Aside from the Mandelbrot set, the most famous fractals are the Julia sets. You've almost definitely seen images of the Julias (like the ones scattered through this post), but what you might not have realized is just how closely related the Julia sets are to the Mandelbrot set.
Remember what the mandelbrot set is? I'll run through a very brief refresher, but if you want more details, you can look at my earlier post about it.
Take a simple quadratic function in the complex plane: f(x)=x2+c, where c is a complex constant. If you iterate f, starting with f(0) - f(0), f2(0) = f(f(0)), f3(0)=f(f(f(0))), ..., then depending on the value of c in the function f, either that series will always stay finite, or it will diverge to infinity.
So - from that, we can think about the family of functions, parameterized by C-values, where fk is the function x2+x. Using that family of functions, we can ask
for any given C, does the iteration series of fc(0) diverge? The Mandelbrot set is the set of c-values for which fc never diverges.
So what's a Julia set? Basically, it's a kind of mirror image of the Mandelbrot. Take the same basic family of functions: f(x)=x2+c. But instead of varying
c, keep C fixed, and vary x. The Julia set for c is the set of x-values for which iterating f
does not diverge. There's an infinite number of Julia sets - one for every possible C.
If you work out the Julia sets for different C-values, you'll find that there are some sets
that are connected, and some that aren't. The non-connected ones are examples of what Mandelbrot called dusts - complex relatives of the Cantor dust; they're generally called Fatou dusts. The connected Julia sets are often strikingly beautiful. Personally, I find some of the dusts even more beautiful than the connected ones.
Knowing that the Julia sets are just a different parameterization of the same basic thing, you'd
expect there to be a deeper relationship between the sets. And you wouldn't be disappointed. The Mandelbrot set is precisely the set of C-values for which the Julia set of fC is connected, and the Fatou dusts are the Julia sets of C-values that are not in the Mandelbrot set.