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.

This kind of result is called "sensitivity to initial conditions", though popular magazines will usually call it by the name of "The Butterfly Effect". It is not the end of the problem, though.

As you look closer and closer at the system, you will notice that the system isn't smooth. It is as seemingly complex on a fine scale as on a large scale. This means that there is no meaningful "gradient" to the function. Not just at one or two points, as happens with discontinuous functions in "normal" maths, but at ALL points in the system. The technical term for this is a Non-Differentiable Function.

But what causes all of this strange behaviour? Surely, there must be a simple explanation, some quirk in the maths. Well, the answer to that is yes, there is. The quirk is called a Strange Attractor.

Simply put, Strange Attractors are anomalous regions in the chaos that seem to pull the chaotic system in their direction. In some ways, this can be likened to gravity, but the effect doesn't always fall off with distance and is not always easily predictable.

There are Strange Attractors in normal mathematics too; we just don't really see them that way, as they are much more benign. Picture the numbers from zero to infinity. Pick one, any one, it can be a fraction or an irrational number, if you like. Now square it, and keep squaring the result. Numbers above 0 and less than 1 will fall towards 0. Numbers below infinity and above 1 will fall towards infinity.

Zero and infinity, then, are attractors for the square function. They will pull numbers towards them, and a stable boundary (1) exists where the pull is equal in both directions.

In chaotic systems, Strange Attractors might appear anywhere. Systems can orbit them, seemingly stable, only to be flung away the next moment. Other times, a chaotic system may stray too close and get pulled into these mathematical black holes. The boundaries of their influence are ill-defined and can seem to intrude into areas controlled by other Strange Attractors.

How does this lead us to fractals—the pictures people see on computers and t-shirts? Well, let's take this one step at a time. The Mandelbrot Set—the best-known fractal of all—is based on the chaotic system Z' = Z^2 + C, where Z and C are complex numbers, and Z' is the value of Z for the next time the calculation is made. The usual way to plot this is to plot the real component of Z as the X coordinate and the imaginary component as the Y coordinate. Then just sit back and watch how the point moves through the system.

There are lots and lots of Strange Attractors hiding in this seemingly trivial equation, and we shall see this in a moment.

If we plot how Z changes, every time we cycle round the equation, we will see one of a number of possibilities:

- It may fall into a Strange Attractor and stop.
- It may orbit one or more Strange Attractors indefinitely, never breaking free of them.
- It may do all kinds of loops and spins around a whole load of Strange Attractors, before escaping to infinity (which is just a regular attractor, not really a Strange Attractor).

When displaying a "classical" Mandelbrot picture, what you are doing is plotting how long the system takes to escape—if it does. Points that do NOT escape (the central plateau of the Mandelbrot Set, for example) are given some otherwise unused value to mark that they did not escape.

Points far from the center tend to escape very easily and quickly, which is why you get fairly boring, bland rings on the outside. Even there, though, you'll notice that as you approach the center, the pull from the Strange Attractors is enough to slow down the escape.

Once we reach the "interesting" part of the set, things go crazy. The function gets pulled in all kinds of directions, with absolutely zero regard for how things were even an infinitesimal distance either side. This is where you get all of the pinwheels, spirals and other strange effects. As we approach the edge of the plateau itself, the chaos worsens as the number and density of Strange Attractors grows.

Eventually, on the plateau itself, nothing can escape. Not even light. Oops, wrong topic. Seriously, within the region of the plateau, ALL starting points will be captured by one or more Strange Attractors. Nothing escapes from this region. If you go back to plotting Z'=Z^2+C, and throw something into that region, you will see that it never comes back out. It really is a mathematical Black Hole, but not due to some gravitational singularity, but rather to a large enough number of powerful enough attractors that nothing escapes.

One of the consequences of having these Strange Attractors pull the system in different directions is that the motion of the system is disjoint. Not just in a few cases, but essentially everywhere. This means that the fractal you produce is also non-differentiable—there is no gradient you can calculate at any point.

But there is another, more curious aspect. Something called "Self Similarity". Self Similarity means that if you zoom into a part of the whole, you will see something that resembles the whole. It won't be exactly the same, but it will be close.

Self Similarity is one thing that makes Fractals so curious. If the system is being thrown around by all these Strange Attractors - which can be anywhere - then how can a small fragment look like the whole? That would imply that Strange Attractors can't be randomly distributed, their distribution and the interaction between them must ALSO be self-similar. They drive the shape, so what the shape does, they must do also.

Of course, things get worse. At no point in a fractal does the pattern repeat itself, so although the Strange Attractors must approximate their positions on different scales, they cannot have exactly the same layout. Nothing repeats in a fractal. Ever.

However, remember that bit about sensitivity to initial conditions? The layout must not only change, it must change in a manner that produces similar results. Normally, with the system being so sensitive and all, any change at all would produce something totally different. Thus, not only must there be change, there must be change that largely (but not entirely) eliminates this sensitivity aspect.

That should be enough to curdle most people's brains at this time. Next story will be on self-similarity in the marketplace and why the free market is really a chaotic system. (This was first shown by Benoit Mandelbrot and now forms a rather obscure but fascinating branch of economic theory.)