Exploration #3

Question: What is the relationship between the Mandelbrot set and filled Julia sets?

Answer: Remember that the Mandelbrot set is a picture of the complex c-plane. Each point in the Mandelbrot set represents a c-value for which the orbit of 0 does not escape under iteration of x² + c. Now we may also draw the corresponding filled Julia set for that c-value. This filled Julia set assumes one of only two possible "shapes" depending upon whether c is chosen in the Mandelbrot set or outside it. If c lies in the Mandelbrot set, then the corresponding filled Julia set is connected, meaning it is just one piece. If c lies outside the Mandelbrot set, then the filled Julia set shatters into infinitely many pieces (what is known technically as a "Cantor set" or, more popularly, "fractal dust").

Exploration: In the Mandelbrot set below, select one of the white dots. These correspond to specific c-values in and around the Mandelbrot set. You will then see the corresponding filled Julia set together with the fate of the orbit of 0. Note that the filled Julia sets from inside the Mandelbrot set assume many different shapes, but they are all connected (= one piece).

Further Exploration: To choose other c-values, you may use the Mandelbrot/Julia Set Applet. This tool will take a few seconds to compute various Julia sets, since, unlike the above images, you really compute each different image.

For Further Information:

The Fractal Geometry of the Mandelbrot Set. I: Periods of the Bulbs. Robert L. Devaney. Especially Section 5. Accessible to high school students with a solid background in complex numbers.
The Mandelbrot and Julia Sets: A Toolkit of Dynamics Activities. Robert L. Devaney. Pages 93-95. Key Curriculum Press. Accessible to advanced high school students.

Devaney home page

Dynamical Systems and Technology Project Home Page