Math

You Can Zoom In On A Fractal Forever

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Even if you're don't know what a fractal is, you've definitely seen one. Fractals are everywhere, from the spirals of galaxies to the endlessly branching sprouts in a floret of broccoli. They're even in your own body. It was only a century ago that mathematicians started describing them, but once they did, it explained a whole lot about the world.

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A Shape In A Shape In A Shape

A fractal is, at its simplest, an infinite pattern. It doesn't take any special technology to create; in fact, you can make one yourself. Try drawing a triangle, then drawing an upside-down triangle within it. At each of the corners around that new triangle, draw an upside-down triangle. Keep doing that until you have triangles within triangles within triangles that are so small, you can't draw anymore. (That just so happens to be a Sierpinski sieve.) If you had infinite vision and an infinitely small pen, you could keep drawing triangles forever. That's the point of a fractal: it looks the same, or at least similar, no matter how much you zoom in.

Despite the fact that a grade schooler can draw one, fractals didn't even get a name until 1975, when mathematician Benoît Mandelbrot coined the term for these seemingly "fractured" shapes. In 1979, he began creating pictures of what came to be called "the most complex object in mathematics," the Mandelbrot set, by using a computer to map out iterative functions — that is, math formulas that always plug the last answer into the next formula. Iterative functions start out remarkably simple, but can quickly balloon into a kaleidoscope of complexity. That means that computers are key. Fractal-like shapes had been described as far back as 1904, but it took computer power to show their infinite characteristics.

Sierpinski colored tetrahedron.

Fractals All The Way Down

Although fractals took a while to be accepted as serious mathematics — in the 1970s, computers weren't fully accepted as serious mathematical tools — it's hard to overstate their importance. That's because most things in nature take on this infinitely complex shape. As Mandelbrot himself wrote, "Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line." You have fractals in your blood vessels, your nerves, and the alveoli in your lungs. Lightning, rivers, seashells, coastlines, hurricanes, and galaxies are all versions of fractals. Nature is frugal, and fractals are efficient: it's less work to use the same pattern over and over than to plan out an entire structure ahead of time.

But fractals aren't just a way to describe nature; they're also a key part of technology. Antennas, computer circuits, even cities work on fractal geometry. With fractals, a few simple rules can create infinite complexity, and that makes them infinitely useful.

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Benoît Mandelbrot, The Father of Fractals

The Mandelbrot Set

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