The following is the second in a two part series of articles based on an interview with Boston University professor Bob Devaney.
Educators often spend time considering what mathematics concepts are appropriate for high school students. According to Professor Devaney, the process of iteration should be taught in high school. The concept comes up in savings and interest problems and Newton's method. Chaos, fractals, talking about the Mandelbrot set: these are fun ways to introduce the iteration. Also, high school students often have seen a bit of complex arithmetic. The Mandelbrot set is a nice illustration of this concept.
Professor Devaney thinks that detailed discussion of dynamics is most appropriate for after school math clubs, or as a last topic at the end of the term when there is often some dead time anyway. However, he says: "Students spend about two or three weeks learning how to factor cubic polynomials; there is no reason why they shouldn't also spend a couple weeks learning about the Mandelbrot set."
As to the actual presentation of the Mandelbrot set appropriate for high school students, here is a general outline. It is perhaps even of interest to people who are not in high school but only know the Mandelbrot set from seeing pictures of it. The explanation should involve quite a bit of computer demonstration, but perhaps this description will give the idea.
Consider the real one dimensional quadratic map f(x)=x*x+1. Iterate zero under this map. The first few iterates are 1,2,5,26. It is fairly clear that under successive iterations, one gets an arbitrarily large number. Now consider the map g(x)=x*x+0. In this case, g(0)=0, so there is a fixed point at the origin. For h(x)=x*x-1, h(0)=-1, and h(-1)=0, so zero is a period two point.
The iterates of the origin under maps of the form of f, g, and h are generally hard to calculate. Putting these maps into a common form, add a parameter, and consider f(x,c)=x*x+c. Consider c=-1.1. If you try this in your head, you see that it is not so easy.
At this stage, one needs a program which graphs iterates of the origin under f. Using this program, experiment with different values of c to see what sort of behavior occurs. At c=-2, the picture becomes chaotic.
Now switch to the complex plane. In other words, consider the function f(z,c) defined above, except that now z and c can be complex numbers. For example, for c=i, I calculate the first two iterates of zero: f(0,i)=i and f(i,i)=-1+i.
The Mandelbrot set is defined to be all the c values for which iterates of the origin stay bounded under iteration. At this point, one should use a computer program to draw the Mandelbrot set. Then, use a program with two windows representing copies of the complex plane: in one window, the user picks a complex c value with a mouse. After the user picks c, the other window shows iterates of the origin. This program gives the user an idea of what it means for iterates to get arbitrarily large.
It is at this point possible to learn about the period of the periodic points in the Mandelbrot set based on the point's placement in one of the bulb shaped parts of the Mandelbrot set. For this and further discussion, I refer to Professor Devaney's books.
The above explanation is a fairly simple and fun way to teach the concepts of iteration, complex arithmetic, and boundedness. It answers the concern that all this popularization of fractals and the Mandelbrot set is just a matter of looking at pretty pictures without any knowledge of the math; although this is not a complete explanation of the mathematics involved, it does seem to give a partial understanding with little needed background.