# Cantellated 6-cubes

(Redirected from Bicantitruncated 6-cube)
 Orthogonal projections in B6 Coxeter plane 6-cube Cantellated 6-cube Bicantellated 6-cube 6-orthoplex Cantellated 6-orthoplex Bicantellated 6-orthoplex Cantitruncated 6-cube Bicantitruncated 6-cube Bicantitruncated 6-orthoplex Cantitruncated 6-orthoplex

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

There are 8 cantellations for the 6-cube, including truncations. Half of them are more easily constructed from the dual 5-orthoplex.

## Cantellated 6-cube

Cantellated 6-cube
Type uniform 6-polytope
Schläfli symbol rr{4,3,3,3,3}
or ${\displaystyle r\left\{{\begin{array}{l}3,3,3,3\\4\end{array}}\right\}}$
Coxeter-Dynkin diagrams

5-faces
4-faces
Cells
Faces
Edges 4800
Vertices 960
Vertex figure
Coxeter groups B6, [3,3,3,3,4]
Properties convex

### Alternate names

• Cantellated hexeract
• Small rhombated hexeract (acronym: srox) (Jonathan Bowers)[1]

### Images

orthographic projections
Coxeter plane B6 B5 B4
Graph
Dihedral symmetry [12] [10] [8]
Coxeter plane B3 B2
Graph
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]

## Bicantellated 6-cube

Cantellated 6-cube
Type uniform 6-polytope
Schläfli symbol 2rr{4,3,3,3,3}
or ${\displaystyle r\left\{{\begin{array}{l}3,3,3\\3,4\end{array}}\right\}}$
Coxeter-Dynkin diagrams

5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups B6, [3,3,3,3,4]
Properties convex

### Alternate names

• Bicantellated hexeract
• Small birhombated hexeract (acronym: saborx) (Jonathan Bowers)[2]

### Images

orthographic projections
Coxeter plane B6 B5 B4
Graph
Dihedral symmetry [12] [10] [8]
Coxeter plane B3 B2
Graph
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]

## Cantitruncated 6-cube

Cantellated 6-cube
Type uniform 6-polytope
Schläfli symbol tr{4,3,3,3,3}
or ${\displaystyle t\left\{{\begin{array}{l}3,3,3,3\\4\end{array}}\right\}}$
Coxeter-Dynkin diagrams

5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups B6, [3,3,3,3,4]
Properties convex

### Alternate names

• Cantitruncated hexeract
• Great rhombihexeract (acronym: grox) (Jonathan Bowers)[3]

### Images

orthographic projections
Coxeter plane B6 B5 B4
Graph
Dihedral symmetry [12] [10] [8]
Coxeter plane B3 B2
Graph
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]

It is fourth in a series of cantitruncated hypercubes:

## Bicantitruncated 6-cube

Cantellated 6-cube
Type uniform 6-polytope
Schläfli symbol 2tr{4,3,3,3,3}
or ${\displaystyle t\left\{{\begin{array}{l}3,3,3\\3,4\end{array}}\right\}}$
Coxeter-Dynkin diagrams

5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups B6, [3,3,3,3,4]
Properties convex

### Alternate names

• Bicantitruncated hexeract
• Great birhombihexeract (acronym: gaborx) (Jonathan Bowers)[4]

### Images

orthographic projections
Coxeter plane B6 B5 B4
Graph
Dihedral symmetry [12] [10] [8]
Coxeter plane B3 B2
Graph
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]

## Related polytopes

These polytopes are part of a set of 63 uniform 6-polytopes generated from the B6 Coxeter plane, including the regular 6-cube or 6-orthoplex.

## Notes

1. ^ Klitzing, (o3o3o3x3o4x - srox)
2. ^ Klitzing, (o3o3x3o3x4o - saborx)
3. ^ Klitzing, (o3o3o3x3x4x - grox)
4. ^ Klitzing, (o3o3x3x3x4o - gaborx)

## 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)". o3o3o3x3o4x - srox, o3o3x3o3x4o - saborx, o3o3o3x3x4x - grox, o3o3x3x3x4o - gaborx