Stericated 6-simplexes

6-simplex	Stericated 6-simplex	Steritruncated 6-simplex
Stericantellated 6-simplex	Stericantitruncated 6-simplex	Steriruncinated 6-simplex
Steriruncitruncated 6-simplex	Steriruncicantellated 6-simplex	Steriruncicantitruncated 6-simplex
Orthogonal projections in A6 Coxeter plane

In six-dimensional geometry, a stericated 6-simplex is a convex uniform 6-polytope with 4th order truncations (sterication) of the regular 6-simplex.

There are 8 unique sterications for the 6-simplex with permutations of truncations, cantellations, and runcinations.

Stericated 6-simplex

Stericated 6-simplex
Type	uniform 6-polytope
Schläfli symbol	t0,4{3,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces	105
4-faces	700
Cells	1470
Faces	1400
Edges	630
Vertices	105
Vertex figure
Coxeter group	A6, [35], order 5040
Properties	convex

Alternate names

• Small cellated heptapeton (Acronym: scal) (Jonathan Bowers)

The vertices of the stericated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,1,1,1,1,2). This construction is based on facets of the stericated 7-orthoplex.

Images

orthographic projections

Ak Coxeter plane	A6	A5	A4
Graph
Dihedral symmetry	[7]	[6]	[5]
Ak Coxeter plane	A3	A2
Graph
Dihedral symmetry	[4]	[3]

Steritruncated 6-simplex

Steritruncated 6-simplex
Type	uniform 6-polytope
Schläfli symbol	t0,1,4{3,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces	105
4-faces	945
Cells	2940
Faces	3780
Edges	2100
Vertices	420
Vertex figure
Coxeter group	A6, [35], order 5040
Properties	convex

Alternate names

• Cellitruncated heptapeton (Acronym: catal) (Jonathan Bowers)

Coordinates

The vertices of the steritruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,1,1,1,2,3). This construction is based on facets of the steritruncated 7-orthoplex.

Images

orthographic projections

Ak Coxeter plane	A6	A5	A4
Graph
Dihedral symmetry	[7]	[6]	[5]
Ak Coxeter plane	A3	A2
Graph
Dihedral symmetry	[4]	[3]

Stericantellated 6-simplex

Stericantellated 6-simplex
Type	uniform 6-polytope
Schläfli symbol	t0,2,4{3,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces	105
4-faces	1050
Cells	3465
Faces	5040
Edges	3150
Vertices	630
Vertex figure
Coxeter group	A6, [35], order 5040
Properties	convex

Alternate names

• Cellirhombated heptapeton (Acronym: cral) (Jonathan Bowers)

Coordinates

The vertices of the stericantellated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,1,1,2,2,3). This construction is based on facets of the stericantellated 7-orthoplex.

Images

orthographic projections

Ak Coxeter plane	A6	A5	A4
Graph
Dihedral symmetry	[7]	[6]	[5]
Ak Coxeter plane	A3	A2
Graph
Dihedral symmetry	[4]	[3]

Stericantitruncated 6-simplex

stericantitruncated 6-simplex
Type	uniform 6-polytope
Schläfli symbol	t0,1,2,4{3,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces	105
4-faces	1155
Cells	4410
Faces	7140
Edges	5040
Vertices	1260
Vertex figure
Coxeter group	A6, [35], order 5040
Properties	convex

Alternate names

• Celligreatorhombated heptapeton (Acronym: cagral) (Jonathan Bowers)

Coordinates

The vertices of the stericanttruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,1,2,3,4). This construction is based on facets of the stericantitruncated 7-orthoplex.

Images

orthographic projections

Ak Coxeter plane	A6	A5	A4
Graph
Dihedral symmetry	[7]	[6]	[5]
Ak Coxeter plane	A3	A2
Graph
Dihedral symmetry	[4]	[3]

Steriruncinated 6-simplex

steriruncinated 6-simplex
Type	uniform 6-polytope
Schläfli symbol	t0,3,4{3,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces	105
4-faces	700
Cells	1995
Faces	2660
Edges	1680
Vertices	420
Vertex figure
Coxeter group	A6, [35], order 5040
Properties	convex

Alternate names

• Celliprismated heptapeton (Acronym: copal) (Jonathan Bowers)

Coordinates

The vertices of the steriruncinated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,1,2,2,3,3). This construction is based on facets of the steriruncinated 7-orthoplex.

Images

orthographic projections

Ak Coxeter plane	A6	A5	A4
Graph
Dihedral symmetry	[7]	[6]	[5]
Ak Coxeter plane	A3	A2
Graph
Dihedral symmetry	[4]	[3]

Steriruncitruncated 6-simplex

steriruncitruncated 6-simplex
Type	uniform 6-polytope
Schläfli symbol	t0,1,3,4{3,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces	105
4-faces	945
Cells	3360
Faces	5670
Edges	4410
Vertices	1260
Vertex figure
Coxeter group	A6, [35], order 5040
Properties	convex

Alternate names

• Celliprismatotruncated heptapeton (Acronym: captal) (Jonathan Bowers)

Coordinates

The vertices of the steriruncittruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,1,2,3,4). This construction is based on facets of the steriruncitruncated 7-orthoplex.

Images

orthographic projections

Ak Coxeter plane	A6	A5	A4
Graph
Dihedral symmetry	[7]	[6]	[5]
Ak Coxeter plane	A3	A2
Graph
Dihedral symmetry	[4]	[3]

Steriruncicantellated 6-simplex

steriruncicantellated 6-simplex
Type	uniform 6-polytope
Schläfli symbol	t0,2,3,4{3,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces	105
4-faces	1050
Cells	3675
Faces	5880
Edges	4410
Vertices	1260
Vertex figure
Coxeter group	A6, [35], order 5040
Properties	convex

Alternate names

• Bistericantitruncated 6-simplex as t_1,2,3,5{3,3,3,3,3}
• Celliprismatorhombated heptapeton (Acronym: copril) (Jonathan Bowers)

Coordinates

The vertices of the steriruncitcantellated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,1,2,3,4). This construction is based on facets of the steriruncicantellated 7-orthoplex.

Images

orthographic projections

Ak Coxeter plane	A6	A5	A4
Graph
Dihedral symmetry	[7]	[6]	[5]
Ak Coxeter plane	A3	A2
Graph
Dihedral symmetry	[4]	[3]

Steriruncicantitruncated 6-simplex

Steriuncicantitruncated 6-simplex
Type	uniform 6-polytope
Schläfli symbol	t0,1,2,3,4{3,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces	105
4-faces	1155
Cells	4620
Faces	8610
Edges	7560
Vertices	2520
Vertex figure
Coxeter group	A6, [35], order 5040
Properties	convex

Alternate names

• Great cellated heptapeton (Acronym: gacal) (Jonathan Bowers)

Coordinates

The vertices of the steriruncicantittruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,1,2,3,4,5). This construction is based on facets of the steriruncicantitruncated 7-orthoplex.

Images

orthographic projections

Ak Coxeter plane	A6	A5	A4
Graph
Dihedral symmetry	[7]	[6]	[5]
Ak Coxeter plane	A3	A2
Graph
Dihedral symmetry	[4]	[3]

Related uniform 6-polytopes

The truncated 6-simplex is one of 35 uniform 6-polytopes based on the [3,3,3,3,3] Coxeter group, all shown here in A₆ Coxeter plane orthographic projections.

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. "6D uniform polytopes (polypeta)". https://bendwavy.org/klitzing/dimensions/polypeta.htm. 

External links

• Polytopes of Various Dimensions
• Multi-dimensional Glossary