Fractional Brownian motion
In my master thesis, I implemented the fractal model fractional Brownian
motion (fBm) to model interstellar clouds. One can generate fBm in an
arbitrary number of dimensions, the background image of this page is an
example of a 2+1 dimensional fBm, i.e. two spatial and one intensity dimension.
Its spatial boundaries are periodic due to the algorithm I used to generate
the image (called FFT filtering).
To display 3+1 dimensional fBm I made a gif animation, which can be seen if
you click on the left black and white icon. The animation is composed of
128 crossections and is 587kB in all. Because of the periodic property of
fBm, this animation is also periodic in time.
The Mandelbrot set
Using fractals as models of interstellar clouds, I got interested in
fractals themself. Using an extremely simple program, I calculated the
Mandelbrot set for the quadratic family and coloured the pixels as a
function of their escape time under iteration.
The boundary of this Mandelbrot set has recently been proven to
Zooming in Mandelbrot
The fractal property of the boundary of the Mandelbrot set means that there
are structures on every scale. Zooming in the boundary one finds several
funny looking things, structures that have been named monsters,
elephant trunks, or whatever. One prominent detail that is reappering
again and again as one is looking deeper and deeper into the Mandelbrot set,
is an almost identical copy of the Mandelbrot set itself.
Spiders in Mandelbrot
This substructure of the Mandelbrot set is often refered to as a
spider. Looking deeper into a spider, one will find smaller spiders,
that in turn contain even smaller spiders and so on, ad infinitum.