JULIA Set: Executable or Text Version

Assembly Language enhanced versions: Executable or Text

The Julia Set is closely related to the famous Mandelbrot set. Both are described by the term coined by the mathematician Benoit Mandelbrot in 1975 - FRACTAL. Although there is a precise mathematical meaning to the term Fractal, in essence it means that no matter how close you "zoom in" to the picture, the picture looks the same; it has the same overall format.

AND...... For looking at ONLY the Mandelbrot Set, use Mandelbrot

This is also Assembly Language enhanced. Text here. Also a 2-colour only version MandelMono

There is a Julia set for every point of the Mandelbrot set - so there's only ONE Mandelbrot set, but infinitely many Julia sets. The JULIA Set is formed as described below:

To get the JULIA Set:

  • Firstly, pick a large number. 1 million will do.

  • Choose some Complex Number C, then, starting with Z=(-1,-1) calculate Znew= Z x Z -C

  • Work out Mod(Znew). Is it larger than 1 million? If not, work out the following.............

  • Znew2 = Znew x Znew - C . Is Mod(Znew2) larger than 1 million? If not, repeat the procedure to get Znew3 and so on..

  • Set a screen up with Z=(-1,-1) at the bottom left corner, and Z=(1,1) at the top right corner, and repeat the above procedure moving from RED -1 to 1 and GREEN -1 to 1 in let's say 1000 steps for each. Colour each of these points according to how many repetitions of the above calculation it took for the Modulus to exceed 1 million. If after a reasonable number of repetitions, say 64 for the number of colour gradations available on a computer, you find that Mod(Znew64) is STILL not bigger than 1 million, colour that pixel BLACK.


You will then get the Julia set associated with the Complex Number C, which is what my program does. You should opt to start off with the Mandelbrot set, and choose the Julia set by clicking on the picture of the Mandelbrot set, then zoom into any particular rectangular area of the Julia Set using the mouse. The Mouse Pointer is turned OFF except once the picture is complete. Then either press -s- to quit the program altogether, or press -m- to turn ON the Mouse Pointer. Once turned ON, choose a rectangular area to zoom into by clicking firstly the LEFT mouse button at one corner, then the RIGHT mouse button for the OPPOSITE corner. One second later, the program resumes with the Mouse Pointer once again turned OFF.

The Lyapunov Exponent: Executable or Text Version

Certain mathematical equations vary in their reaction to errors in initial conditions. The Lyapunov Exponent provides a way to map this susceptibility graphically. Alexander Lyapunov was a Russian mathematician who died in 1918. Sensitivity to errors is one of the hallmarks of mathematical chaos, along with repeated cycles and "mixing" behaviour.

The simplest situation showing a very complex pattern of chaotic / non-chaotic behaviour is recursive application of the Logistic Equation:

P(X) = p.X.(a-X)

Those interested in the detailed mathematics explained accessibly should read the article Extracting Beauty from Chaos " by Cambridge University researcher Andy Burbanks. Just as Andy Burbanks does, my program uses TWO versions of the above Logistic Equation, P(X) and Q(X), with a 2-D picture plotted of p versus q.

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