# Cantellated 7-simplex

 Orthogonal projections in A7Coxeter plane 7-simplex Cantellated 7-simplex Bicantellated 7-simplex Tricantellated 7-simplex Birectified 7-simplex Cantitruncated 7-simplex Bicantitruncated 7-simplex Tricantitruncated 7-simplex

In seven-dimensional geometry, a cantellated 7-simplex is a convex uniform 7-polytope, being a cantellation of the regular 7-simplex.

There are unique 6 degrees of cantellation for the 7-simplex, including truncations.

## Cantellated 7-simplex

Cantellated 7-simplex
Type uniform polyexon
Schläfli symbol t0,2{3,3,3,3,3,3}
Coxeter-Dynkin diagram
6-faces
5-faces
4-faces
Cells
Faces
Edges 1008
Vertices 168
Vertex figure 5-simplex prism
Coxeter groups A7, [3,3,3,3,3,3]
Properties convex

### Alternate names

• Small rhombated octaexon (acronym: saro) (Jonathan Bowers)[1]

### Coordinates

The vertices of the cantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,0,1,1,2). This construction is based on facets of the cantellated 8-orthoplex.

### Images

orthographic projections
AkCoxeter plane A7 A6 A5
Graph
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [5] [4] [3]
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## Bicantellated 7-simplex

Bicantellated 7-simplex
Type uniform polyexon
Schläfli symbol t1,3{3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges 2520
Vertices 420
Vertex figure
Coxeter groups A7, [3,3,3,3,3,3]
Properties convex

### Alternate names

• Small birhombated octaexon (acronym: sabro) (Jonathan Bowers)[2]

### Coordinates

The vertices of the bicantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,1,1,2,2). This construction is based on facets of the bicantellated 8-orthoplex.

### Images

orthographic projections
AkCoxeter plane A7 A6 A5
Graph
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [5] [4] [3]
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## Tricantellated 7-simplex

Tricantellated 7-simplex
Type uniform polyexon
Schläfli symbol t2,4{3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges 3360
Vertices 560
Vertex figure
Coxeter groups A7, [3,3,3,3,3,3]
Properties convex

### Alternate names

• Small trirhombihexadecaexon (stiroh) (Jonathan Bowers)[3]

### Coordinates

The vertices of the tricantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,1,2,2,2). This construction is based on facets of the tricantellated 8-orthoplex.

### Images

orthographic projections
AkCoxeter plane A7 A6 A5
Graph
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [5] [4] [3]
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## Cantitruncated 7-simplex

Cantitruncated 7-simplex
Type uniform polyexon
Schläfli symbol t0,1,2{3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges 1176
Vertices 336
Vertex figure
Coxeter groups A7, [3,3,3,3,3,3]
Properties convex

### Alternate names

• Great rhombated octaexon (acronym: garo) (Jonathan Bowers)[4]

### Coordinates

The vertices of the cantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,0,1,2,3). This construction is based on facets of the cantitruncated 8-orthoplex.

### Images

orthographic projections
AkCoxeter plane A7 A6 A5
Graph
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [5] [4] [3]
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## Bicantitruncated 7-simplex

Bicantitruncated 7-simplex
Type uniform polyexon
Schläfli symbol t1,2,3{3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges 2940
Vertices 840
Vertex figure
Coxeter groups A7, [3,3,3,3,3,3]
Properties convex

### Alternate names

• Great birhombated octaexon (acronym: gabro) (Jonathan Bowers)[5]

### Coordinates

The vertices of the bicantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,1,2,3,3). This construction is based on facets of the bicantitruncated 8-orthoplex.

### Images

orthographic projections
AkCoxeter plane A7 A6 A5
Graph
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [5] [4] [3]
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## Tricantitruncated 7-simplex

Tricantitruncated 7-simplex
Type uniform polyexon
Schläfli symbol t2,3,4{3,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges 3920
Vertices 1120
Vertex figure
Coxeter groups A7, [3,3,3,3,3,3]
Properties convex

### Alternate names

• Great trirhombihexadecaexon (acronym: gatroh) (Jonathan Bowers)[6]

### Coordinates

The vertices of the tricantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,2,3,4,4). This construction is based on facets of the tricantitruncated 8-orthoplex.

### Images

orthographic projections
AkCoxeter plane A7 A6 A5
Graph
Dihedral symmetry [8] [[7]] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [[5]] [4] [[3]]
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## Related polytopes

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## Notes

1. ^ Klitizing, (x3o3x3o3o3o3o - saro)
2. ^ Klitizing, (o3x3o3x3o3o3o - sabro)
3. ^ Klitizing, (o3o3x3o3x3o3o - stiroh)
4. ^ Klitizing, (x3x3x3o3o3o3o - garo)
5. ^ Klitizing, (o3x3x3x3o3o3o - gabro)
6. ^ Klitizing, (o3o3x3x3x3o3o - gatroh)
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## 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, editied 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, 7D, uniform polytopes (polyexa) x3o3x3o3o3o3o - saro, o3x3o3x3o3o3o - sabro, o3o3x3o3x3o3o - stiroh, x3x3x3o3o3o3o - garo, o3x3x3x3o3o3o - gabro, o3o3x3x3x3o3o - gatroh
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