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# Re: OT hypercube (was: Con-other)

From: Lars Mathiesen Sunday, June 1, 2008, 12:42
2008/5/31 Mark J. Reed <markjreed@...>:
> My sophomore year in college I had a roommate (another computer geek) > who had written a basic program to display a projection of an > N-dimensional cube for N up to 7 (the limit came from the BASIC > language's limit on array subscripts). You could turn perspective on > or off, and rotate the figure arbitrarily in its space before the > projection (hypercubes rotate around planes, instead of lines, and > there are six such planar "axes").
Define rotate... if you rotate a 3-cube around a line, you have one degree of freedom (strictly: The Lie group of isometric maps from the embedding 3-space to itself that map the line to itself and restrict to the identity map on the line, is 1-dimensional). You can also rotate the 3-cube around a point, getting three degrees of freedom. Similarly, you can rotate a 4-cube around a point with six degrees of freedom, around a line with three, and around a plane with one. But of course, if you're viewing a computer animation, at any given time the 4-cube will in fact be rotating around a specific plane. And yes, the space of (antisymmetric) bivectors in the tangent space to a fourdimensional manifold does have six dimensions, and thus an orthonormal basis of six elements -- you could call those (a set of) axes of the vector space correponding to the projective(?) space of the planes themselves. Lars