
Type	uniform 7-polytope
Schläfli symbol	t_0,2,4{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
A_k Coxeter plane	A₇	A₆	A₅
Graph
Dihedral symmetry	[8]	[7]	[6]
A_k Coxeter plane	A₄	A₃	A₂
Graph
Dihedral symmetry	[5]	[4]	[3]

Bistericantellated 7-simplex

Bistericantellated 7-simplex
Type	uniform 7-polytope
Schläfli symbol	t_1,3,5{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
A_k Coxeter plane	A₇	A₆	A₅
Graph
Dihedral symmetry	[8]	[7]	[6]
A_k Coxeter plane	A₄	A₃	A₂
Graph
Dihedral symmetry	[5]	[4]	[3]

Stericantitruncated 7-simplex

stericantitruncated 7-simplex
Type	uniform 7-polytope
Schläfli symbol	t_0,1,2,4{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
A_k Coxeter plane	A₇	A₆	A₅
Graph
Dihedral symmetry	[8]	[7]	[6]
A_k Coxeter plane	A₄	A₃	A₂
Graph
Dihedral symmetry	[5]	[4]	[3]

Bistericantitruncated 7-simplex

bistericantitruncated 7-simplex
Type	uniform 7-polytope
Schläfli symbol	t_1,2,3,5{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
A_k Coxeter plane	A₇	A₆	A₅
Graph
Dihedral symmetry	[8]	[7]	[6]
A_k Coxeter plane	A₄	A₃	A₂
Graph
Dihedral symmetry	[5]	[4]	[3]

Steriruncinated 7-simplex

Steriruncinated 7-simplex
Type	uniform 7-polytope
Schläfli symbol	t_0,3,4{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
A_k Coxeter plane	A₇	A₆	A₅
Graph
Dihedral symmetry	[8]	[7]	[6]
A_k Coxeter plane	A₄	A₃	A₂
Graph
Dihedral symmetry	[5]	[4]	[3]

Steriruncitruncated 7-simplex

steriruncitruncated 7-simplex
Type	uniform 7-polytope
Schläfli symbol	t_0,1,3,4{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
A_k Coxeter plane	A₇	A₆	A₅
Graph
Dihedral symmetry	[8]	[7]	[6]
A_k Coxeter plane	A₄	A₃	A₂
Graph
Dihedral symmetry	[5]	[4]	[3]

Steriruncicantellated 7-simplex

steriruncicantellated 7-simplex
Type	uniform 7-polytope
Schläfli symbol	t_0,2,3,4{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
A_k Coxeter plane	A₇	A₆	A₅
Graph
Dihedral symmetry	[8]	[7]	[6]
A_k Coxeter plane	A₄	A₃	A₂
Graph
Dihedral symmetry	[5]	[4]	[3]

Bisteriruncitruncated 7-simplex

bisteriruncitruncated 7-simplex
Type	uniform 7-polytope
Schläfli symbol	t_1,2,4,5{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
A_k Coxeter plane	A₇	A₆	A₅
Graph
Dihedral symmetry	[8]	[[7]]	[6]
A_k Coxeter plane	A₄	A₃	A₂
Graph
Dihedral symmetry	[[5]]	[4]	[[3]]

Steriruncicantitruncated 7-simplex

steriruncicantitruncated 7-simplex
Type	uniform 7-polytope
Schläfli symbol	t_0,1,2,3,4{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
A_k Coxeter plane	A₇	A₆	A₅
Graph
Dihedral symmetry	[8]	[7]	[6]
A_k Coxeter plane	A₄	A₃	A₂
Graph
Dihedral symmetry	[5]	[4]	[3]

Bisteriruncicantitruncated 7-simplex

bisteriruncicantitruncated 7-simplex
Type	uniform 7-polytope
Schläfli symbol	t_1,2,3,4,5{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
A_k Coxeter plane	A₇	A₆	A₅
Graph
Dihedral symmetry	[8]	[[7]]	[6]
A_k 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

Klitizing, (x3o3o3o3x3o3o - sco)
Klitizing, (o3x3o3o3o3x3o - sabach)
Klitizing, (x3x3o3o3x3o3o - cato)
Klitizing, (o3x3x3o3o3x3o - bacto)
Klitizing, (x3o3x3o3x3o3o - caro)
Klitizing, (o3x3o3x3o3x3o - bacroh)
Klitizing, (x3x3x3o3x3o3o - cagro)
Klitizing, (o3x3x3x3o3x3o - bacogro)
Klitizing, (x3o3o3x3x3o3o - cepo)
Klitizing, (x3x3x3o3x3o3o - capto)
Klitizing, (x3o3x3x3x3o3o - capro)
Klitizing, (o3x3x3o3x3x3o - bicpath)
Klitizing, (x3x3x3x3x3o3o - gecco)
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

External links

Fundamental convex regular and uniform polytopes in dimensions 2–10
Family	A_n	B_n	I₂(p) / D_n	E₆ / E₇ / E₈ / F₄ / G₂	H_n
Regular polygon	Triangle	Square	p-gon	Hexagon	Pentagon
Uniform polyhedron	Tetrahedron	Octahedron • Cube	Demicube		Dodecahedron • Icosahedron
Uniform polychoron	Pentachoron	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
Uniform n-polytope	n-simplex	n-orthoplex • n-cube	n-demicube	1_k2 • 2_k1 • k₂₁	n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds

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[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)

A7 polytopes
t₀	t₁	t₂	t₃	t_0,1	t_0,2	t_1,2	t_0,3
t_1,3	t_2,3	t_0,4	t_1,4	t_2,4	t_0,5	t_1,5	t_0,6
t_0,1,2	t_0,1,3	t_0,2,3	t_1,2,3	t_0,1,4	t_0,2,4	t_1,2,4	t_0,3,4
t_1,3,4	t_2,3,4	t_0,1,5	t_0,2,5	t_1,2,5	t_0,3,5	t_1,3,5	t_0,4,5
t_0,1,6	t_0,2,6	t_0,3,6	t_0,1,2,3	t_0,1,2,4	t_0,1,3,4	t_0,2,3,4	t_1,2,3,4
t_0,1,2,5	t_0,1,3,5	t_0,2,3,5	t_1,2,3,5	t_0,1,4,5	t_0,2,4,5	t_1,2,4,5	t_0,3,4,5
t_0,1,2,6	t_0,1,3,6	t_0,2,3,6	t_0,1,4,6	t_0,2,4,6	t_0,1,5,6	t_0,1,2,3,4	t_0,1,2,3,5
t_0,1,2,4,5	t_0,1,3,4,5	t_0,2,3,4,5	t_1,2,3,4,5	t_0,1,2,3,6	t_0,1,2,4,6	t_0,1,3,4,6	t_0,2,3,4,6
t_0,1,2,5,6	t_0,1,3,5,6	t_0,1,2,3,4,5	t_0,1,2,3,4,6	t_0,1,2,3,5,6	t_0,1,2,4,5,6	t_{0,1,2,3,4,5,6}