Fractals

Why on earth should you care about fractals today? You know, the type of image that seems to go on forever in a recursive pattern, constantly forming new branches on itself in a hypnotizing manner:

It turns out that fractals are pretty neat, and they’re all over the place in nature. Remember how excited I was to share spontaneous synchronization with you? From fireflies to subatomic particles to astronomical systems, all of these systems come about because of the same underlying mechanism, and nature is brutal in its efficiency: once it has an idea, it runs with it everywhere.

Zoom in on any coastline, and you’ll see something remarkable. Really—any coastline anywhere will do. As you get closer to a detail on a particular pattern, you start to notice that there’s another, much smaller pattern embedded in that other pattern. You zoom in to see it, only to notice that there is another similar pattern embedded in that pattern, and on and on it goes:

Oh yeah: it doesn’t matter how far down you go or how close you get.

Now, GIFs are not fractals. They literally can’t be! That’s because when you pause the moving image, you can see that the resolution just isn’t there, and if you zoom in, you’ll only see pixels at a higher resolution (until the GIF causes the “smaller” image to emerge).

Even still, we can start to get a good idea of what they represent by looking at this classic GIF, based on the “Yo, Dawg” meme:

This GIF really does a great job of showing a recursive loop, and it draws upon itself—a feature called self-similarity—in order to create this repetition. The tiny head is embedded in his hair, and there’s another tiny head embedded in that little head.

Of course, the GIF doesn’t have turtles all the way down. Images and GIFs I can share with you here will only be of limited use from now on, since we’re talking about actual fractals. You know those famous nesting dolls, where one smaller doll is inside of the bigger doll, and then there’s an even smaller doll inside of that smaller one? With fractals, this repetition happens forever.

How, then, are fractals so precise? How do they go all the way down?

It all starts with a few simple rules, and it finishes with iteration. The rules are usually pretty straightforward, like starting with a basic geometric shape like a triangle, and then you transform it. That really just means that you perform some kind of action that changes the shape, like maybe adding a second small triangle off to the right side or something.

Then, the rule is applied again, once the cycle is complete. You apply the same rule to the new shape, and a tad more complexity appears. The next iteration makes it even more complex, and the layers just keep adding as you iterate.

While you’re enjoying your morning coffee, think about how fractals are out there, everywhere. Our blood vessels use the fractal framework to maximize their distribution of oxygen throughout our body, and root systems use fractals to soak up as much of the water and nutrients in the soil as possible.

Last thing: I kind of lied to you earlier when I said that fractals in nature are turtles all the way down. That’s not entirely true, since the laws of physics don’t allow for things smaller than a certain point, as Moore’s Law makes plain. Quantum mechanics means that there is an actual bottom with no turtles.

Still, my explanation should at least get our brains cooking a little bit on fractals today. For readers among us who are mathematically inclined, is there more context you can add today? Folks, don’t miss the comments section here—it’s the best thing about this place!

What fractals do you notice that we haven’t discussed so far?