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6-cube Template:CDD	Rectified 6-cube Template:CDD	Birectified 6-cube Template:CDD
Birectified 6-orthoplex Template:CDD	Rectified 6-orthoplex Template:CDD	6-orthoplex Template:CDD
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-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	Template:CDD Template:CDD
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:

$(0,\pm 1,\pm 1,\pm 1,\pm 1,\pm 1)$

Birectified 6-cube

Birectified 6-cube
Type	uniform polypeton
Schläfli symbol	2r{4,3,3,3,3}
Coxeter-Dynkin diagrams	Template:CDD Template:CDD
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:

$(0,0,\pm 1,\pm 1,\pm 1,\pm 1)$

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.

Notes

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

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.
• Template:KlitzingPolytopes o3x3o3o3o4o - rax, o3o3x3o3o4o - brox,

External links

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• Template:GlossaryForHyperspace
• Polytopes of Various Dimensions
• Multi-dimensional Glossary