# The Mandelbrot Set

definition:

The Mandelbrot set is the set M of complex numbers c=x+iy for which
the series

{ z_{0} = 0 , z_{1} = c , ... , z_{n+1} = (z_{n})^{2}
+ c , ... } is bounded.

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:

(dedicated to S.Z.)

## 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'*.

Last update: 1996-12-04

**Your comments are welcome!** Mail to Patrick
Hahn (phahn@vo.lu)

Homepage
http://www2.vo.lu/homepages/phahn/default.htm