Nice Pictures from Mathematics

We're probably safely over the worst of the Chaos mania. When Fractals first began to appear in the media, it was as though the world realized for the first time that mathematics produced pretty pictures. Ha! Remember circles? They're pretty, aren't they? Anyway. It is rather sad how really clever people have such a hard time explaining technical matters in non-technical language. I'm not criticizing the attempts, but some are most definitely more successful than others. Here's an example that isn't entirely satisfactory to my mind, but possibly quite okay to anyone who simply likes to feel that they have gotten familiar with higher mathematics.

A presentation by a British mathematician

A System The idea of a system is a very general one. It is simply something that can be in any one of several states. Only examples can help to understand this idea, because of its extreme generality. (1) A six-sided die on a flat surface. The different states are which number is showing, from one to six. (2) A light. The states are "on" or "off". (You can think of other states, such as "flickering", if you want. Throw them in into the collection of states. Every choice of the set of all possible states makes a different system.) (3) A driver on a highway. The states might be: (*) looking for a rest area. (*) Looking for an exit. (*) Looking for a gasoline exit. (*) Trying to pass. (*) Cruising. Most people, when driving, are in one of these states. Of course it is an oversimplification to insist that a driver only exists in these states, but mathematics is about simplification. (4) A disc drive. It can be reading, writing, or idling. The examples are numerous. A dynamical system is a system that goes from one state into another state in exactly the same order every time. It is as though each state had a forwarding address: "Now go to state x!" a little like a treasure hunt. Obviously, if (as in all the examples I gave) there are only a finite number of states, the dynamical system will not be very interesting. In contrast, if there are an infinite number of states, what the system is doing can be fascinating. Where do all the colors come in? Suppose all the states of our system can be represented by the points in a square. Then every point (state) knows where the system will go next. In other words, right after the system is in that state, it must always go to the same next state, just like the stops on a school bus route. The system is said to be deterministic, because given that the system is at State A, its entire future is known. Joining the dots, we get a trace of all the states that the system visits.

• For instance, all the states from which the system goes on to hit the upper edge of the square can be colored red,
• All the states from which the system goes on to hit the lower edge of the square can be colored blue,
• States starting at which the system ends up at the right edge can be colored green,
• States starting at which the system ends up at the left edge can be colored yellow, and
• States from where the system never hits the boundary but keeps going round and round can be colored grey, say.

This gives you an idea of what goes on in this coloring business. One of the best known fractal figures is a coloring of the plane determined by how many state changes are needed before the state leaves the circle of radius 2 around the point (0,0). (In that example, it is known that if the system leaves the 2-circle, it will not return.) See below. The Mandelbrot set shown here is a representation of something a little more sophisticated than merely a map of starting values from which some system heads off into the sunset. (It is actually a map of an entire family of systems, coloring the systems according to a particular characteristic.) Pretty Pictures from Simple Math Simple algebra can provide pretty pictures too, and has the benefit of being easy to understand. (The very complexity of fractals and dynamical systems attracts some people undoubtedly.) For example, all the points on the XY plane for which

lie on a circle. (A circle of radius 3, with a center at O.) But we can easily color every point depending on how much x^2 + y^2 is. If it is exactly 9, we could color it black, and if it is between 0and 1, we could color it blue, and so on, and we would get a pattern of concentric rings. Well, why not change the formula? What about a minus sign? x^2 - y^2 = 9 might give us some pretty pictures, too! Here's the result, below on the left. The points that are colored yellow correspond to points that satisfy the equation, the other colors represent points for which the left side is larger than 9 or smaller than 9, by different amounts. It is possible to view the behavior of this formula in 3 dimensions. The result is known as a saddle-surface, for obvious reasons. The flat plane indicates the zero level.