Speculations on Fractal Geometry in Spatial Dynamics

Abstract

The most exciting science always stands on the frontier between the known and the unknown, and in the last decade in the effort to push our understanding of real and natural systems with all their imperfections to the limit of our analytical powers, fractals and their geometry have attracted very widespread attention. The idea that the geometry of Euclid is but a special case of a more general geometry in which the irregular and the fractional rather than the regular and the integral represent the norm, is appealing in its intuitive sense, and thus fractal geometry has spread like wildfire throughout the sciences. Applications of course abound. Although science is ordered in that its theories deal with phenomena which are represented systematically and with regularity, its practice through applications is anything but for real systems seem to be at the other end of the spectrum. Everywhere we look from the stock market to the form of river systems, from crystal growth to the way we pack circuits onto a silicon chip, shapes which can only be described in fractional or fractal dimension seem to exist.

“Mandelbrot has attracted the attention of scientists on the ubiquity of fractal shapes among natural objects. This was an important and fruitful contribution. What is still missing in general is an understanding of how fractal shapes arise.”

David Ruelle (1991) Chance and Chaos, Princeton University Press, Princeton, NJ, p. 178