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Re: OT hypercube (was: Con-other)

From:Mark J. Reed <markjreed@...>
Date:Sunday, June 1, 2008, 15:30
I think it was clear that by "rotation" I meant what you called 1
degree of freedom.  At least in 3-space, and so I assume higher spaces
as well, multiple-freedom rotations can be described as an ordered
composite of single-freedom rotations...

On 6/1/08, Lars Mathiesen <thorinn@...> wrote:
> 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 >
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