When you think of fractals, you might think of Grateful Dead posters and T-shirts, all pulsating with rainbow colors and swirling similarity. Fractals, first named by mathematician Benoit Mandelbrot in 1975, are special mathematical sets of numbers that display similarity through the full range of scale — i.e., they look the same no matter how big or how small they are. Another characteristic of fractals is that they exhibit great complexity driven by simplicity — some of the most complicated and beautiful fractals can be created with an equation populated with just a handful of terms. Scroll down towards the bottom to see how this works.
The Mandelbrot Set
One of the things that attracted me to fractals is their ubiquity in nature. The laws that govern the creation of fractals seem to be found throughout the natural world. Pineapples grow according to fractal laws and ice crystals form in fractal shapes, the same ones that show up in river deltas and the veins of your body. It's often been said that Mother Nature is a hell of a good designer, and fractals can be thought of as the design principles she follows when putting things together. Fractals are hyper-efficient and allow plants to maximize their exposure to sunlight and cardiovascular systems to most efficiently transport oxygen to all parts of the body. Fractals are beautiful wherever they pop up, so there's plenty of examples to share.
A great example of how fractals can be constructed with just a few terms is my favorite fractal, the Mandelbrot Set. Named for its discoverer, the previously mentioned mathematician Benoit Mandelbrot, the Mandelbrot Set describes a fantastical shape that displays amazing self-similarity no matter what scale it is looked at and can be rendered with this simple equation:
zn+1 = zn2 + c
I won't get into the technicalities of the equation here (you can read this infographic I made about how to render the Mandelbrot Set if you want to dive into more specifics), but basically it means that you take a complex number, square it, and then add itself to the product, over and over again. Do it enough times, translate those numbers to colors and locations on a plane, and baby, you've got yourself a beautiful fractal!
Here's what I mean by fractals looking the same throughout the scale. This shows a zoom into a smaller region on the larger Mandelbrot Set. Notice anything similar between where you start and where you end?
Illustration: Shea Gunther
For an extreme example of how this works, check out this video showing a super deep zoom into the Mandelbrot Set.
Besides the Mandelbrot Set, there are scores of other types of fractals. Here are a few of the more well-known fractals.
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