Fractal Friday: The Mandelbrot Set

To kick off Fractal Friday, it’s the classic fractal we all know and love (or are sick of seeing on desktop backgrounds) – the Mandelbrot set!

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The basic Mandelbrot, zoomed in. Adorable.

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That’s more like it. 

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This, on the other hand, is getting ridiculous

But where does it come from?

A quick summary for those of you who are less keen on the mathematical side: a fractal is a shape with self-similarity. As you zoom in, everything keeps looking basically the same. You can see this with trees – a trunk splits into branches, branches split into twigs, twigs split into smaller twigs, so no matter how close you look the pattern stays the same.

Images of the Mandelbrot set fall into this category (see link to slightly mind-melting video at the end for a demonstration). What the image actually represents is a particular type of complex numbers. Complex numbers are any numbers that have a real part (an ordinary number, like 2, 0.67855, π, and so on) and an imaginary part (an ordinary number multiplied by the square root of -1, i). If we think of a general complex number c, square it, and add it to itself, then keep doing that forever (iterate it), two things can happen. One, the size of the complex number will grow rapidly and it’ll become infinite; two, it will stay finite. The parts of the image coloured black represent this second case, the complex numbers that won’t become infinitely large no matter how many times you square them and add them to themselves.

(Put more mathematically: z → z2 + c, starting from z = 0, does not diverge to infinity.)

What about the pretty colours surrounding it? Partly that just helps it look nice, but the colours also serve to represent how quickly each number ‘escapes’ – since no complex number with a real or imaginary part greater than 2 can be in the Mandelbrot set, often a computer program is used to check, after each iteration (i.e. every time the number is squared and added to itself), whether or not either of these values has exceeded 2. If it has, the point representing that number is excluded from the Mandelbrot set, and assigned a colour based on how fast it reached the ‘escape’ value.

Other fun things about the Mandelbrot set: if you look closely into the gap where the two larger ‘circles’ meet, you’ll see what’s known as the ‘sea horse valley’ (so called as the spiral patterns resemble sea horses). The ‘dip’ at the far right of the shape is known as the elephant valley, though the resemblance is a little more dubious.

Zooming into the Mandelbrot set. I accept no responsibility for any brains melted in the course of watching this video. 

For more about the Mandelbrot set, see this and this.

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