Monday, April 11, 2005

The coolest theorem

Today we learned The Brouwer fixed-point theorem for the disc:
If f:B2 --> B2 is continuous, then there exists a point x in B2 such that f(x) = x.
This says that if you have a circle (the inside too, not just the outside edge), and you take the points on that circle and move them around in a continuous way, you will always have a point that doesn't move.

More impressively, take two identical sheets of paper and lay them on top of each other. Crumple one up and set it on top of the other one. There is some point that is directly above the point where it started. You can flip the crumpled paper over, move it around -- there will always be a point that doesn't move.

"But," you say, "I thought you said it was true for a disc! A piece of paper is not a disc." Silly, you're thinking that circles and rectangles are different! But they're topologically equivalent, just like you and a pencil eraser (unless you have pierced ears).

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