# Cantellated 6-simplexes

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 Orthogonal projections in A6 Coxeter plane 6-simplex Cantellated 6-simplex Bicantellated 6-simplex Birectified 6-simplex Cantitruncated 6-simplex Bicantitruncated 6-simplex

In six-dimensional geometry, a cantellated 6-simplex is a convex uniform 6-polytope, being a cantellation of the regular 6-simplex.

There are unique 4 degrees of cantellation for the 6-simplex, including truncations.

## Cantellated 6-simplex

Cantellated 6-simplex
Type uniform 6-polytope
Schläfli symbol rr{3,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces 35
4-faces 210
Cells 560
Faces 805
Edges 525
Vertices 105
Vertex figure 5-cell prism
Coxeter group A6, [35], order 5040
Properties convex

### Alternate names

• Small rhombated heptapeton (Acronym: sril) (Jonathan Bowers)[1]

### Coordinates

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

[2]

## Bicantellated 6-simplex

Bicantellated 6-simplex
Type uniform 6-polytope
Schläfli symbol r2r{3,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces 49
4-faces 329
Cells 980
Faces 1540
Edges 1050
Vertices 210
Vertex figure
Coxeter group A6, [35], order 5040
Properties convex

### Alternate names

• Small prismated heptapeton (Acronym: sabril) (Jonathan Bowers)[3]

### Coordinates

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

## Cantitruncated 6-simplex

cantitruncated 6-simplex
Type uniform 6-polytope
Schläfli symbol tr{3,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces 35
4-faces 210
Cells 560
Faces 805
Edges 630
Vertices 210
Vertex figure
Coxeter group A6, [35], order 5040
Properties convex

### Alternate names

• Great rhombated heptapeton (Acronym: gril) (Jonathan Bowers)[4]

### Coordinates

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

## Bicantitruncated 6-simplex

bicantitruncated 6-simplex
Type uniform 6-polytope
Schläfli symbol t2r{3,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces 49
4-faces 329
Cells 980
Faces 1540
Edges 1260
Vertices 420
Vertex figure
Coxeter group A6, [35], order 5040
Properties convex

### Alternate names

• Great birhombated heptapeton (Acronym: gabril) (Jonathan Bowers)[5]

### Coordinates

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

 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. ^ Klitizing, (x3o3x3o3o3o - sril)
2. ^ Klitzing, (x3o3x3o3o3o - sril)
3. ^ Klitzing, (o3x3o3x3o3o - sabril)
4. ^ Klitzing, (x3x3x3o3o3o - gril)
5. ^ Klitzing, (o3x3x3x3o3o - gabril)

## 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) x3o3x3o3o3o - sril, o3x3o3x3o3o - sabril, x3x3x3o3o3o - gril, o3x3x3x3o3o - gabril