## Friday, May 07, 2004

### Hypershapes

Here's a new saying to try:
"It's better than a badly angled million dimensional hypercube in the eye."
I'll bet you never heard that particularly nerdly variation of the "sharp stick in the eye" saying before.

Hypercubes are nasty things. Consider a unit-square. Each of the sides is one half unit from the center. However, the corners are farther from the center than the sides: about 1.4 times as far. The corners of a cube are even farther relative to the sides: about 1.7 times as far. For a four dimensional hypercube the ratio is 2.0.

For a million dimensional hypercube, the ratio is 1000. What this means is that if you take a 3 dimensional projection of the hypercube tilted at just the wrong set of angles, you can get one really nasty, spikey shape with some points coming 1000 times farther out from the center than others. Let's just say you wouldn't want to sit on one by mistake. If this blog supported pictures, I'd post the shapes here for lower dimensional hypercubes, so you could see what I mean, but I can't, so I won't.

One last thing about hypercubes. The number of vertices is 2 N where N is the number of dimensions. For a 1000 dimensional hypercube, there are far more vertices than subatomic particles in the universe.

Hyperspheres are as sexy as hypercubes are nasty. They are wonderfully smooth everywhere. Just as a sphere is more smooth and sexy than a circle, a 4 dimensional hypersphere is even sexier than either of those. An infinite dimensional hypersphere is infinitely sexy. You just want to touch one. It makes me drool just thinking about it. Well, maybe not, but you get the idea.

I know what you're thinking now that you've read the above, admittedly rather off-the-wall, paragraphs. Either Bret has lost his hyper-marbles completely, or has just jumped off the nerdly deep end. Have no fear, there's method to my madness. I'm going to write, over the next couple of years, a series of posts describing everything from Marxism to Bushisms as interactions of vectors in high dimensional hyperspaces. I'll try to introduce the concepts in hyper-byte sized chunks. Bear with me, as it will take a while to get to the meat. The meat is not necessarily useful, but it is (I think) interesting.