Sexy Jihad

Kuro5hin has written the first of what will be a whole line of articles on fractals for beginners. An excerpt:

In order to cover fractals in any meaningful way, I am going to first examine what lies behind them. The first, and most important, of these elements is “Chaos Theory”, a branch of mathematics that is rather better known than it is understood.

Chaos Theory covers a class of mathematical systems that don’t follow the usual rules. (Anarchistic equations!) Normally, mathematical functions are relatively well-behaved. You can find the gradient at a given point, for example. If you change the values you start with (your initial conditions) by just a little, you will change the values you end up with by just a little and in a predictable way.

Chaotic systems, on the other hand, throw the rules out the window, without opening the window first. For small enough initial conditions, they behave themselves and generally go to some steady state value. As you increase the values, though, something happens. The system will start to oscillate with period 2. Keep increasing the value, and it stays at that frequency, until you cross a specific threshold. Then the frequency doubles. It keeps on doing this, with the intervals getting shorter each time.

Translated into English, this means that if you take some function and then use the results as the new inputs to the same function, it will first bounce between two values. Then, as you increase the initial conditions, it will bounce between four possible values. Then eight, sixteen, thirty-two and so on. It will always be a power of two and it will always be in that order.

At this point, a careful observer—call them Mitchell Feigenbaum for the sake of argument—may notice that the ratio between the thresholds is fixed for that function. In other words, if your start point is P0, you first double frequency at P1 you double again at P2, and double a third time at P3, then (P1-P0)/(P2-P1) = (P2-P1)/(P3-P2), and so on for all of the times the frequency doubles.

A really good observer will then calculate this constant. It will be about 4.669211660910299067185320382047. Later, you might try different chaotic functions. You will get exactly the same behaviour and exactly the same ratio. The Feigenbaum’s universal constant applies to ALL chaotic systems, in that interval where oscillation occurs. There are no exceptions and there are even fewer useful explanations.

OK, so what happens when we go beyond oscillation? Then we enter the realm of pure chaos. The system will be totally deterministic (it is pure mathematics, after all, and there are no random elements to it) but it is completely unpredictable. It will never repeat, it will never settle down, it will never do anything you might expect. If the initial conditions change at all, no matter how slightly, the chaos completely changes. It is almost as if you are looking at two completely different systems, not the same system with a very minor tweak.

Stuff like this shows that math really is fun. I wish I had more stuff like this in high school, instead of totally useless classes like gym and shop class…..

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