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.