Fractals

About 1988 or so I had a Philips MSX2 computer (Z80, 8-bit, 3.77 MHz processor, 128 kB RAM and separate video display processor with 128 kB video RAM, 1 (one) single sided - single density diskdrive (360 kB) and Borland Pascal). My father got interested in fractals, he read an article in Scientific American, 'The Fractal Geometry of Nature' by Benoit B. Mandelbrot and some other books on the subject.

We worked out an algorithm for computing the pictures of the Mandelbrot Set. After that we had a problem. The algorithm was easily implemented, and we had the whole thing up and running in (hang on to your seat!) BASIC. This was (of course!) extremely slow. We did have a pascal compiler but it was exclusively text-oriented and could perform no graphics functions whatsoever.

During the following half year we found a book describing the fuctions of the MSX2 Video Display Processor and wrote a number of functions to acces the MSX2 graphics. This resulted in our first and (to our experience) very fast Mandelbrot Picture.

We saw in the course of some ten minutes the emergence of the full Mandelbrot set in 16 colours and 512x212 pixels.

In the meantime I had a new Philips MSX2 computer with 2 (two) double sided - single density diskdrives (each 720 kB) and video extention (video superimpose, realtime frame grabbing etc.). Quite nifty, especially in the nineteen eighties.

Eventually I realised that MSX was a dead-end and switched to a 486, 33 MHz, 2MB and 40MB and immediately implemented my Mandelbrot program for the PC. The speed was astonishing (that was 1991 or so), where the MSX2 needed some 10 minutes for the full Mandelbrot set, the 486 did it in seconds. Now I have a Pentium at home and a SGI O2 workstation at work, which is a different story altogether.

Now I use FractInt, a freeware program of the Stone-Soup-Group. It can calculate myriads of fractal types and is very fast. My own Mandelbrot program is in deep-freeze state.

To keep everybody happy I have both Text links to Mandelbrot Pictures and a High Graphics Gallery of clickable thumbnails of the same pictures.

Apparently I am not the only one to think my fractals are attractive, since I made it into the university newspaper (Universiteits Krant) of the University of Groningen (I know, it isn't the New York Times, but hey, it is a first step) on a centerfold special on "Art in Science" (this is the html version, a pdf version is only available on the campus).

In addition, the front page of the "Invariant Magazine" (Issue 16, 2005) of the Student Mathematical Society "invariants" of the University of Oxford features one of my images, and has a comprehensive article on "Drawing the Mandelbrot Set" (by Martin Churchill, on page 27; info at invar@herald.ox.ac.uk).

Some links to other sites, which also feature pages with extended background on what fractals are and what can be and/or is done with them:

I have not had time yet to read all of them, so if you find things you think are especially interesting or have some other kind of comment on these links, please let me know.

This is the reply I sent to the following mail I got from Georgina Bruni (I should edit this to make it a nice normal piece of text):

Dear Anton

Can you explain - in simple terms - I'm not a mathematition - what the Julia Set is? What does it mean? How can it be used in science?

Regards
Georgina Bruni

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Well, I am no mathematician either, but I can give it a try. What you do to calculate an image of the julia set is the following: For every point in a plane (i.e. a pixel on your screen) you do the following: evaluate a formula (the coordinates of the current point go in, new coordinates come out), if the outcome is below a certain treshold, put the outcome back in and repeat until the outcome is above the treshold. You should count the number of repeats you made. If you do this for every point (pixel) on your plane (screen), you now have a repeat-count for every pixel. The final thing is to assign to each different count number a colour. If you do this properly you will end up with a nice picture.

More mathematically, the Julia set is closely related to the Mandelbrot set (both are fractal-type sets). Strictly speaking, both sets are the points on a plane (for mandelbrot, actually the full julia set is 4 dimensional) for which the counter I desctibed above goes to infinity. Usually this is approximated (ofcourse) as some large number. The real fun is that, since these sets are fractals, you can go on magnifying parts of them indefinitely and discover new things all the time.

As for a meaning, I don't know, a mathematician might give you some clue, but it is a bit like asking for the meaning of a sine curve or parabola. As far as I know the Julia and Mandelbrot Sets as such are not usefull (in science or elsewhere), but fractals are. They are used in sophisticated image compression algorithms, where an image is converted to a formula which should be evaluated as the Julia or Mandelbrot formula's are, to re-genereate the image.

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Last modified: December 15, 2005

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