The Mandelbrot Set

definition:

The Mandelbrot set is the set M of complex numbers c=x+iy for which the series
{ z₀ = 0 , z₁ = c , ... , z_n+1 = (z_n)² + c , ... } is bounded.

A demo program in Pascal More theory about Mandelbrot and Julia sets

The Mandelbrot set is quasi-self-similar: by zooming in details, there are always reappearing the same structures as in the initial image; but on the other hand, every region has its particular 'specialities'.

The belly of the Mandelbrot set

is rich of stars and spirals.

one of the first spirals I decovered.

specimen with particulary twisted arms.
Fractint's color-cycling option shows the 'rotation effect'

After these random discoveries, I searched sytematically (see trick 1) to find spirals with 20 arms:

Encouraged by this success, I continued my investigations (see trick 2) to find 'megaspirals' like this one:

The region of the head

has structures ending with long fibres like fishbones.

The next two images were catched nearby a bay in the head of the Mandelbrot set. The mathematical accuracy has been decreased by choosing an insufficiant number of iterations in order to get more beautiful pictures:

Dead trees, or picture from a science fiction film?

maple leaf?
Use color-cycling to see the 'autumn-effect'.