Fractals: How Monsters Became Beautiful

Do you know what the circumference of the British isle is? Or the length of any coastline? The answer might surprise you and it will be our starting point in a quest for the origin of fractals.

_{source: Wikipedia}

Mathematics is weird sometimes. Did you know that "infinite" often isn't enough? I'll try to explain in a way I hope you can visualize. Take the circle; one of the most basic geometric shapes we know of. Now imagine starting with a regular shape with 4 straight lines: a square. With each addition of sides to that shape, first 5, then 6, 7, 100, 1000 and so on, the shape wil resemble a circle more and more. A regular shape with 1000000 sides is damn near a perfect circle already. But the perfect circle is a regular shape with infinite sides. Now imagine that you can draw from the center of the shape lines to all corners, all infinite of them; so you would need infinite lines to cover the whole shape. That should be enough for normal people, but the mathematician asks: what if I draw a much larger circle around the first one? Then the infinite lines needed to cover the smaller circumference suddenly isn't enough anymore to cover the larger one!

So, strangely enough, the answer to the question in the intro is either "infinite" or "it depends on the size of the stick you measure it with". And the smaller the ruler or stick, the longer a line you will measure. And if you can zoom in infinitely, you will always find smaller indentations in the coastline which you hadn't considered before. If I lost you here already, don't worry too much; watch the video at the end of this article and all will become clear!

Another exploration into the infinite were the so called "Mathematical Monsters". It started in the late 1800's when German mathematician Georg Cantor created the first monster, or actually the first fractal:

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Another great early monster is Koch's Snowflake, where you start with a regular triangle and then remove the middle third from each side and bridge the gap with two new sides of equal length. You can iterate this action forever; just zoom in on the smallest, latest addition, and repeat. So to the eye the circumference of Koch's Snowflake looks finite, but mathematically it is infinite.

_{source: Wikipedia}

This continuous iteration of actions is represented in mathematics by Fractal Equations, the most famous one being the Mandelbrot Set. This is a rather simple equation with one oddity: the "equals" sign works both ways. This means that after you input a number on one side, the output on the other side becomes the input on the first side. And this you iterate ad infinitum, with one side of the equation ever nearing, but never reaching zero, and the other side nearing but never reaching infinity.

Fractals are nothing more than all input and output from these equations put on a graph, and lighting them with a color and intensity representing the distance from previous plot-points and the speed at which they move towards zero or infinite. A true fractal is therefore a visually pleasing image that can be zoomed in on forever. This is also why the structures of these beautiful images could be examined only after the invention of fast computers; these iterations have to be done trillions of times to produce any order out of the chaos of numbers. Just look at some of the magic of the most famous fractal of them all:

Like I said in the beginning: there's so much more to explore about Mandelbrot and the Mathematical Monsters. There's for example the way that fractals are all around us in nature: the way branches fork off from the tree and twigs fork off from the branch and leaves fork off from the twigs... Fractals are the stuff of life; they represent a mathematical explanation of how simple principles and simple building blocks naturally evolve into complex bodies with complex interactions.

So, please, if you can spare the time to be a little amazed and if you haven't seen it yet, watch the story of Benoit Mandelbrot in this video:

This article was in a large part inspired by @soundwavesphoton, who regularly surprises us with the products of his "fractalized mind", go check out his blog.

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