OK, I can't claim to really understand what this thing is doing, but the Hyperspace Star Polytope Slicer is an awesome thing to watch. It's a Java applet that does 4-dimensional animation of slices of polytopes (whatever that is!). Luckily, you don't have to grok the math to enjoy the show: fiddle with the controls, and watch the pretty shapes.

Here's a cross-eyed stereoscopic pair sample: