In six-dimensional
geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, 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
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
In geometry, a Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called ''coordinates'', which are the signed distances to the point from two fixed perpendicular o ...
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
In geometry, a Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called ''coordinates'', which are the signed distances to the point from two fixed perpendicular o ...
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, a Coxeter element is an element of an irreducible Coxeter group which is a product of all simple reflections. The product depends on the order in which they are taken, but different orderings produce conjugate elements, which hav ...
, including the regular
6-cube or
6-orthoplex.
Notes
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
wiley.com
*** (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.
* o3o3o3o3x4o - rax, o3o3o3x3o4o - brox
External links
*
Polytopes of Various Dimensions
{{Polytopes
6-polytopes