Aatish Bhatia in Empirical Zeal:
…as I watched this miniature world self-assemble on my windshield like an alien landscape, I wondered about the physics behind these patterns. I learned later that these patterns of ice are related to a rich and very active current area of research in math and physics known as universality. The key mathematical principles that belie these intricate patterns lead us to some unexpected places, such as coffee rings, growth patterns in bacterial colonies, and the wake of a flame as it burns through cigarette paper.
Let’s start with a simple example. Imagine a game similar to Tetris, but where you only have one kind of block – a 1 x 1 square. These identical blocks fall at random, like raindrops. Here’s a question for you. What pattern of blocks would you expect to see building up at the bottom of the screen?
You might guess that since the blocks are falling randomly, you should end up with a smooth, uniform pile of blocks, like the piles of sand that collect on a beach. But this isn’t what happens. Instead, in our make-believe Tetris world, you end up with a rough, jagged skyline, where tall towers sit next to deep gaps. A tall stack of blocks is just as likely to sit next to a short stack as it is to sit next to another tall stack.
This doesn’t look much like what I saw on my windshield. For one thing, there aren’t any gaps or holes. But we’ll get to that later.
This Tetris world is an example of what’s known as a Poisson process, and I’ve written about these processes before. The main point is that randomness doesn’t mean uniformity. Instead, randomness is typically clumpy, just like the jagged skyline of Tetris blocks that you see above, or like the clusters of buzzbombs dropped over London in World War II.