Monday, August 17, 2009

Klein Bottles and Kool Stuff

According to wikipedia, in mathematics, the Klein bottle is a certain non-orientable surface, i.e., a surface (a two-dimensional manifold) with no distinct "inner" and "outer" sides. Other related non-orientable objects include the Möbius strip and the real projective plane. Whereas a Möbius strip is a two-dimensional surface with boundary, a Klein bottle has no boundary. (For comparison, a sphere is an orientable surface with no boundary.)

What that means, for those of you that don’t speak math, is something rather mind boggling. It’s a 3 dimensional object that has no dimensions.

Wait, let’s back up and see if we can make this make sense. Most of us played with Mobius strips when we were kids. For the benefit of those that didn’t, a Mobius strip is a long strip of paper that is given a half-twist then taped together. It makes a sort of slightly twisted figure eight. You place the strip on the table, and start drawing a line down the center of the inside, moving the paper along as you go. After a while, without raising your pencil, the line you are drawing gets back to where it started in one continuous loop. And when you look at the strip you see that you’ve drawn on both the inside and the outside, all without breaking contact with the paper. Pretty neat trick, that.

Well a Klein bottle works the same way except that you give a sphere a half twist and after you do that, no matter where you put the liquid, and you CAN fill it up with liquid, all the volume is on the outside of the sphere. There’s no inside any longer. Figure that one out. After all, it’s still a three-dimensional object. It still had height, depth and width. It just doesn’t have any volume any longer.

There are lots of things like this Klein bottle in math. Strange abnormalities that make fun of the solid world we live in and most of us more mundane types take comfort in the fact that they’re all fantasies and daydreams that scientists just theorize about. But the Klein bottle is real and you can buy mugs and bottles right now. Not that they’re easy to clean. You’d think that a mug which keeps all it’s liquid on the outside would be a snap to wash up after you’re finished with it, but unfortunately gravity exists so in order to keep all that liquid from hitting the floor the outside where you pour the beer is inside, umm under… well maybe behind… there’s glass in the way! The liquid winds up between two surfaces and you can’t get to them all that easily.

Sounds confusing? Maybe, but think just how freaked out your dinner guests will be the next time you serve them their favorite libation in a Klein mug and tell them to be careful, all the liquid’s on the outside!

For more information, or to order your very own Klein bottle, just go to

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