Video Find of the Day
Here's a video that graphically shows the way the Mandelbrot set M describes the character of Julia sets of the quadratic mappings "complex number goes to itself squared plus c" by the position of the complex (planar) parameter "c" relative to M. That doesn't make sense to most of you, does it? You're welcome! Enjoy the pretty pictures!
The Julia set
The next video does without the attempt at explanation and just shows the Julia sets transforming as the parameter c varies. Most of the Julia sets shown correspond to values of c near the boundary of the Mandelbrot set. You can know that because in those cases the Julia set is connected but doesn't appear to enclose area. Those that enclose area show black regions. Their parameter c lies in the interior of the Mandelbrot set. The Julia sets that fall apart into a myriad pieces (called "Fatou dust") are those whose parameter lies outside the Mandelbrot set.
The thing that gets me about all this is that in 1971 I took a course in higher analysis that dealt with Julia sets. We had to learn all their weird properties without being able to see any of these pictures! How did we get by?