Rectified 6-cubes

(Redirected from Birectified 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 6-polytope
Schläfli symbol t1{4,34} or r{4,34}
${\displaystyle \left\{{\begin{array}{l}4\\3,3,3,3\end{array}}\right\}}$
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:

${\displaystyle (0,\ \pm 1,\ \pm 1,\ \pm 1,\ \pm 1,\ \pm 1)}$

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]

Birectified 6-cube

Birectified 6-cube
Type uniform 6-polytope
Coxeter symbol 0311
Schläfli symbol t2{4,34} or 2r{4,34}
${\displaystyle \left\{{\begin{array}{l}3,4\\3,3,3\end{array}}\right\}}$
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:

${\displaystyle (0,\ 0,\ \pm 1,\ \pm 1,\ \pm 1,\ \pm 1)}$

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]

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.

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)". o3x3o3o3o4o - rax, o3o3x3o3o4o - brox,