Julia Hayton
May 25, 2015 8:00 am

Let’s say you’re sitting in front of a blank piece of paper when you get the urge to draw a triangle. It happens! Sometimes a circle or a square or a dodecahedron simply won’t do!

You fill the paper with a triangle:

It’s a pretty good triangle (a nice, even equilateral), but couldn’t it be even MORE triangle-y? Couldn’t it have a few more lines and angles and pointy bits? Of course it could! And you aren’t going to be satisfied until this is the MOST TRIANGLE-Y TRIANGLE THAT EVER TRIANGLED.

So you draw another triangle inside the triangle, this time pointing the other direction:

But hold up, now there are even MORE point-up triangles to draw new triangles inside. You keep drawing point-down triangles in the point-up triangles:

And more, and more, and more!

Somewhere around your 100th triangle it dawns on you that this will never, ever end. No matter how many triangles you’ve already drawn, you will always be able to add more triangles within triangles.

You have created an unstoppable, infinitely-expanding monster. In other words, a fractal! These are the weird, awesome shapes that appear in nature that get you totally hooked on spotting them everywhere.

It’s a beautiful thing:

This is a classic fractal called the Sierpinski triangle, named after the Polish mathematician Waclaw Sierpinski, who studied its fractal properties. It’s one of the most famous fractals out there because it gives such a clear demonstration of what makes a fractal a fractal (that, and because it looks cool, which is the main reason non-mathematicians like myself find fractals so interesting).

But what does define a fractal? Simply put, a fractal is a pattern repeated at different scales. In the case above, we started with a triangle and a simple rule: draw new triangles inside every triangle, again and again. No matter how many new triangles you add or how small they get, you’re always going to be able to continue the pattern (but you would need either an infinitely large piece of paper or an infinitely fine-tipped pen to draw the whole thing; check your local Target).

That’s it, that’s a fractal! And from these humble beginnings grow shapes of infinite complexity. It’s downright poetic.

To see another classic fractal in action, let’s go back to the same triangle we started with and switch the rule. You’ve already crammed the triangle full of other triangles; that’s yesterday’s news, you’re so past that now. Instead, you wonder: what would happen if you added angles to the outside of the original, so that every flat edge sprouts a new pointy tip? 

To make a long story short, what happens is Koch’s snowflake:

Koch’s snowflake is an example of a fractal curve—unlike the triangle we started with, where the first triangle stays the same size and shape but gets packed with new triangles, this snowflake stays empty. All the magic happens on the edges.

Every time the snowflake’s edge sprouts a new set of angles, the length of the snowflake’s perimeter gets a little longer. And since you can add new angles forever, the snowflake’s perimeter is infinitely long, no matter what size of snowflake you started with. Your first triangle could be drawn on a post-it note, but that doesn’t matter. If you follow the rules of the fractal, you’ll still end up with a snowflake whose edge is infinitely long.

Unfortunately, you can’t get infinite shapes in real life (these pesky things called the “laws of reality” get in the way; they’re a real bummer). This means you can’t find “true” fractals just lying around, but you can find natural examples of patterns that repeat at different scales, just like a fractal—they just won’t be able to expand infinitely, because the universe isn’t infinitely small or infinitely large (as far as we know, at least).

But natural fractals still look really cool, and finding a fractal pattern in nature is like stumbling onto one of nature’s easter eggs: a sneaky secret that’s just waiting for someone observant to notice it.

A particularly awesome natural fractal is found in Romanesco broccoli, which would get my enthusiastic vote for Veggie Beauty Pageant champion: 

Romanesco broccoli buds don’t just clump together in roughly circular clusters like normal broccoli. Instead, their buds form tight, slanted spirals, and as they get bigger they grow new, tinier buds with the same spiral pattern. Like any fractal, Romanesco broccoli is self-similar at different scales: zoom in on a Romanesco broccoli bud, and you’ll find the same spiral shape as the whole head of broccoli has. Zoom in further, and you’ll find it again. Suddenly, instead of looking at one head of broccoli, you’re looking at thousands of tiny spirals, all building up to something bigger, all part of the same larger pattern.

Fractals mess with our ideas of beginnings and endings; when does a river with a thousand different branchings become a stream, become a creek, become a trickle? When you think of fractals this way, you can see fractals everywhere: in every pattern that goes from big to small, and small to big, the same cycle repeating into infinity.

[Images via Wikimedia Commons and here]

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