The Wonder-World of Mandelbrot
The Mandelbrot fractal set is defined by the following iteration on complex
z := z * z + c
Those c values belong to the set for which the above iteration does not
converge to infinity. The canonic set is defined when z started from zero.
Other starting values produce distorted versions of the set. The Julia
sets are defined on the z value as domain with a fixed c value.
It is easy to see that the iteration converges to infinity if the absolute value
of z reaches 2. Therefore it is a usual technique to color the points outside
the set according to the number of iterations before z reached 2. The
points of the set are colored black. The table above contains the canonic
Mandelbrot set (in the middle) with some interesting parts magnified.
Click on the images to see a large resolution (1262x984) version of them.
My Programs to Draw the Mandelbrot/Julia Sets
As I am a programmer and love these fractal images I wrote programs
to calculate and draw the Mandelbrot set for every types of computers
I worked on (e.g. Commodore 64, Enterprise, PC, SGI etc.), and I used various
languages (Basic, Pascal, C, Assembly). I cannot provide you with all those
programs simply because I do not have most of them around any more. However,
I am happy to present some of them for your pleasure and entertainment.
- XJulia Very fast Julia
set generator program with algorithm and source.
The Julia set is generated by a mouse click on the Mandelbrot set and it
is so fast, that you can "fly" over the Mandelbrot set and see a
REALTIME movie (20 frames/sec) of the corresponding
- XMandel An X-Windows Mandelbrot set generator
program with source. Contains a "smart" random demo, which finds nice regions
to zoom in.
- ZsManJul Mandelbrot/Julia generator
program for PC(VGA)
Similar features to the above programs, surprising speed from a PC.
Links to other Fractal sites
If you are new to fractals I recommend to have a look at the
Fractal Lesson page
by Cynthia Lanius, which is a good introduction to fractals in general.
David E. Joyce has a good
description of Julia and Mandelbrot sets.
On-line Image Generators
Images and Movies
Other fractal related sites
On March 28, 1996.