Runcinated 6-simplexes

6-simplex	Runcinated 6-simplex	Biruncinated 6-simplex
Runcitruncated 6-simplex	Biruncitruncated 6-simplex	Runcicantellated 6-simplex
Runcicantitruncated 6-simplex	Biruncicantitruncated 6-simplex
Orthogonal projections in A6 Coxeter plane

In six-dimensional geometry, a runcinated 6-simplex is a convex uniform 6-polytope constructed as a runcination (3rd order truncations) of the regular 6-simplex.

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

Runcinated 6-simplex

Runcinated 6-simplex
Type	uniform 6-polytope
Schläfli symbol	t0,3{3,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces	70
4-faces	455
Cells	1330
Faces	1610
Edges	840
Vertices	140
Vertex figure
Coxeter group	A6, [35], order 5040
Properties	convex

Alternate names

• Small prismated heptapeton (Acronym: spil) (Jonathan Bowers)

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

Images

orthographic projections

Ak Coxeter plane	A6	A5	A4
Graph		150px	150px
Dihedral symmetry	[7]	[6]	[5]
Ak Coxeter plane	A3	A2
Graph	150px	150px
Dihedral symmetry	[4]	[3]

Biruncinated 6-simplex

biruncinated 6-simplex
Type	uniform 6-polytope
Schläfli symbol	t1,4{3,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces	84
4-faces	714
Cells	2100
Faces	2520
Edges	1260
Vertices	210
Vertex figure
Coxeter group	A6, [[35]], order 10080
Properties	convex

Alternate names

• Small biprismated tetradecapeton (Acronym: sibpof) (Jonathan Bowers)

Coordinates

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

Images

orthographic projections

Ak Coxeter plane	A6	A5	A4
Graph		150px	150px
Symmetry	[[7]](*)=[14]	[6]	[[5]](*)=[10]
Ak Coxeter plane	A3	A2
Graph	150px	150px
Symmetry	[4]	[[3]](*)=[6]

Note: (*) Symmetry doubled for A_k graphs with even k due to symmetrically-ringed Coxeter-Dynkin diagram.

Runcitruncated 6-simplex

Runcitruncated 6-simplex
Type	uniform 6-polytope
Schläfli symbol	t0,1,3{3,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces	70
4-faces	560
Cells	1820
Faces	2800
Edges	1890
Vertices	420
Vertex figure
Coxeter group	A6, [35], order 5040
Properties	convex

Alternate names

• Prismatotruncated heptapeton (Acronym: patal) (Jonathan Bowers)

Coordinates

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

Images

orthographic projections

Ak Coxeter plane	A6	A5	A4
Graph		150px	150px
Dihedral symmetry	[7]	[6]	[5]
Ak Coxeter plane	A3	A2
Graph	150px	150px
Dihedral symmetry	[4]	[3]

Biruncitruncated 6-simplex

biruncitruncated 6-simplex
Type	uniform 6-polytope
Schläfli symbol	t1,2,4{3,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces	84
4-faces	714
Cells	2310
Faces	3570
Edges	2520
Vertices	630
Vertex figure
Coxeter group	A6, [35], order 5040
Properties	convex

Alternate names

• Biprismatorhombated heptapeton (Acronym: bapril) (Jonathan Bowers)

Coordinates

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

Images

orthographic projections

Ak Coxeter plane	A6	A5	A4
Graph		150px	150px
Dihedral symmetry	[7]	[6]	[5]
Ak Coxeter plane	A3	A2
Graph	150px	150px
Dihedral symmetry	[4]	[3]

Runcicantellated 6-simplex

Runcicantellated 6-simplex
Type	uniform 6-polytope
Schläfli symbol	t0,2,3{3,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces	70
4-faces	455
Cells	1295
Faces	1960
Edges	1470
Vertices	420
Vertex figure
Coxeter group	A6, [35], order 5040
Properties	convex

Alternate names

• Prismatorhombated heptapeton (Acronym: pril) (Jonathan Bowers)

Coordinates

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

Images

orthographic projections

Ak Coxeter plane	A6	A5	A4
Graph		150px	150px
Dihedral symmetry	[7]	[6]	[5]
Ak Coxeter plane	A3	A2
Graph	150px	150px
Dihedral symmetry	[4]	[3]

Runcicantitruncated 6-simplex

Runcicantitruncated 6-simplex
Type	uniform 6-polytope
Schläfli symbol	t0,1,2,3{3,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces	70
4-faces	560
Cells	1820
Faces	3010
Edges	2520
Vertices	840
Vertex figure
Coxeter group	A6, [35], order 5040
Properties	convex

Alternate names

• Runcicantitruncated heptapeton
• Great prismated heptapeton (Acronym: gapil) (Jonathan Bowers)

Coordinates

The vertices of the runcicantitruncated 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 runcicantitruncated 7-orthoplex.

Images

orthographic projections

Ak Coxeter plane	A6	A5	A4
Graph		150px	150px
Dihedral symmetry	[7]	[6]	[5]
Ak Coxeter plane	A3	A2
Graph	150px	150px
Dihedral symmetry	[4]	[3]

Biruncicantitruncated 6-simplex

biruncicantitruncated 6-simplex
Type	uniform 6-polytope
Schläfli symbol	t1,2,3,4{3,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces	84
4-faces	714
Cells	2520
Faces	4410
Edges	3780
Vertices	1260
Vertex figure
Coxeter group	A6, [[35]], order 10080
Properties	convex

Alternate names

• Biruncicantitruncated heptapeton
• Great biprismated tetradecapeton (Acronym: gibpof) (Jonathan Bowers)

Coordinates

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

Images

orthographic projections

Ak Coxeter plane	A6	A5	A4
Graph		150px	150px
Symmetry	[[7]](*)=[14]	[6]	[[5]](*)=[10]
Ak Coxeter plane	A3	A2
Graph	150px	150px
Symmetry	[4]	[[3]](*)=[6]

Note: (*) Symmetry doubled for A_k graphs with even k due to symmetrically-ringed Coxeter-Dynkin diagram.

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.
• Richard Klitzing, 6D, uniform polytopes (polypeta) x3o3o3x3o3o - spil, o3x3o3o3x3o - sibpof, x3x3o3x3o3o - patal, o3x3x3o3x3o - bapril, x3o3x3x3o3o - pril, x3x3x3x3o3o - gapil, o3x3x3x3x3o - gibpof

External links

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