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, [3^5], order 5040
Properties	convex

Alternate names

• Small prismated heptapeton (Acronym: spil) (Jonathan Bowers)^[1]

Coordinates

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
Dihedral symmetry	[7]	[6]	[5]
Ak Coxeter plane	A3	A2
Graph
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, [[3^5]], order 10080
Properties	convex

Alternate names

• Small biprismated tetradecapeton (Acronym: sibpof) (Jonathan Bowers)^[2]

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
Symmetry	[[7]]^(*)=[14]	[6]	[[5]]^(*)=[10]
Ak Coxeter plane	A3	A2
Graph
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, [3^5], order 5040
Properties	convex

Alternate names

• Prismatotruncated heptapeton (Acronym: patal) (Jonathan Bowers)^[3]

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
Dihedral symmetry	[7]	[6]	[5]
Ak Coxeter plane	A3	A2
Graph
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, [3^5], order 5040
Properties	convex

Alternate names

• Biprismatorhombated heptapeton (Acronym: bapril) (Jonathan Bowers)^[4]

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
Dihedral symmetry	[7]	[6]	[5]
Ak Coxeter plane	A3	A2
Graph
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, [3^5], order 5040
Properties	convex

Alternate names

• Prismatorhombated heptapeton (Acronym: pril) (Jonathan Bowers)^[5]

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
Dihedral symmetry	[7]	[6]	[5]
Ak Coxeter plane	A3	A2
Graph
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, [3^5], order 5040
Properties	convex

Alternate names

• Runcicantitruncated heptapeton
• Great prismated heptapeton (Acronym: gapil) (Jonathan Bowers)^[6]

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
Dihedral symmetry	[7]	[6]	[5]
Ak Coxeter plane	A3	A2
Graph
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, [[3^5]], order 10080
Properties	convex

Alternate names

• Biruncicantitruncated heptapeton
• Great biprismated tetradecapeton (Acronym: gibpof) (Jonathan Bowers)^[7]

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
Symmetry	[[7]]^(*)=[14]	[6]	[[5]]^(*)=[10]
Ak Coxeter plane	A3	A2
Graph
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.

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, (x3o3o3x3o3o - spil)
2. Klitzing, (o3x3o3o3x3o - sibpof)
3. Klitzing, (x3x3o3x3o3o - patal)
4. Klitzing, (o3x3x3o3x3o - bapril)
5. Klitzing, (x3o3x3x3o3o - pril)
6. Klitzing, (x3x3x3x3o3o - gapil)
7. Klitzing, (o3x3x3x3x3o - gibpof)

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.
• 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