 The magazine of the Melbourne PC User Group Mandelbrot and Fractal Geometry Ian McDowell |
The Mandelbrot article in PC Update of January/February 1990 attracted more than usual interest, and the Mandelbrot Dial Help brings more phone calls than the writer's SMART Integrated Package assistance used to. Thus a follow-up article of a more general kind should prove popular and informative.
If the body of this article seems complex, read the last two paragraphs now to find out why you may be pleased to get involved with it.
Benoit Mandelbrot is rightly famed for his fundamental discovery c1967 and after, i.e. the set of all complex numbers which form a sequence generated by squaring the last member of the sequence and adding it to itself, such that the sequence does not diverge. Divergence becomes very slow near the boundaries of the set. Incredible images emerge when each pixel on a computer display is coloured according to the rate of divergence of the sequence based on the complex number corresponding to it. There is an order to seemingly chaotic behaviour, such as epidemics, the course of a river, a flag flapping in the wind, a disturbed asteriodal orbit, or even Melbourne weather.
Mandelbrot investigated many other fractal phenomena. These are distinguished by the characteristic that the nature of the fractal images is independent of scale or magnification. His "The Fractal Geometry of Nature" (Freeman, New York, 1977) is the generic text. In it he shows that fractal behaviour is found in the study of clustering of galaxies, turbulence of liquids, movement of commodity prices, corruption of data transmission, cloud formations, size and distribution of moon craters, snowflake patterns and much more.
He coined the word "fractal" from Latin "frangere", to break, or "fractus", irregular. Using the example of the length of a broken coastline, he defined the fundamental equation of fractal geometry to be as follows. Total length equals the product of length of measure to the power of (1-D) and the number of measurements. For fractal behaviour, the scaling factor D is constant and fundamental. Its value for a given study is usually between one and two (D~1.5 for a coastline). It is calculated directly for mathematical constructs such as Koch curves (D~1.26) or Sierpinski carpets (D~1.89). It may be estimated from double logarithmic plots of total length as a function of length of measure, for models of natural phenomena.
He introduced us to other unusual mathematical terms such as dust, curd, whey, pertile. His practical results include the finding that charting commodity prices is futile; if they are to be studied, it must be in terms of the fractal scaling factor and its implications, and not by the standard techniques of mathematical statistics. He rejected the famous term "strange attractor", which arose from Lorenz' classic study of the behaviour of water wheels, but it seems destined to remain with us.
He drew attention to the work of mathematicians few of us had ever heard of, such as Bachelier, Cantor, Fatou, Feigenbaum, Fournier, Hausdorff, Hubbard, Julia, Levy, Koch, Myrberg, Peano, Richardson, Sierpinski and Zipf.
Generation of Mandelbrot images and exploration at increasing levels of magnification is probably the popular mathematical recreation among computer people today. The Scientific American has published about a dozen articles relating to fractals over the last twenty years. Enthusiasts have made it easy for us to participate in the enjoyment of this pastime. Proceed in the following way.
A group of about twenty such enthusiasts have put together a detailed program called FRACTINT, and placed it in the public domain. They do not ask for a shareware payment. The program is available from the Melb PC User Group library. It is incredibly sophisticated. It displays not only a suite of variations on the basic Mandelbrot set, but inversions, colour variations, spectacular changing colour displays, Barnsley ferns (see his secondary-level "Fractals Everywhere"), bifurcation, plasma clouds, Newton attractors, lambda sets, three dimensional images, fractal landscapes, iterated function sets (IFS) and much, much more.
For those of us who used Theron Wierenga's program, FRACTINT is very much faster by reason of a variety of programming short cuts. It does not save to disk as images are generated, but holds them in screen memory, to save only if required, or to proceed with a zoom. It generates Julia sets corresponding to a Mandelbrot starting point in seconds. It displays optionally the points corresponding to the sequence as the program generates them; they bear a marked resemblance to spiral galaxies. It does not require the preparation of color masks; simply select the display which makes the most spectacular use of the dwell values, and save it for reuse.
The writer has photographed, enlarged and framed some dramatic Mandelbrot and Julia images. Decorate your home in terms of your computing inclinations! Amaze your friends! You, too, can be a happy Mandelbrot fractal explorer within days!
Reprinted from the April 1990 issue of PC Update, the magazine of Melbourne PC User Group, Australia
