All of us, at one time or another, have wondered about socks that go missing in the laundry. A pair of socks go in, but often, only one of the pair comes back out again.
Where did the missing sock go? Dieter Kleinfelter, a topologist with Grampius University, has the answer: the sock is still there, along with all of the other missing socks.
"The answer is quite simple, at least for me", says Kleinfelter. "The sock becomes a Klein flask."
A Klein flask, also known as a Klein bottle, is, essentially, a bottle -- but it has an inside, but no outside.
"You see, when you place a sock in the laundry, it is in an aqueous solution, where it is rotated randomly along 'n' axes, and eventually, along 'n' dimensions. When aligned properly, the sock becomes a Klein flask."
"It has an internal volume, but no external surface. Yet, the sock is still there. If you have a sturdy washing machine, and wash your socks regularly, there could be dozens, or even hundreds, of socks inside of the washing machine."
Is there a way to retrieve the socks? No, according to Kleinfelter.
"Supposing you could find the missing socks, which is a mathematical impossility", says Kleinfelter, "you would have yet another insurmountable problem. You would have to turn them right-side out. If you could do such an impossible thing, you would then have a sock with an outside, but no inside."
"You would, in practical effect, have no room to put your foot. The sock would actually constitute nothing more than a theoretical geometrical line of indeterminate length."
This would seem to nonetheless result in an increase of mass of the washing machine because of its invisible contents, but Kleinfelter says, again, no.
"There is no increase in mass because, with no external surface, the socks have actually rotated -- and I use the term advisedly -- rotated into an alternate universe", Kleinfelter says.
"You'll never see those socks again."