Maybe you didn’t notice, but the poster hanging up around campus for Dorian Small‘s recent concert in Norman was highly mathematical. It was the Klein Bottle Fish:
Dorian Small is unorientable.
What is a Klein Bottle? It’s a non-orientable surface; which means there is no uniform way to assign directions on the surface. In particular, that means the Klein Bottle has no inside! But that doesn’t stop people from selling Klein Bottle bottles and Klein steins!
Another more easily imagined example of a unorientable surface is a Mobius strip:
The Mobius Strip
If you were to paint a Mobius strip by starting somewhere, merrily painting along until you’ve done one side, you’d discover that you’ve actually painted the whole thing! The same is true for a Klein Bottle. These videos give a very nice explanation:
P.S. For a neat trick, make a paper Mobius strip and cut it down the center all the way around. But before you actually do it, double points if you can figure out what will happen!
