Percolation.

I made a movie recently for my advisor. The movie is so pretty, that I thought I’d share it here: may I present to you randomly drawn dots, where two dots are the same color when they touch!

I’ll be a bit more explicit: a dot is drawn at a random location; if it does not overlap any previous dots, it gets a new color. Otherwise, the dot takes the color of the component it touches. Sometimes a new dot connects many components, and in this case, the new component takes on the color of the largest among the old components.

There’s a lot of neat questions to be asked about such a process: for instance, after drawing n dots, how many components should we expect to see? As you can see in the movie, when you draw only a few dots, most of those dots are isolated and have their own color; but after drawing a ridiculously large number of dots, they are all connected and the same color. And inbetween, something more interesting happens.

Here’s an example of “something more interesting” taken from a larger picture than the above movie: