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

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Posts: 128
Registered: 12/8/04
Re: 4-dimensional cube
Posted: May 11, 2000 4:44 PM
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joss <> writes:
> We have speculated on what the two dimensional shadow of a
> four-dimensional cube would look like, but can't get any
> nearer to closure...

The points in a d-dimensional cube are { xi ei } where
0 <= xi <= 1 and ei are an orthonormal basis.

Projecting this down to a k-dimensional 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 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
4-cube 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 4-cube into 3-space usually
looks like a rhombic dodecahedron.
David Eppstein UC Irvine Dept. of Information & Computer Science

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