The theorem that provides continuity for space, time, and airport maps

Ever seen the little 'You Are Here' dot on a map of the airport, or the mall, or any other structure? That's a demonstration of the Brouwer Fixed-Point Theorem. Violently shaken martini shakers and crumpled pieces of paper are also demonstrations. It's all over the place. Take a look at one of the weirdest, and most powerful, proofs out there.

Luitzen Egbertus Jan Brouwer was born in Holland in 1881. He studied math at the University of Amsterdam, and in 1912 came up with a funny little thing he called the Brouwer Fixed-Point Theorem. A practical demonstration of it involves two pieces of paper, both of the same size. Put one directly on top of the other and stare at it for a second. Then pick up the top paper. Crumple it up, however much you like, and throw it contemptuously down on top of the second piece again. (Note: It doesn't have to be contemptuous, but how many times do you get a chance to be contemptuous in the name of science? Why waste an opportunity?) According to the Brouwer Fixed-Point Theorem, there is at least one point on the crumpled paper that is exactly over the spot where it was before you crumpled it up. Even if you mangle those two spaces, one point stays fixed.

The theorem works as a function of time, as well as of space. Grab a container full of liquid — a closed one that no liquid can spill out of — and shake it, stir it, any way you want it. Go to town on it. When you're done, and it's all settled, there will still be one point exactly where it was before any of that started. There are two conditions that need to be met for this to work. You can't sneakily take pieces off the paper or droplets out of the container. You also can't open the container to let liquid splash around or scatter the paper. Other than that, you can go nuts. The shape of the object you crumple, squish, or stir doesn't matter. You can shake it as long as you want, as much as you want. Time and violence give you no power over the Brouwer Fixed-Point Theorem.

And it's a good thing they don't, because you count on the theroem every day. It's a kickstart for many new and arcane topological concepts, and one topological concept that has been around as long as memory serves and is used by everyone. Any map, whether it's a map at a mall kiosk, or on one of the walls of an airport, or printed with the highest technological precision possible, or drawn on a bar napkin by a drunk, guides you with one central concept: at least one point on the map lines up exactly with one point on the actual place where you are standing.

'You are Here,' the map says. 'Here' on the map lines up exactly with 'Here' in real life. As you can see, the bathrooms, on the map, are just left of the point marked 'Here.' Since the point where you are standing lines up with the point on the map, if you proceed to the left in real life, you will find that bathroom you need so desperately.

And this is the amazing thing about the theorem. Obviously, the bathrooms on the map do not line up with where they are in real life. The map is a great deal smaller than real life. That part of the representation is manipulated and mangled. But that doesn't matter. As long as you can line up, according to the Fixed-Point Theorem, the point where you are standing in real life and the point where you would be standing on the map, you can navigate your way around the mall, airport, bar, or city where you happen to be. Fixed point theorems, intuitively figured out by generations of people, have guided our thinking since before they were ever articulated as mathematical concepts. We're all mathematicians - some people, like Brouwer, just came up with a written language for us.

Image: CCC

Via Stanford, Mudd Math Fun Facts, Planet Math, San Jose State University, and Physics of Finance.