Rectified 6-cubes

6-cube	Rectified 6-cube	Birectified 6-cube
Birectified 6-orthoplex	Rectified 6-orthoplex	6-orthoplex
Orthogonal projections in A6 Coxeter plane

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-cube 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} [math]\displaystyle{ \left\{\begin{array}{l}4\\3, 3, 3, 3\end{array}\right\} }[/math]
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.

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

[math]\displaystyle{ (0,\ \pm1,\ \pm1,\ \pm1,\ \pm1,\ \pm1) }[/math]

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} [math]\displaystyle{ \left\{\begin{array}{l}3, 4\\3, 3, 3\end{array}\right\} }[/math]
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)
• Rectified 6-demicube

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:

[math]\displaystyle{ (0,\ 0,\ \pm1,\ \pm1,\ \pm1,\ \pm1) }[/math]

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)". https://bendwavy.org/klitzing/dimensions/polypeta.htm.  o3x3o3o3o4o - rax, o3o3x3o3o4o - brox,

External links

• Weisstein, Eric W.. "Hypercube". http://mathworld.wolfram.com/Hypercube.html. 
• Polytopes of Various Dimensions
• Multi-dimensional Glossary