Alexander Horned Sphere

I thought, since for the next couple of days we’ll probably have a broader audience than we usually do, that I would do posts on some more elementary topics than usual. I’m going to try to explain an odd example from topology in elementary terms, without (hopefully) butchering the math too much.

Put a rubber band around a balloon, and then blow it up. By stretching the rubber band, you can take it off the balloon. Now imagine that you first tie the balloon in a complicated balloon animal shape. Can you still get the rubber band off? You can imagine that the balloon is tied so tightly that there isn’t enough room to squeeze the rubber band by, but this is a perfectly flat mathematical rubber band we’re talking about here; no spot is too tight to squeeze through. Given that, can you still get the rubber band off?

The mathematical answer is no. Alexander discovered a counterexample in 1924 now known as the Alexander horned sphere. You can twist the balloon into a strange fractal shape with infinitely many interlocking horns so that the rubber band cannot be pushed past all of the horns. (The practical answer is yes, since you can’t really twist a balloon into a fractal shape, and to interlock the horns I think you have to surreptitiously cut up the balloon and glue it back together when no one is looking.)