The Mandelbrot Set
By Robert Knetsch
Rob has had many years of teaching Physics and Math in the private school system and loves abstract thinking and the history of new math discoveries
In the past few decades, fractals have become a popular visual tool to observe the complexity of mathematics based on quite simple rules. Fractals are usually able to be observed when an iterative process is used. This means that the result of a function is generated as a function of inputting the previous value of the function. It also means using complex numbers, useful concepts to apply and observe interesting results, the Mandelbrot Set being the most famous of them.
This article describes how the Mandelbrot Set came about, which is the plotted image the set of numbers that converge when the previous input is continually squared. That is, when fx+1=fx approaches a constant value (or alternates between values), where fx=x2. What is a challenge in this case is that x is a complex number rather than just a real number (so usually mathematicians do not use x, but z instead.) The value for x is ax + bi, where a and b are real numbers and i2 is -1 (called an imaginary number). An example of a complex number would be 0.5+4i.
As an example, suppose we took the complex number c1 = 0.1 + 0.5i. Squaring this results in the number c2=-0.24+0.1i. Once again, squaring this will produce c3= 0.0476 – 0.048i. Eventually, with subsequent iterations, the output of the iterative function (magnitude of c) will approach 0. In other words, the function converges on 0. Other numbers will not do this. Try, as an example, c=2+3i. You will find that this eventually “blows up” to a very large number (i.e., approaches infinity), not converging on a constant value. We consider any number to be a part of the Mandelbrot set if the number converges.
The Mandelbrot Set is the set of all (complex) numbers that converge on fx+1=fx2+c, where c is the complex number you are interested in and f0 = 0. It is a deceptively simple rule that results in a highly complex (literally!) graph of the set which, if you plotted the complex numbers that are part of the set as “black” and those that are not as “white,” you would get the graph as shown (in the complex plane).
With this simple rule, the resulting graph has too many properties to list in this article; some of them comprise the fact that it is infinitely “zoomable”, and certain patterns are constantly repeated. Another counterintuitive property is that it has infinite perimeter, but a finite area!
For teachers who are interested in teaching fractals, the relatively simple task of teaching complex numbers, combined with a relatively straightforward computer program can be a rewarding and fascinating area of exploration for high school math and computer science students.