The Patterns of Chaos

magine you wish to measure the coastline of, for example, Great Britain. For speed, you might try to approximate by laying down some extremely long rulers - say 100 miles each - which might give you a figure of 3000 miles (for the sake of an example). Obviously you could arrive at a more accurate figure by using shorter rulers. One mile rulers would allow you to follow the shoreline around many of the bays and promontories and part way up some of the larger rivers, thus you would arrive at a considerably larger figure, maybe 10,000 miles. Using ten foot rulers, every river would be followed more or less to its source and back, perhaps giving a figure of 40,000 miles. By this stage, measurements of a classical geometrical figure of similar proportions, an ellipse for instance, would be becoming more consistent, but our coastline just carries on growing. You could go even further, and measure around every little inch-wide wrinkle, or even around every grain of sand.
Britain has an emphatically finite area, but this exercise progressively reveals that the length of its perimeter tends towards infinity! In mathematics, a shape which displays this kind of property - retaining complexity and detail as the magnification is increased - is called a fractal.

The concept of fractals was known about during the last century, but was limited to simple, repetitive geometrical constructions. They came of age in the 1970s, when the French mathematician Benoit Mandelbrot, working at an IBM research facility, began to use computers to plot the escape points of simple orbital formulae. The results were beautiful, abstract patterns which display the self-similarity common among fractal images, whereby distinctive shapes occur again and again as the pattern is zoomed into (as demonstrated in the images on this page).

Fractals are an elegant demonstration of the related mathematics of chaos theory, in which massively complex results can be obtained by following simple rule sets. Apart from their aesthetic and intellectual appeal, these areas are finding practical applications. Fractal formulae are being used for image compression, and chaos mathematics is being successfully used to optimise manufacturing processes.

Future articles will look at how fractal images are actually calculated, and explore some more of the many different patterns which have been discovered.