JuliaDomna
version 1.0
by Ernest Tomlinson

JuliaDomna (named for a famous Julia, the wife of the Roman emperor Septimius Severus) is a screen saver module for Duncan Wilcox's Blanket, which draws a succession of Julia sets.  To use it, drop the add-on into the "Screen Savers" directory containing all the other Blanket modules.

Julia sets are related to the well-known Mandelbrot set, which is governed by the following recurrence relation in the complex plane:

z(n+1) = [z(n)]^2 + c

As everyone should know, the Mandelbrot set comprises all complex numbers c for which the sequence z(0), z(1), . . . , z(n), . . . does not "blow up"; in practice, the recurrence relation is applied iteratively to z = x + iy, starting with x = y = 0, either until |z| exceeds some small value (usually chosen to be 2, although the value is hardly important) or until a maximum number of iterations (in the hundreds to thousands) have been executed.

A Julia set comprises all complex numbers z(0) for which the same sequence, derived from the same recurrence relation, does not "blow up".  The same practical criteria are applied to determine membership in the set.  In this case, c is a fixed complex constant; the connectivity and appearance of the Julia set depends on the value of c.  If c should happen to lie within the Mandelbrot set, the resulting Julia set will be of a piece; if c lies outside the Mandelbrot set, the corresponding Julia set will be split into a number of disconnected pieces--in the extreme case, the set comprises only a scattering of "dust".
When drawing the set in the coordinate plane, colors can be assigned to points based on the number of iterations required before z exceeds its bounds.

JuliaDomna makes a screen saver of this by repeated drawing Julia sets from randomly chosen constants c.  Every time one set is completed, drawing starts on another set.  The system color table is used to select colors for drawing the set--since the first thirty-two entries in the color table are various shades of grey, that's mostly what you'll see.  I may work out some scheme for user specification of color tables to get different effects (e. g. the red-and-orange "fire" scheme that seems commonly to be used for color plates of the Mandelbrot and Julia sets.)  No configurability is at the moment implemented.

Ernest S. Tomlinson
e-mail:  etomlins@rohan.sdsu.edu
web "page":  http://www.ugcs.caltech.edu/~et/