4 Dimensions Unfolding
						Here's something you don't see every day . . . . It's a projection in 3-space of a hypercube rotating in 4-dimensional space. Another name for it is a tesseract. This cartoon was cooked up using Turbo Pascal 3 back in the Dark Ages of Computing when Turbo Pascal was pretty neat. I bought Turbo Pascal Version 1.0 as soon as it came out, on a tip from Richard Chycoski who now works in Silicon Valley. I got Borland's telephone number from Byte.  Thinking I was dealing with a big-deal 24/7 operation, I rang them at 8 PM. The person who answered was one of the programmers working overtime. He was a little surprised but took my order and shipped me the product right away. It was amazing to have a full compiler that worked so fast and could do so much.
	Rotating a tesseract? Easy, if you extend the concept of a pivot matrix to 4 dimensions. You are seeing a central-axis projection. It is analogous to casting the shadow of a wire-frame cube onto a screen using a lamp located on an axis connecting the midpoints of two opposite faces. In 4 dimensions, the central projection gives you the classic "cube-in-a-cube" effect. I have chosen for my axis of rotation one that produces a turning-inside-out effect. For the viewer in 3-space, seeing a 4D object in rotation induces strange reactions. In the 60's when graphics-ready computers were finally available, the first thing programmers did was to use them to read in their huge collections of IBM cards with pinup pictures punched onto them. The second thing was to design tesseracts and project them on-screen in various states of rotation. Full animation was the next step. Here I have only reinvented the wheel, or actually the tesseract. So it's nothing original . . . . Visualizing 4D objects is not physically possible, but a human can navigate 4-space with a little practise. Read Shadows of Reality (Tony Robbin) and find out how. Animating a tesseract? Easy? Yes, but very tedious. The Pascal program displayed the frames continuously and could not write them to file. This GIF file was created from 45 snapshots.  I pressed Print Screen key over and over as the executable ran in a DOS session under Win98. Each time, the Clipboard data had to be copied into an image editor, cropped, and saved. The frames were then imported into ImageReady to make the final GIF.