# Rectified 6-cubes

(Redirected from Rectified 6-cube)
 Orthogonal projections in A6 Coxeter plane 6-cube Rectified 6-cube Birectified 6-cube Birectified 6-orthoplex Rectified 6-orthoplex 6-orthoplex

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

There are unique 6 degrees of rectifications, the zeroth being the 6-cube, and the 6th and last being the 6-orthoplex. Vertices of the rectified 6-cube are located at the edge-centers of the 6-cube. Vertices of the birectified 6-ocube are located in the square face centers of the 6-cube.

## Rectified 6-cube

Rectified 6-cube
Type uniform polypeton
Schläfli symbol r{4,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces 76
4-faces 444
Cells 1120
Faces 1520
Edges 960
Vertices 192
Vertex figure 5-cell prism
Petrie polygon Dodecagon
Coxeter groups B6, [3,3,3,3,4]
D6, [33,1,1]
Properties convex

### Alternate names

• Rectified hexeract (acronym: rax) (Jonathan Bowers)

### Construction

The rectified 6-cube may be constructed from the 6-cube by truncating its vertices at the midpoints of its edges.

### Coordinates

The Cartesian coordinates of the vertices of the rectified 6-cube with edge length √2 are all permutations of:

$(0,\ \pm1,\ \pm1,\ \pm1,\ \pm1,\ \pm1)$

### 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]

## Birectified 6-cube

Birectified 6-cube
Type uniform polypeton
Schläfli symbol 2r{4,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces 76
4-faces 636
Cells 2080
Faces 3200
Edges 1920
Vertices 240
Vertex figure {4}x{3,3} duoprism
Coxeter groups B6, [3,3,3,3,4]
D6, [33,1,1]
Properties convex

### Alternate names

• Birectified hexeract (acronym: brox) (Jonathan Bowers)

### Construction

The birectified 6-cube may be constructed from the 6-cube by truncating its vertices at the midpoints of its edges.

### Coordinates

The Cartesian coordinates of the vertices of the rectified 6-cube with edge length √2 are all permutations of:

$(0,\ 0,\ \pm1,\ \pm1,\ \pm1,\ \pm1)$

### 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 polypeta generated from the B6 Coxeter plane, including the regular 6-cube or 6-orthoplex.

 β6 t1β6 t2β6 t2γ6 t1γ6 γ6 t0,1β6 t0,2β6 t1,2β6 t0,3β6 t1,3β6 t2,3γ6 t0,4β6 t1,4γ6 t1,3γ6 t1,2γ6 t0,5γ6 t0,4γ6 t0,3γ6 t0,2γ6 t0,1γ6 t0,1,2β6 t0,1,3β6 t0,2,3β6 t1,2,3β6 t0,1,4β6 t0,2,4β6 t1,2,4β6 t0,3,4β6 t1,2,4γ6 t1,2,3γ6 t0,1,5β6 t0,2,5β6 t0,3,4γ6 t0,2,5γ6 t0,2,4γ6 t0,2,3γ6 t0,1,5γ6 t0,1,4γ6 t0,1,3γ6 t0,1,2γ6 t0,1,2,3β6 t0,1,2,4β6 t0,1,3,4β6 t0,2,3,4β6 t1,2,3,4γ6 t0,1,2,5β6 t0,1,3,5β6 t0,2,3,5γ6 t0,2,3,4γ6 t0,1,4,5γ6 t0,1,3,5γ6 t0,1,3,4γ6 t0,1,2,5γ6 t0,1,2,4γ6 t0,1,2,3γ6 t0,1,2,3,4β6 t0,1,2,3,5β6 t0,1,2,4,5β6 t0,1,2,4,5γ6 t0,1,2,3,5γ6 t0,1,2,3,4γ6 t0,1,2,3,4,5γ6

## 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, 6D, uniform polytopes (polypeta) o3x3o3o3o4o - rax, o3o3x3o3o4o - brox,