# Cantellated 6-simplexes

(Redirected from Bicantellated 6-simplex)
 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}
or ${\displaystyle r\left\{{\begin{array}{l}3,3,3,3\\3\end{array}}\right\}}$
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 2rr{3,3,3,3,3}
or ${\displaystyle r\left\{{\begin{array}{l}3,3,3\\3,3\end{array}}\right\}}$
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}
or ${\displaystyle t\left\{{\begin{array}{l}3,3,3,3\\3\end{array}}\right\}}$
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 2tr{3,3,3,3,3}
or ${\displaystyle t\left\{{\begin{array}{l}3,3,3\\3,3\end{array}}\right\}}$
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.

## 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.
• Klitzing, Richard. "6D uniform polytopes (polypeta)". x3o3x3o3o3o - sril, o3x3o3x3o3o - sabril, x3x3x3o3o3o - gril, o3x3x3x3o3o - gabril