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Topic:
4dimensional cube
Replies:
6
Last Post:
May 13, 2000 12:54 AM




Re: 4dimensional cube
Posted:
May 11, 2000 4:44 PM


joss <ian_partridge.joss_taitNOiaSPAM@ukgateway.net.invalid> writes: > We have speculated on what the two dimensional shadow of a > fourdimensional cube would look like, but can't get any > nearer to closure...
The points in a ddimensional cube are { xi ei } where 0 <= xi <= 1 and ei are an orthonormal basis.
Projecting this down to a kdimensional space simply consists of choosing images for each of the ei and making a set of the same form { xi image(ei)  0 <= xi <= 1 }. Of course it need not be a cube because the images of the images of the ei need not be orthonormal. Such a projection is known as a zonohedron or zonotope; see http://www.ics.uci.edu/~eppstein/junkyard/zono.html for some web pointers about this stuff.
In the particular case of a projection into the plane, you get a centrally symmetric polygon with up to 2d sides, where each side is parallel to one of the images of one of the ei. For a projection of a 4cube onto a plane, it's either a square, a hexagon, or an octagon. The square and hexagon cases only happen for certain special projection directions in which two of the ei project to parallel vectors or one of them projects to zero. The projection of a 4cube into 3space usually looks like a rhombic dodecahedron.  David Eppstein UC Irvine Dept. of Information & Computer Science eppstein@ics.uci.edu http://www.ics.uci.edu/~eppstein/



