In six-dimensional
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, a rectified 6-cube is a convex
uniform 6-polytope, being a
rectification
Rectification has the following technical meanings:
Mathematics
* Rectification (geometry), truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points
* Rectifiable curve, in mathematics
* Recti ...
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
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 are all permutations of:
:
Images
Birectified 6-cube
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 are all permutations of:
:
Images
Related polytopes
These polytopes are part of a set of 63
uniform 6-polytopes generated from the B
6 Coxeter plane
In mathematics, the Coxeter number ''h'' is the order of a Coxeter element of an irreducible Coxeter group. It is named after H.S.M. Coxeter.
Definitions
Note that this article assumes a finite Coxeter group. For infinite Coxeter groups, there a ...
, including the regular
6-cube or
6-orthoplex.
Notes
References
*
H.S.M. Coxeter
Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century.
Biography
Coxeter was born in Kensington t ...
:
** 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,
*** (Paper 22) H.S.M. Coxeter, ''Regular and Semi Regular Polytopes I'',
ath. Zeit. 46 (1940) 380-407, MR 2,10*** (Paper 23) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes II'',
ath. Zeit. 188 (1985) 559-591*** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'',
ath. Zeit. 200 (1988) 3-45*
Norman Johnson ''Uniform Polytopes'', Manuscript (1991)
** N.W. Johnson: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D.
* o3x3o3o3o4o - rax, o3o3x3o3o4o - brox,
External links
*
Polytopes of Various Dimensions
{{Polytopes
6-polytopes