Fractals and the Mandelbrot set

[Alex K]

I taught some of my 12th graders about fractal dimension the other day as a kind of bonus topic, and that prompted me to look at what people are doing with fractals nowadays. They are kind of out of fashion after being totally en vogue in the 70s, but I still find them as fascinating as ever.

What I find fascinating about them is that these whole worlds of infinite complexity and detail arise from the simplest mathematical prescriptions one can imagine.

So for example, if you take a number in the complex plane, c, and then plug that into the formula

z=z^2+c

again and again, you can let the computer draw those points c with a computer that yield a finite result, and if you want you can color-code how quickly the numbers diverge. This is what you get zooming in, just from the formula z=z^2+c





Note that you encounter tiny copies of the same similar structures again and again at smaller and smaller orders of magnitude, but they *never* repeat exactly. If that doesn't make you think, I don't know what to tell you.

When B. Mandelbrot discovered this formula, he hacked it into the best computer with a plotter that he could find, and this is what he got:



I don't think he imagined what this structure would turn out to be.