Danskin's theorem

Imagine you have a piece of paper that's kind of wavy -- it goes up and down in some places, and back down again. Now imagine you draw a line across the paper, from left to right, that touches all the highest parts of the up-and-down waves. If you draw another line that's as close as possible to the first one, but touches all the lowest parts of the waves, you've just drawn what's called an "envelope" -- like an envelope around the up-and-down waves.

Now, let's say you have a bunch of circles. Maybe they're all different sizes and in different places on your paper. And imagine you draw a line around all the circles, touching each one at just one point. This is called an "external tangent" line -- it's like hugging all the circles from the outside.

You might think that these two lines -- the envelope of the wavy paper and the external tangent line around the circles -- don't have anything to do with each other. But in fact, a mathematician named Danzer figured out that they're connected in a very cool way. He proved something called "Danskin's theorem," which says that if you measure the distance between the two lines at any point on your paper, and plot those distances over and over again for different points, you'll end up with another one of those wavy lines! This time, though, the waves go up and down in a very specific way -- they're actually the "envelope" of all the circles you started with.

So, to summarize: Danskin's theorem says that if you draw an envelope of a wavy line on your paper and an external tangent line around a bunch of circles, you'll get a new wavy line that's the envelope of those circles. And you can figure this out by measuring the distances between the two lines at different points and plotting them -- you'll get the new wavy line that way. Cool, huh?