Re: OT hypercube (was: Con-other)

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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Mark J. Reed <markjreed@...>