# Runcinated 6-cubes

(Redirected from Biruncitruncated 6-cube)
 Orthogonal projections in B6 Coxeter plane 6-cube Runcinated 6-cube Biruncinated 6-cube Runcinated 6-orthoplex 6-orthoplex Runcitruncated 6-cube Biruncitruncated 6-cube Runcicantellated 6-orthoplex Runcicantellated 6-cube Biruncitruncated 6-orthoplex Runcitruncated 6-orthoplex Runcicanti-truncated 6-cube Biruncicanti-truncated 6-cube Runcicanti-truncated 6-orthoplex

In six-dimensional geometry, a runcinated 6-cube is a convex uniform 6-polytope with 3rd order truncations (runcination) of the regular 6-cube.

There are 12 unique runcinations of the 6-cube with permutations of truncations, and cantellations. Half are expressed relative to the dual 6-orthoplex.

## Runcinated 6-cube

 Runcinated 6-cube Type Uniform 6-polytope Schläfli symbol t0,3{4,3,3,3,3} Coxeter-Dynkin diagram 4-faces Cells Faces Edges 7680 Vertices 1280 Vertex figure Coxeter group B6 [4,3,3,3,3] Properties convex

### Alternate names

• Small prismated hexeract (spox) (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]

## Biruncinated 6-cube

 Biruncinated 6-cube Type Uniform 6-polytope Schläfli symbol t1,4{4,3,3,3,3} Coxeter-Dynkin diagram 4-faces Cells Faces Edges 11520 Vertices 1920 Vertex figure Coxeter group B6 [4,3,3,3,3] Properties convex

### Alternate names

• Small biprismated hexeractihexacontatetrapeton (sobpoxog) (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]

## Runcitruncated 6-cube

 Runcitruncated 6-cube Type Uniform 6-polytope Schläfli symbol t0,1,3{4,3,3,3,3} Coxeter-Dynkin diagram 4-faces Cells Faces Edges 17280 Vertices 3840 Vertex figure Coxeter group B6 [4,3,3,3,3] Properties convex

### Alternate names

• Prismatotruncated hexeract (potax) (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]

## Biruncitruncated 6-cube

 Biruncitruncated 6-cube Type Uniform 6-polytope Schläfli symbol t1,2,4{4,3,3,3,3} Coxeter-Dynkin diagram 4-faces Cells Faces Edges 23040 Vertices 5760 Vertex figure Coxeter group B6 [4,3,3,3,3] Properties convex

### Alternate names

• Biprismatotruncated hexeract (boprag) (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]

## Runcicantellated 6-cube

 Runcicantellated 6-cube Type Uniform 6-polytope Schläfli symbol t0,2,3{4,3,3,3,3} Coxeter-Dynkin diagram 4-faces Cells Faces Edges 13440 Vertices 3840 Vertex figure Coxeter group B6 [4,3,3,3,3] Properties convex

### Alternate names

• Prismatorhombated hexeract (prox) (Jonathan Bowers)[5]

### 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]

## Runcicantitruncated 6-cube

 Runcicantitruncated 6-cube Type Uniform 6-polytope Schläfli symbol t0,1,2,3{4,3,3,3,3} Coxeter-Dynkin diagram 4-faces Cells Faces Edges 23040 Vertices 7680 Vertex figure Coxeter group B6 [4,3,3,3,3] Properties convex

### Alternate names

• Great prismated hexeract (gippox) (Jonathan Bowers)[6]

### 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]

## Biruncitruncated 6-cube

 Biruncitruncated 6-cube Type Uniform 6-polytope Schläfli symbol t1,2,3,4{4,3,3,3,3} Coxeter-Dynkin diagram 4-faces Cells Faces Edges 23040 Vertices 5760 Vertex figure Coxeter group B6 [4,3,3,3,3] Properties convex

### Alternate names

• Biprismatotruncated hexeract (boprag) (Jonathan Bowers)[7]

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

## Notes

1. ^ Klitzing, (o3o3x3o3o4x - spox)
2. ^ Klitzing, (o3x3o3o3x4o - sobpoxog)
3. ^ Klitzing, (o3o3x3o3x4x - potax)
4. ^ Klitzing, (o3x3o3x3x4o - boprag)
5. ^ Klitzing, (o3o3x3x3o4x - prox)
6. ^ Klitzing, (o3o3x3x3x4x - gippox)
7. ^ Klitzing, (o3x3x3x3x4o - boprag)

## 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)". o3o3x3o3o4x - spox, o3x3o3o3x4o - sobpoxog, o3o3x3o3x4x - potax, o3x3o3x3x4o - boprag, o3o3x3x3o4x - prox, o3o3x3x3x4x - gippox, o3x3x3x3x4o - boprag