Leo came to me the other day and told me that he couldn’t put on his pants because they were “Klein pants”. Um…whaaaaa?
Turns out that he had managed to get one leg inserted into the other in such a way that it really did seems like his pants were working something was a Klein Bottle: You could put your legs down different leg holes at the top, and they would come out in the same place!
Now, one obvious way to get this effect is to simply put your legs into the bottoms instead of the top, and then they’ll both come out the single waist. But the tangle that Leo had managed to get his pants into was actually more (unintentionally) sophisticated than that. It really did seem for a moment like there was something impossible about what he’d managed to do with them.
I thought about taking a picture of it, but it’s sort of hard to see; like trying to see what’s special about a Klein bottle if it wasn’t clear, so instead I’ll simply explain that the way you get this effect: Simply stick one leg down the other leg from the inside. The effect is that if you start by putting both legs into different holes, as normal, they end up coming out the same hole at the bottom! This is initially very weird, but it’s obvious what’s going on once you look at it for a moment…well, so is a Klein Bottle, I guess.
[Edit: A reader points out that all topology is “soft” by definition, in that anything that it’s specified as a topological assertion, in, by definition, flexible! This is why you get to do things like turn spheres inside out, morph mugs, and such-like transformations, as long as you retain their defining properties.]