Exploration #1

Question: What is the Mandelbrot set?

Answer: The Mandelbrot set is a picture in the complex "c-plane" of the fate of the orbit of 0 under iteration of the function x2 + c. A c-value is in the Mandelbrot set if the orbit of 0 under iteration of x2 + c for the particular value of c does not tend to infinity. If the orbit of 0 tends to infinity, then that c-value is not in the Mandelbrot set.

Question: Hey man, slow down! Wanna give me that in English? (please...)

Answer: OK, here we go again. You give me a complex number, call it c. Then I will tell you whether c is in the Mandelbrot set. OK? Here's how I do it. I start with the function x2 + c. Then I take 0 and put it into the function. That is, I compute 02 + c which gives c. Then I do this again, using the output from the previous computation as the input for the next. That is, I now use c and compute c2 + c. Then I do it again: I get (c2 + c)2 + c. And so forth. I generate a list of complex numbers in this way. If these complex numbers (called the orbit of 0) get larger and larger (or further and further away from the origin), then your choice of c is NOT in the Mandelbrot set. But if this is not the case (the orbit stays "bounded"), then c IS in the Mandelbrot set. Are you happy now?

In each of the following five examples, observe the following information:

Example 1: c = 0 .

Example 2: c = -1 .

Example 3: c = i .

Example 4: c = - 2 .

Example 5: c = 2 i .

Question 1A: Is c = 3 in the Mandelbrot set? Answer

Question 1B: Is c = 1 in the Mandelbrot set? Answer

Question 1C: Is c = 1 + i in the Mandelbrot set? Answer

Thus we see that the following c-values are in the Mandelbrot set:

whereas the following c-values are not in the Mandelbrot set:

We may therefore begin to "paint" the picture of the Mandelbrot set by coloring complex numbers in the c-plane according to the following rule:

Thus we have the following picture so far:

Further Exploration:

The Java applet below allows you to click on a point in or around the Mandelbrot set and then view the fate of the orbit of 0. By fate of the orbit we mean the eventual behavior of the orbit of 0: We do not plot all of the points on the orbit of 0. We merely display the points number 25, 26, 27, etc. on the orbit of 0.

Click here to open the Mandelbrot Set Iterator applet.

Please note that your browser must be Java-enabled in order to use this applet. All recent versions of Netsapce or Internet Explorer are Java-enabled.

For Further Information:

Devaney home page

Dynamical Systems and Technology Project Home Page