The Mandelbrot Set

Student: So, I've played with Julia Sets and found some really cool pictures, but none of them were that Mandelbrot set! How do I get that?

Mentor: The Mandelbrot set was discovered much later than Julia sets. Julia and Fatou were looking at Julia sets right after World War I, but until computers came along and got easier to program -- in the 1960s and 1970s -- mathematicians didn't know much about what Julia sets looked like and how many different kinds there were. In the 1970s, Benoit Mandelbrot was interested in looking at highly irregular forms in geometry, such as coastlines, mountain range shapes, and coral formations, and looked again at Julia sets with the aid of a computer.

• Julia Sets can be fractals, in that they have fractional dimension and self-similarity.
• Julia Sets are either all one piece -- like the circle for Z^2 + (0,0) -- or completely dust -- like the dots we get for Z^2 + (1/2,0).
• There is a pattern to which Julia sets are all one piece and which are dust. We only have to look at what happens to the point (0,0) when we iterate: If (0,0) escapes, then the Julia set is dust, if (0,0) doesn't escape, then the Julia set is connected.

Student: So where does the Mandelbrot set fit in?

Mentor: Take the set of all functions f(Z) = Z^2 + C and look at all of the possible C points and their Julia sets. If the Julia set is connected color the C point black, if the Julia set is dust, don't color the C point. Here's what you get:

Student: So if its Julia set is connected, C is in the set?

Mentor: That's right. People have made the picture even more beautiful by adding colors to the points outside the set. Most people add colors by making the points different colors depending on how many iterations it takes for the (0,0) point to get out of the circle of radius 2 -- remember that this is how we tell that the Julia set is dust. Try experimenting with the Mandelbrot Set yourself!

Student: So how is the Mandelbrot set related to fractals? I know many of the Julia sets are fractals.

Mentor: The Mandelbrot set is fractal. Look at its edges! They are very complicated. And they have self-similarity -- try zooming in using the Mandelbrot Set .

Student: So the Mandelbrot set is really a picture of how all the Julia sets for f(Z) = Z^2 + C behave?

Mentor: Yes! There are other Mandelbrot sets too. For example, we might consider looking at f(Z) = Z^3 + C instead. It is neat to see how the 3rd power -- or other powers for that matter -- changes the Julia sets and the Mandelbrot set!