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.