# Runcinated 6-simplexes

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

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)[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, [[35]], 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 Ak 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)[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, [35], 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, [35], 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, [35], 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, [[35]], 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 Ak graphs with even k due to symmetrically-ringed Coxeter-Dynkin diagram.

## 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 [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)". x3o3o3x3o3o - spil, o3x3o3o3x3o - sibpof, x3x3o3x3o3o - patal, o3x3x3o3x3o - bapril, x3o3x3x3o3o - pril, x3x3x3x3o3o - gapil, o3x3x3x3x3o - gibpof