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How Fractals Work

Before They Were Fractals
Katsushika Hokusai used the fractal concept of self-similarity in his painting "The Great Wave Off Kanagawa" the early 1800s.
Katsushika Hokusai used the fractal concept of self-similarity in his painting "The Great Wave Off Kanagawa" the early 1800s.

When most people think about fractals, they often think about the most famous one of them all, the Mandelbrot Set. Named after the mathematician Benoit Mandelbrot, it's become practically synonymous with the concept of fractals. But it's far from being the only fractal in town.

We mentioned the fern earlier, which represents one of nature's simple and limited fractals. Limited fractals don't go on indefinitely; they only display a few iterations of congruent shapes. Simple and limited fractals are also not exact in their self-similarity -- a fern's leaflets may not perfectly mimic the shape of the larger frond. The spiral of a seashell and the crystals of a snowflake are two other classic examples of this type of fractal found in the natural world. While not mathematically exact, they still have a fractal nature.

Early African and Navajo artists noticed the beauty in these recursive patterns and sought to emulate them in many aspects of their everyday lives, including art and town planning [source: Eglash, Bales]. As in nature, the number of recursive iterations of each pattern was limited by the scale of the material they were working with.

Leonardo da Vinci also saw this pattern in tree branches, as tree limbs grew and split into more branches [source: Da Vinci]. In 1820, Japanese artist Katsushika Hokusai created "The Great Wave Off Kanagawa," a colorful rendering of a large ocean wave where the top breaks off into smaller and smaller (self-similar) waves [source: NOVA].

Mathematicians eventually got in on the act as well. Gaston Julia devised the idea of using a feedback loop to produce a repeating pattern in the early 20th century. Georg Cantor experimented with properties of recursive and self-similar sets in the 1880s, and in 1904 Helge von Koch published the concept of an infinite curve, using approximately the same technique but with a continuous line. And of course, we've already mentioned Lewis Richardson exploring Koch's idea while trying to measure English coastlines.

These explorations into such complex mathematics were mostly theoretical, however. Lacking at the time was a machine capable of performing the grunt work of so many mathematical calculations in a reasonable amount of time to find out where these ideas really led. As the power of computers evolved, so too did the ability of mathematicians to test these theories.

In the next section, we'll look at the mathematics behind fractal geometry.