Failure of Traditional Models

Abstract

The fractal concept was formally introduced into the physical sciences by Beniot Mandelbrot over 20 years ago and has since then captured the imagination of a generation of scientists. Mandelbrot had, of course, been working on the development of the idea for over a decade before he was finally willing to expose his brainchild to the scrutiny of the scientific community at large. His first monograph on fractals [16] brings together the experimental and physical arguments that undermine the traditional picture of the physical world. Since the time of Lagrange (1759) it has been accepted that celestial mechanics and physical phenomena are, by and large, described by smooth, continuous, and unique functions. This belief is part of the conceptual infrastructure of the physical sciences. The evolution of physical processes is modeled by systems of dynamical equations and the solutions to such equations are continuous and dif-ferentiable at all but a finite number of points. Therefore the phenomena being described by these equations were thought to have these properties of continuity and differentiability as well. Thus, the solutions to the equations of motion such as the Euler-Lagrange equations, or Hamilton’s equations, are analytic functions and such functions were thought to represent physical phenomena in general.