16 January 2007

Brouwer's shopping mall diorama theorem

This is a math post, but it also involves some audience participation. There's a crafts project. It may also require some driving. Ready?

Pick any closed, contiguous region of the universe—like, say, the nearest mall. Draw a map of it. Or you can make a diorama, if you're just that fond of the mall, or of making dioramas.

Go ahead. It doesn't have to be to scale.

While you're working, I'll say something profoundly obvious. The whole idea of a map, of course, is that every place in that part of the real world corresponds to exactly one spot on the map.

Done? Good. Now take the map (or model) and put it inside the closed region of space that it represents. That is, go to the mall. Brouwer's fixed-point theorem says that the map now has a fixed point: there's a point on the map that is actually at the very location that it represents.

This works no matter how large or small your map is. If your map is the size of the entire food court, and you take it there and spread it out on the floor, there will be a spot somewhere in the food court that exactly lines up with the corresponding spot on the map. Shift the map a little bit, and that spot won't line up anymore—but some other spot will. Always. You can turn the map around to face the wrong way. You can hold your 3-D model upside down. It doesn't matter. In fact, this works even if your map is not drawn to scale or if things are totally the wrong shape. There are only two requirements regarding accuracy. First, your map can't leave anything out. So if you forgot to draw the Banana Republic, you have to mentally squeeze it in between Orange Julius and The Icing where it belongs. Second, your map must be continuous. That is, any path someone might take from one point to another in the mall has to make a continuous path (without any “jumps”) on your map as well.

In the language of topology, any continuous function that maps a closed ball in Rn into itself has a fixed point. I have no idea why this works. Amazing.

It may have occurred to you that there already are nice, large maps conveniently located throughout the mall. Brouwer's theorem applies to those maps, too. In fact, in honor of Brouwer, the fixed points of these maps are always clearly marked, usually with a red dot or an arrow. Next time you're in a mall, take a look.

No comments: