# Cantellated 7-orthoplexes

 Orthogonal projections in B6 Coxeter plane 7-orthoplex Cantellated 7-orthoplex Bicantellated 7-orthoplex Birectified 7-orthoplex Cantitruncated 7-orthoplex Bicantitruncated 7-orthoplex Cantellated 7-cube Bicantellated 7-cube Tricantellated 7-cube Cantitruncated 7-cube Bicantitruncated 7-cube Tricantitruncated 7-cube

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

There are ten degrees of cantellation for the 7-orthoplex, including truncations. Six are most simply constructible from the dual 7-cube.

## Cantellated 7-orthoplex

Cantellated 7-orthoplex
Type uniform 7-polytope
Schläfli symbol rr{3,3,3,3,3,4}
Coxeter-Dynkin diagram
6-faces
5-faces
4-faces
Cells
Faces
Edges 7560
Vertices 840
Vertex figure
Coxeter groups B7, [4,3,3,3,3,3]
Properties convex

### Alternate names

• Small rhombated hecatonicosoctaexon (acronym: sarz) (Jonathan Bowers)[1]

### Images

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph
Dihedral symmetry [14] [12] [10]
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph
Dihedral symmetry [8] [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]

## Bicantellated 7-orthoplex

Bicantellated 7-orthoplex
Type uniform 7-polytope
Schläfli symbol 2rr{3,3,3,3,3,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups B7, [4,3,3,3,3,3]
Properties convex

### Alternate names

• Small birhombated hecatonicosoctaexon (acronym: sebraz) (Jonathan Bowers)[2]

### Images

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph
Dihedral symmetry [14] [12] [10]
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph
Dihedral symmetry [8] [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]

## Cantitruncated 7-orthoplex

Cantitruncated 7-orthoplex
Type uniform 7-polytope
Schläfli symbol tr{3,3,3,3,3,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges 8400
Vertices 1680
Vertex figure
Coxeter groups B7, [4,3,3,3,3,3]
Properties convex

### Alternate names

• Great rhombated hecatonicosoctaexon (acronym: garz) (Jonathan Bowers)[3]

### Images

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph too complex
Dihedral symmetry [14] [12] [10]
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph
Dihedral symmetry [8] [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]

## Bicantitruncated 7-orthoplex

Bicantitruncated 7-orthoplex
Type uniform 7-polytope
Schläfli symbol 2tr{3,3,3,3,3,4}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges 30240
Vertices 6720
Vertex figure
Coxeter groups B7, [4,3,3,3,3,3]
Properties convex

### Alternate names

• Great birhombated hecatonicosoctaexon (acronym: gebraz) (Jonathan Bowers)[4]

### Images

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph too complex
Dihedral symmetry [14] [12] [10]
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph
Dihedral symmetry [8] [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]

## Related polytopes

These polytopes are from a family of 127 uniform 7-polytopes with B7 symmetry.

## Notes

1. ^ Klitizing, (o3o3o3o3x3o4x - sarz)
2. ^ Klitizing, (o3o3o3x3o3x4o - sebraz)
3. ^ Klitizing, (o3o3o3o3x3x4x - garz)
4. ^ Klitizing, (o3o3o3x3x3x4o - gebraz)

## 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. 1966
• Klitzing, Richard. "7D uniform polytopes (polyexa)". - o3o3o3o3x3o4x - sarz, o3o3o3x3o3x4o - sebraz, o3o3o3o3x3x4x - garz, o3o3o3x3x3x4o - gebraz