Friday, March 5, 2010

Greetings from the internet, plus why I am called Fractal David

Well, guess who started a blog. I think some of my thoughts have been bouncing around in my oversized head for a bit too long. This blog will be the outlet for them, hopefully for the general enlightenment of all who read it. Hopefully after partaking you will somehow feel less confused than before. (or perhaps more?) Anyway, I feel under a good deal of pressure to make my first post a good one, so I'll begin with a subject that is very dear to me:

Fractals are cool. I mean it--really, really cool. Objects in real life are limited in how complex they can be--once you look at them closely enough, they're just atoms, and not far beyond that is all theoretical physics and speculation. Fractals, on the other hand, are different: they're infinitely complex. No matter how far you zoom in on one, they never get any simpler; you can never get down to their smallest constituent because they go on forever.

Some of the simplest fractals are generated by simple processes. The Cantor set, for instance, starts with a line. Remove the middle third of this line. Repeat the process for the two thirds of the line you have left, giving you four shorter lines. Repeat this process on each line ad nauseum until all that's left is a fine dust of infinitely many infinitely short lines. The first few steps of this process look like this:

Pretty cool, and infinitely complex, but not terribly interesting. Other fractals of this kind include the Koch curve, the Peano curve, and Sierpinski's triangle, which I encourage (but don't force) you to look up.

But other processes can make much different, more interesting fractals. By choosing a complex number and repeatedly squaring the result and adding the original number, a fractal can be made out of the points that don't become infinitely large: the Mandelbrot set, which looks something like this:
Much more interesting than a collection of line fragments, in my opinion. The Mandelbrot set has a fascinating geography unlike anything in real life, showing various infinite patterns wherever you zoom in along its boundary. Coolest of all is that smaller versions of the whole set are hidden away in it--the tiny black dot on the uppermost 'branch' of the set is one. This video illustrates just how ridiculous the Mandelbrot set is better than my ramblings ever could. If you don't feel like sitting through the whole 10 minutes (it gets kind of repetitive), skip to the end, where after zooming in to a magnification that dwarfs the size of the universe, we finally get back to a minuscule version of the whole set. I honestly cheered the first time I saw that.

Anyway, this infinite complexity and self-copying make fractals amazing. I don't doubt I could spend my life trying to understand why such a simple process can generate something so far beyond human imagination as the Mandelbrot set. But it's always fascinating to try. Exploring the Mandelbrot set is a hobby that never gets old.

So hopefully you're now at least a fraction as excited about fractals as I am. In conclusion, hello. And until next time, goodbye.

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