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

I have done 4D-geometry visualization long before raytracing. That was
wireframe for red/green glasses. I haven't done it in povray.

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

I always prefered perspective projection.

Anyway, you seem to want to use a solid 3D object. You will lose a lot of
the 4D information that way. The 4D object has vertices, edges, faces,
and hyperfaces as "visual" elements of its geometry. By your method you
lose all vertices, edges, and faces that are projected to the inside of
the 3D model. You also lose all hyperfaces alltogether.

For povray I could think of several alternatives:

1. Do a wireframe model using spheres and cylinders -- or one large
    spheresweep (after all the graph of the hypercube does have a Eulerian
    cycle). Vertcies and edges will be clearly visible, faces and hyperfaces
    sort of.
2. Show the faces as semitransparent polygons. Possibly of different colours.
    Vertices, edges, and faces are visible, hyperfaces sort of. The faces will
    maybe be too many to not be confusing.
3. Show the hyperfaces by differently coloured media.

Or, maybe, use a combination.

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

No. and that is not a feature unique to 4D->3D. An example in 2D-1D:
Consider the normal vectors (1,1) and (1,-1) and work it out. Another
example: Take the sets {(0,0),(1,1)} and {(0,1),(1,0)}. Projection, then
intersection yields {0,1}, while the other way around you obtain the empty
set. The first collects all 1D-points that can be extended to some point
in each of the original sets. The second collects all 1D-point that have
one extension that pertains to both sets simultaneously.


-- 
merge{#local i=-11;#while(i<11)#local
i=i+.1;sphere{<i*(i*i*(.05-i*i*(4e-7*i*i+3e-4))-3)10*sin(i)30>.5}#end
pigment{rgbt 1}interior{media{emission x}}hollow}//  Mark Weyer

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