Rotating Four-Dimensional Shapes
A few years ago (and by the time you read this, it might be many years ago; I don't remember the year) an article appeared in Scientific American describing writing a program that would display 4-dimensional objects rotating in 4-space by showing their two-dimensional shadows in two stereoscopic views so you could get a 3-dimensional picture (Ow. The explanation sounds more complicated than the actual procedure!) I also remembered having seen a program ages back that showed a 4-cube (tesseract) in just such a way. So I did it. I remember it took no little doing to compute the co-ordinates of the vertices of a simplex (the 4-D analogue to the tetrahedron), though of course the hypercube was easy.
Note: Since writing this page, I have discovered a similar page and applet on the net (by one Michael Gibbs), that are roughly six to ten times cooler than this one you're looking at. It does many of the same things, and a lot mine doesn't do, and it does it with many more polytopes. I intend to add another polytope or two to this page, so there should be more to play with here, but don't miss out on seeing Michael's.
The controls are a little less than intuitive, sorry about that. You have four buttons for rotation, one for each plane it can rotate in. The 1-axis (I think) runs horizontally, the 2-axis vertically, the 3-axis toward you, and the 4-axis... well, perpendicular to the others. So the rot12 button will rotate the shape one increment (I think 5 degrees) in the plane of the screen (which is defined by the 1-axis and the 2-axis). If the run checkbox is selected, instead of moving the shape one increment, it increments its rate of rotation by one increment. So it will start spinning and keep going. If you hit the same button again, it will go faster, and so on. If the neg checkbox is selected, the buttons do the same thing but opposite. So they rotate the shape counterclockwise (instead of clockwise) or slow the rate of rotation and eventually set it going the other way, and so forth. You can use the stop button to stop any rotations, and the pull-down menu to select which shape you want (which also stops rotations and returns it to its initial configuration, so you may want to select the current one again to do that). The shapes available are the hypercube (4-dimensional cube) and the simplex (5 equidistant points in 4-space, analogous to an equilateral triangle in 2-space or a regular tetrahedron in 3-space).
Give it a shot. Remember to do the funky stereo-view thing with your eyes, if you can. If not, hey, you just have twice as much to study.
Let me know what you think of this. If you want to see the code... well, it's not really ready for public consumption, but drop me a line and we can probably arrange it.