Julia Sets & Self-Squared Dragons

These images were made by the Fracture screensaver.

These are images of Julia Sets and Self-Squared Dragons. Unlike the Mandelbrot Set, which is a unique mathematical object, there are an infinite number of Julia Sets (each defined by a complex parameter called c) and Self-Squared Dragons (each defined by a complex parameter called lambda — λ). Each of these sets is related mathematically to the Mandelbrot Set; in a sense, the Mandelbrot Set is a "dictionary" or a "catalog" of all of the infinitely many Julia and Dragon sets. The Julia and Dragon sets tend to be simpler and more predictable in structure than the Mandebrot Set; their resultant symmetry can sometimes make them very beautiful.

Mathematically, Julia Sets are defined by the iteration of the formula z ← z² + c, just as with the Mandelbrot Set. However, here c is the parameter for the Julia Set being imaged, and never changes, while z's initial value is the point on the complex plane being colored. A little reflection will reveal that the fate of the point (0, 0) in the Julia Set with parameter c will be exactly the same as the fate of the point c in an image of the Mandelbrot Set. A similar but more complicated relationship exists for the Dragons, which use the formula z ← λz(1-z). This is the sense in which the Mandelbrot Set is a catalog; it is a collection of the (0, 0) points of all of the Julia Sets. It turns out that the fate of that (0, 0) point is very representative of the fates in store for the other points in a Julia Set, so one may use the Mandelbrot Set as a collection of "summaries" of all of the Julia Sets.

All of these images are copyright © 2001 Ben Haller. Personal use of these images is allowed; all other use, including any kind of redistribution or reproduction of these images, is forbidden without the express, written consent of Ben Haller.