Re: OT hypercube (was: Con-other)

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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