The Beauty of Fractals

No more than six years ago! Only ten and twenty-odd years ago! On many days, I find it hard to believe that only six years have passed since I first saw and described the structure of the beautiful set which is celebrated in the present book, and to which I am honored and delighted that my name should be attached. No more than twenty-odd years have passed since I be­came convinced that my varied forays into unfashionable and lonely corners of the Unknown were not separate enterprises. No one had seen any unity between them, other than provided by my personality; yet, around 1964, they showed promise of consolidating one day into an organized field, which I proceeded to investigate systematically. And no more than ten years have passed since my field had consolidated enough to justify writing a book about it, hence giving it a name, which led me to coin the word fractal geometry. The beauty of many fractals is the more extraordinary for its having been wholly unexpected: they were meant to be mathematical diagrams drawn to make a scholarly point, and one might have expected them to be dull and dry. It is true that the poet wrote that Euclid gazed at beauty bare, but the full and continuing appreciation of the beauty of Euclid demands hard and long train­ing, and perhaps also a special gift. To the contrary, it seems that nobody is in­different to fractals. In fact, many view their first encounter with fractal geome­try as a totally new experience from the viewpoints of aesthetics as well as science. From these viewpoints, fractals are indeed as new as can be.

Quadratic Julia sets, and the Mandelbrot set, arise in a mathematical situation which is extremely simple, namely from sequences of complex numbers defined inductively by the relation where c is a complex constant. I must say that, in 1980, whenever I told my friends that I was just starting with J.H. Hubbard a study of polynomials of degree 2 in one complex variable (and more specifically those of the form z↦z²+c). they would all stare at me and ask: Do you expect to find anything new? It is, however, this simple family of polynomials which is responsible for producing these objects which are so complicated — not chaotic, but on the contrary, rigorously organized according to sophisticated combinatorial laws.

Art critics in the centuries to come will, I expect, look back on our age and come to conclusions quite different than our own experts. Most likely the painters and sculptors esteemed today will nearly have been forgotten, and instead the appearance of electronic media will be hailed as the most significant turn in the history of art. The debut of those first halting and immature attempts to achieve that ancient goal, namely the pictorial expression and representation of our world, but with a new media, will finally be given due recognition.