## How are these nice pictures arising altogether?

The whole calculation is based on the behaviour of recursive formulas with
complex numbers (numbers in the form a + bj where j=sqrt(-1) i.e. defined
square root of -1). Similar the number straight line of real numbers (our
normal, ordinary numbers like 1, 2, 3, 4.766, 3.1415926... etc.), all complex
numbers form a plane, the *Gauss's number plane* called from the famous
mathematician Carl-Friederich Gauss (1777-1855). Each one of these picture are
resulted from the simple formula

z(n+1) = z(n)² + c

where z(0) (start value) is the picture constant, c the point of the plane.
The number of iterations will counted until the condition **ABS(z(x))
> 2** occurs. A colour depending to x will be chosen from a free
defined colour scale e.g. x=1 => red, x=2 => orange, x=3 => yellow,
x=4 => green x=5 => cyan x=6 => blue x=7 => magenta x=8 => rot,
x=9 => orange etc. i.e. an endless repetition of this colour cycle. On many
points the referenced condition above never occurs (x=infinite) => e.g.
black (In the practice the computation will be stopped after an enough highly
chosen boundary as it called *depth of computation* xmax [e.g. 1000] to
avoid an endless loop during the computing process). This procedure will be
taken on each point (screen pixel on the computer), so they form these pictures
at the finish of computation.

This chaotic behaviour was already discovered from the French mathematician
Gaston Julia but his knowledge's found only a further interest since the age of
graphic computers.

Go back to the first
picture

© 1996, 1998 by Andreas Meile