Mathematical Notes: Poetry and art out of geometric chaos

FRACTALS SEEM far more special than most general shapes of mathematics because they are characterised by so-called "symmetries". Broadly speaking, mathematical and natural fractals are shapes whose roughness and fragmentation neither tend to vanish nor fluctuate up and down, but remain unchanged as one zooms in continually and examination is refined. Hence, the structure of every piece holds the key to the whole structure.

Fractal geometry is conveniently viewed as a language, and it has proven its value by its uses. Its uses in art and pure mathematics, being without "practical" application, can be said to be poetic. Its uses in various areas of the study of materials and of other areas of engineering are examples of practical prose. Its uses in physical theory, especially in conjunction with the basic equations of mathematical physics, combine poetry and high prose.

To the layman, fractal art tends to seem simply magical, but no mathematician can fail to try and understand its structure and its meaning. Between the extremes of the excessive geometric order of Euclid and of the geometric chaos of the most general mathematics, can there be a middle ground of "organised" or "orderly" geometric chaos? To provide such a middle ground is the ambition of fractal geometry. The fact that fractals are difficult to grasp and slow to develop does not make them any less fascinating.

Pure mathematics certainly does exist as one of the remarkable activities of Man, it certainly is different in spirit from the art of creating pictures by numerical manipulation, and it has indeed proven that it can thrive in splendid isolation - at least over brief periods. Nevertheless, the interaction between art, mathematics, and fractals confirms what is suggested by almost all earlier experiences. Over the long haul mathematics gains by not attempting to destroy the "organic" unity that appears to exist between seemingly disparate but equally worthy activities of man, the abstract and the intuitive.

So how did fractals come to play their roles of "extracting order out of chaos"? The algorithms that generate fractals are typically so extraordinarily short as to look positively dumb. This means they must be called "simple". Their fractal outputs, to the contrary, often appear to involve structures of great richness. A priori one would have expected that the construction of complex shapes would necessitate complex roles.

What is the special feature that makes fractal geometry perform in such an unusual manner? The answer is very simple. The algorithms are recursive, and the computer code written to represent them involves "loops". That is, the basic instructions are simple, and their effects can be followed easily. But let these simple instructions be followed repeatedly. The process of iteration effectively builds up an increasingly complicated transform, whose effects the mind can follow less and less easily. Eventually one reaches something that is "qualitatively" different from the original building block.

Many fractals have been accepted as works of a new form of art. Some are "representational", others totally abstract. Yet all strike almost everyone in forceful, almost sensual, fashion. The artist, the child, and the "man in the street" never seem to have seen enough, and they never expected to have seen anything of this sort from mathematics. Neither had the mathematician expected his field to interact with art in this way.

Eugene Wigner has written about "the unreasonable effectiveness of mathematics in the natural sciences". To this line I have been privileged to add a parallel statement concerning "the unreasonable effectiveness of mathematics as creator of shapes that man can marvel about and enjoy".

Benoit Mandelbrot is the author of `The Fractal Geometry of Nature' (W.H. Freeman and Co, pounds 45.95) and `Multifractals and 1/F Noise' (Springer- Verlag, pounds 26.95)