Alexander Horned Sphere
January 12th, 2008 by WaltI 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.)
January 12th, 2008 at 9:39 pm
The proof that you have to surreptitiously cut up the balloon is almost more interesting than the proof that the horned sphere exists in the first place. Make the rubber band into a perfectly thin one (as well as flat) and look at the sheet it traces out as you slide it off the original, round balloon. Now, as you twist up the balloon (without cutting it) you also twist up that sheet. At the end you have the path telling you how to remove the band from the balloon. Thus in order to make it so you have to cut the band, you have the cut the balloon.
January 14th, 2008 at 6:04 am
Does this apply to condoms?
January 15th, 2008 at 8:57 pm
Setting up Alexander’s Horned Sphere in the context of rubber bands and balloons and then revealing that the analogy or image or whatever really makes no sense whatsoever…
What’s it like? It’s like something…
Oh yeah. Since Alexander’s sphere was inspired by Antoine’s necklace, and Antoine was a blind WWI veteran, you might say the excellent pedagogical device with the rubber bands is sort of like offering to help a blind veteran across the street, and then pushing him under a bus.
January 15th, 2008 at 9:22 pm
Condom usage is an applied science, Robert. Try it, and let us know how it works out.