Wednesday, January 16, 2008

Klein bottle

So last quarter in advanced multivariable calculus, we learned about a thing called the Klein Bottle.

This quarter, in cognitive psych, we were talking about "geons", or "geometric icons", which according to one theory, are basic building blocks by which we recognize and categorize objects. Geons include things like ... rectilinear surfaces, wedges, cones, torus sections ... =)

The mention of geons today made me remember klein bottles--cause we don't really have a geon that fits a klein bottle, per se. Which is to say, you have to concentrate a little to visualize or coneptualize a klein bottle, since it's not found in nature.

So let me attempt to describe a klein bottle for you. Let us start with a mobius strip, which is a sort of geon for a klein bottle. Have you ever made a mobius strip? It was discovered by a mathematician in the 19th century. They're really easy to make. You simply cut a long narrow strip of paper, give it a half twist (just twist one end 180 degrees), and then tape the ends together. I remember making one as a child, and finding it fascinating. It's fascinating for various reasons. It only has *one* surface. If you don't give the twist, and tape the ends together, the strip you get has two surfaces--an inside and an outside. But with the twist, it has only one surfact--no inside and no outside. If you run your finger along the single surface of the mobius strip, you will see that your finger will come back to the point you started having touched all along it's single surface. The other kewl thing about mobius strips is that if you cut one in half lengthwise, it just become a single mobius strip twice as long, instead of falling into two halves. Very kewl. I dearly hope I haven't altogether lost you yet.

So a mobius strip has one surface and one edge. A Klein bottle has one surface and no edges. Very kewl. You can theoretically compose a klein bottle which does not intersect itself by giving it a twist out in the fourth dimension. Of course we can't actually do this (yet). But it's fun to try to think about. You can buy a hand blown glass klein bottle on ebay. Very kewl. It has no inside and no outside--it's all one surface. In mathematics we call it "non-orientable"--which is to say you can't really define "up" or "down" from the surface.

You can even buy various versions of glass klein bottles at Then you could keep it on your mantel, and show it off to your friends, and prove to everyone what a nerd you are. =)

No comments: