Hi folks.

Firstly, has anyone looked into it? (4D geometry in POV-Ray that is.)

Part of the problem is that the fourth dimension is a little bit 
mind-numbing to think about!

Anyway, I'd like to use POV-Ray to draw animations of a hypercube 
rotating in 4 dimensions, orthographically projected into 3 dimensions.

If I'm not mistaken, a hypercube can be defined as the set of all points 
that satisfy
   -k < (x, y, z, w) < +k
It can also be defined as the intersection of 8 hyperplanes - and this 
lents itself for to rotation.

Suppose we have a hyperplane
   Ax + By + Cz + Dw - E = 0
Then the "normal" would be <A, B, C, D> and the distance from the origin 
would be E. (In 2D, a normal defines a line. In 3D, a normal defines a 
plane. So presumably in 4D a normal defines a hyperplane...)

If (say) <A, B, C, D> = <0, 1, 0, 0>, then an orthographic projection 
into 3D gives us an ordinary plane with normal <0, 1, 0>.

However...

As far as I can tell, if you rotate that hyperplane off the axis even 
slightly (in such a way that D <> 0), then the 3D projection of that 
hyperplane fills the entire 3D space. That would mean that if I do any 
double rotation by a small angle, *all* 8 projections would fill all of 
3D space, resulting in... well... nothing.

Question: is the intersection of the 3D projections of the 8 hyperplanes 
equal to the 3D projection of the intersection of the 8 hyperplanes?

(Question: Does that even make sense?!)

Thoughts / suggestions?
Andrew @ home.

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