# Cantellated 8-simplexes

(Redirected from Bicantellated 8-simplex)
 Orthogonal projections in A8 Coxeter plane Cantellated 8-simplex Bicantellated 8-simplex Tricantellated 8-simplex Cantitruncated 8-simplex Bicantitruncated 8-simplex Tricantitruncated 8-simplex

In eight-dimensional geometry, a cantellated 8-simplex is a convex uniform 8-polytope, being a cantellation of the regular 8-simplex.

There are six unique cantellations for the 8-simplex, including permutations of truncation.

## Cantellated 8-simplex

Cantellated 8-simplex
Type uniform 8-polytope
Schläfli symbol rr{3,3,3,3,3,3,3}
Coxeter-Dynkin diagram
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges 1764
Vertices 252
Vertex figure 6-simplex prism
Coxeter group A8, [37], order 362880
Properties convex

### Alternate names

• Small rhombated enneazetton (acronym: srene) (Jonathan Bowers)[1]

### Coordinates

The Cartesian coordinates of the vertices of the cantellated 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,0,0,0,1,1,2). This construction is based on facets of the cantellated 9-orthoplex.

### Images

orthographic projections
Ak Coxeter plane A8 A7 A6 A5
Graph
Dihedral symmetry [9] [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [5] [4] [3]

## Bicantellated 8-simplex

Bicantellated 8-simplex
Type uniform 8-polytope
Schläfli symbol r2r{3,3,3,3,3,3,3}
Coxeter-Dynkin diagram
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges 5292
Vertices 756
Vertex figure
Coxeter group A8, [37], order 362880
Properties convex

### Alternate names

• Small birhombated enneazetton (acronym: sabrene) (Jonathan Bowers)[2]

### Coordinates

The Cartesian coordinates of the vertices of the bicantellated 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,0,0,1,1,2,2). This construction is based on facets of the bicantellated 9-orthoplex.

### Images

orthographic projections
Ak Coxeter plane A8 A7 A6 A5
Graph
Dihedral symmetry [9] [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [5] [4] [3]

## Tricantellated 8-simplex

tricantellated 8-simplex
Type uniform 8-polytope
Schläfli symbol r3r{3,3,3,3,3,3,3}
Coxeter-Dynkin diagram
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges 8820
Vertices 1260
Vertex figure
Coxeter group A8, [37], order 362880
Properties convex

### Alternate names

• Small trirhombihexadecaexon (acronym: satrene) (Jonathan Bowers)[3]

### Coordinates

The Cartesian coordinates of the vertices of the tricantellated 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,0,0,1,1,2,2). This construction is based on facets of the tricantellated 9-orthoplex.

### Images

orthographic projections
Ak Coxeter plane A8 A7 A6 A5
Graph
Dihedral symmetry [9] [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [5] [4] [3]

## Cantitruncated 8-simplex

Cantitruncated 8-simplex
Type uniform 8-polytope
Schläfli symbol tr{3,3,3,3,3,3,3}
Coxeter-Dynkin diagram
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter group A8, [37], order 362880
Properties convex

### Alternate names

• Great rhombated enneazetton (acronym: grene) (Jonathan Bowers)[4]

### Coordinates

The Cartesian coordinates of the vertices of the cantitruncated 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,0,0,0,1,2,3). This construction is based on facets of the bicantitruncated 9-orthoplex.

### Images

orthographic projections
Ak Coxeter plane A8 A7 A6 A5
Graph
Dihedral symmetry [9] [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [5] [4] [3]

## Bicantitruncated 8-simplex

Bicantitruncated 8-simplex
Type uniform 8-polytope
Schläfli symbol t2r{3,3,3,3,3,3,3}
Coxeter-Dynkin diagram
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter group A8, [37], order 362880
Properties convex

## Alternate names

• Great birhombated enneazetton (acronym: gabrene) (Jonathan Bowers)[5]

### Coordinates

The Cartesian coordinates of the vertices of the bicantitruncated 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,0,0,1,2,3,3). This construction is based on facets of the bicantitruncated 9-orthoplex.

### Images

orthographic projections
Ak Coxeter plane A8 A7 A6 A5
Graph
Dihedral symmetry [9] [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [5] [4] [3]

## Tricantitruncated 8-simplex

Tricantitruncated 8-simplex
Type uniform 8-polytope
Schläfli symbol t3r{3,3,3,3,3,3,3}
Coxeter-Dynkin diagram
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter group A8, [37], order 362880
Properties convex
• Great trirhombated enneazetton (acronym: gatrene) (Jonathan Bowers)[6]

### Coordinates

The Cartesian coordinates of the vertices of the tricantitruncated 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,0,1,2,3,3,3). This construction is based on facets of the bicantitruncated 9-orthoplex.

### Images

orthographic projections
Ak Coxeter plane A8 A7 A6 A5
Graph
Dihedral symmetry [9] [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [5] [4] [3]

## Related polytopes

This polytope is one of 135 uniform 8-polytopes with A8 symmetry.

## Notes

1. ^ Klitizing, (x3o3x3o3o3o3o3o - srene)
2. ^ Klitizing, (o3x3o3x3o3o3o3o - sabrene)
3. ^ Klitizing, (o3o3x3o3x3o3o3o - satrene)
4. ^ Klitizing, (x3x3x3o3o3o3o3o - grene)
5. ^ Klitizing, (o3x3x3x3o3o3o3o - gabrene)
6. ^ Klitizing, (o3o3x3x3x3o3o3o - gatrene)

## 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. "8D uniform polytopes (polyzetta)". x3o3x3o3o3o3o3o - srene, o3x3o3x3o3o3o3o - sabrene, o3o3x3o3x3o3o3o - satrene, x3x3x3o3o3o3o3o - grene, o3x3x3x3o3o3o3o - gabrene, o3o3x3x3x3o3o3o - gatrene