So I saw how flat planes can intersect at a point in 4D (in 3D they're either parallel or intersect along a line), or are skew (neither parallel nor intersecting); how hyperplanes (3D “planes”, as spaces in 4D) intersect and interact, how rotations work a little more, etc...
Nothing that I learned is actually new. The geometry of four dimensions is very well-studied, and the mathematics is not terrifically complicated. Quick visits to Wikipedia will tell you all about the known 4D (and higher) regular polytopes (which actually is an interesting topic; 4D has an unusually high six regular convex polytopes, whereas 3D has five (the Platonic solids), and everything above 4 has only three), and all kinds of details about how things can be arranged in many higher dimensions... It's interesting reading, but for me, I'm trying to learn to “picture” it better, improve my intuition about such spaces (I still have to put “picture” in scare quotes, because it isn't quite picturing it). I seem to be most successful with the “red/blue” dimensional that I describe on my hypersphere page, and which I read about in Ian Stewart's book Flatterland. Picture the fourth dimension as a color, in this case running from red through purple to blue. Things have to be in the same location and color to be in contact, etc. It's reasonably successful.
Anyway, something else occurred to me about this. We can't really picture four dimensions particularly well, in part, I think, because we have specialized mental machinery dedicated to picturing spatial relationships that we're trying to cram it into, and that machinery is of course designed for three dimensions. But if we were doing this all the “hard” way, actually keeping track of co-ordinates of vertices and whatnot and doing the various trigonometric calculations to compute rotations and everything, there is nothing intrinsically harder about doing it for four dimensions than three. Well, there is more to keep track of, but nothing qualitatively different. So a putative artificial intelligence, something that had access to its own thought-processes and modeling systems, could presumably think in four (or five, etc) dimensions just as comfortably as in three, though perhaps a bit slower.
This sort of plays into the feeling you get when you first study this sort of thing, of “Gee, we could access this fourth dimension if we only knew which direction it was,” as though our limited experience with only three dimensions was all that was stopping us. But a computer (even a non-intelligent one) thinking in four dimensions perceives it just as clearly as it does three. Even if you “knew what direction” it was, it doesn't help you point anything there if you don't have any accessible physical processes that exert a force with any component that way. Something to ponder.