An Introduction to Fractals

I recently wrote an article about fractals and realized that they would be a perfect topic for the blog. In terms of combining beauty and mathematics, it’s hard to beat fractals. There’s something for everyone, whether you want to stretch your math skills, learn about some of the most fascinating and least understood systems in our world, or just look at pretty, pretty pictures. (Why not all three?)

Fractals get their name from the Latin word for “broken” and are generally distinguished by being self-similar, meaning that the same patterns repeat at different scales. This means that as beautiful as a lot of fractal patterns are as static images, they’re even better if you can interact with them to zoom in on different parts and see the details. Here’s an image from a zoom-in sequence in the Mandelbrot set, one of the most famous fractal sets (fractals generated by a mathematical definition or function). The entire thing looks sort of like a bug, and you can see here how that shape repeats over and over.

800px-Mandel_zoom_08_satellite_antenna(WolfgangBeyer).jpg

Step 8 of a zoom sequence in the Mandelbrot set, magnified over a million times relative to the original image! (Image created by Wolfgang Beyer in Ultra Fractal 3, 2006. Licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license.)

In case you were too mesmerized by the image to read the caption, this is magnified more than a million times! I highly recommend you go through the whole zoom sequence to get a sense of the detail we’re talking about with this kind of fractal. What’s really amazing to me is that the Mandelbrot set is really just the graph of a set of numbers that meet a pretty simple definition. Other sets like the Julia set have different definitions that result in distinctive graphs, with their own repeating motifs and shapes.

In addition to being gorgeously repetitive and complex at the same time, self-similarity means that all this complexity happens on smaller and smaller scales, so the whole thing takes up about the same space as the simple original shape. As an example, I’ve illustrated the Koch snowflake below. You can draw this yourself by drawing a triangle and adding triangles one-third of the size pointing outward from each of the sides. Repeat until you’re tired of drawing teeny triangles. Check it out:

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First four iterations of a Koch snowflake.

Notice how beyond the orange triangles, every step fills in blank space, but doesn’t really add to the page size required for the drawing. If your paper is big enough to draw the first few iterations, it’ll accommodate all the following iterations you could possibly want to draw. (Try it and see how far you get!)

As I implied at the beginning, fractals are way more than a beautiful mathematical idea. Self-similar patterns show up a lot in nature—think of ferns or broccoli, for example. They’re also a huge part of systems that are chaotic, which are studied and described by chaos theory. This term gets thrown around a lot, but it doesn’t mean what a lot of people probably think it does based on our usual meaning for “chaos.”

Chaos theory is sometimes referred to in pop culture as the “butterfly effect,” referring to the idea that a butterfly flapping its wings can result in a typhoon halfway around the world. I don’t like this analogy that much, because (to me, at least) it makes it sound sort of hand-wavy (wing-flappy?) and chaotic in the way we usually use the word, with things not making much sense. That’s not really what chaotic systems are like at all.

An analogy I like better is one of those line jumble puzzles (I’m not sure if there’s a technical name for these) where you have to follow the correct line to join objects on either side. Two lines may start out right next to each other, but that’s no guarantee that they’ll end up anywhere close. All it takes is one little mistake and you’re way off course, leading you to a totally wrong answer.

Chaos Line Puzzle.png

Either I need my parents to give me “the talk” again, or I messed up.

This illustrates what chaos really means in these systems—there are paths laid out, but those paths depend on very precise initial conditions. If your starting conditions are off just a little bit, your final state will be totally different from what it would be with slightly different initial conditions. This means that the biggest problem with predicting chaotic systems is not that they don’t make sense, but that any uncertainty about your starting point makes it really hard to tell where you’re going to end up.

With that description, it’s not a surprise that weather falls in the category of chaotic systems, leaving meteorologists with predictions that start out reasonable in the very short term and tend to become more and more inaccurate with time. (Remember to cut them some slack for this!) Other chaotic fields of study include traffic patterns, finance, and group psychology (psychohistory, maybe?). The more we know how to recognize it, the more we see it.

The widespread presence of chaotic systems in nature and human interactions definitely poses challenges, but the good news is that research in chaos theory is increasing our understanding of the features involved in these systems, as well as finding new ways to apply those insights to problems in medicine, engineering, biology, and more. It’s a fascinating field with a lot to offer, and fractals and other features are an important part of understanding the dynamics of the world around us. It doesn’t hurt that they’re so satisfying to look at, either!

[Obviously, this is just the most general introduction to fractals and chaotic systems! Read more about the Mandelbrot set mathematics or, of course, check out the Wikipedia page. For information on fractals in general, Fractal Foundation has a ton of great resources including fractal software (best computer-related time-waster ever!).

Chaos theory is a huge subject to cover, but there’s a lot online, like this text or this brief discussion. If you’re looking for books, you can start with the classic Chaos by James Gleick. Or you can watch this water-drop experiment courtesy of Dr. Ian Malcolm!]

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