Stericated 6-simplexes

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

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

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)^[1]

Coordinates

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)^[2]

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)^[3]

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)^[4]

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)^[5]

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)^[6]

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)^[7]

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)^[8]

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.

A6 polytopes
t0	t1	t2	t0,1	t0,2	t1,2	t0,3	t1,3	t2,3
t0,4	t1,4	t0,5	t0,1,2	t0,1,3	t0,2,3	t1,2,3	t0,1,4	t0,2,4
t1,2,4	t0,3,4	t0,1,5	t0,2,5	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	t0,1,4,5	t0,1,2,3,4	t0,1,2,3,5	t0,1,2,4,5	t0,1,2,3,4,5

Notes

1. Klitzing, (x3o3o3o3x3o - scal)
2. Klitzing, (x3x3o3o3x3o - catal)
3. Klitzing, (x3o3x3o3x3o - cral)
4. Klitzing, (x3x3x3o3x3o - cagral)
5. Klitzing, (x3o3o3x3x3o - copal)
6. Klitzing, (x3x3o3x3x3o - captal)
7. Klitzing, ( x3o3x3x3x3o - copril)
8. Klitzing, (x3x3x3x3x3o - gacal)

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. "6D uniform polytopes (polypeta)".

External links

• Polytopes of Various Dimensions
• Multi-dimensional Glossary

v t e Fundamental convex regular and uniform polytopes in dimensions 2–10
Family	An	Bn	I2(p) / Dn	E6 / E7 / E8 / F4 / G2	Hn
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	1k2 • 2k1 • k21	n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds