JeanFrançois GOUYET 

Fractal structures were discovered by mathematicians over a century ago and have been used as subtle examples of continuous but nonrectifiable curves, that is, those whose length cannot be measured, or of continuous but nowhere differentiable curves, that is, those for which it is impossible to draw a tangent at any their points. Benoît Mandelbrot was the first to realize that many shapes in nature exhibit a fractal structure, from clouds, trees, mountains, certain plants, rivers and coastlines to the distribution of the craters on the moon. The existence of such structures in nature stems from the presence of disorder, or results from a functional optimization. Indeed, this is how trees and lungs maximise their surface/volume ratios. This volume, which derives from a course given for the last three years at the Ecole Supérieure d’Electricité, should be seen as an introduction to the numerous phenomena giving rise to fractal structures. It is intended for students and for all those wishing to initiate themselves into this fascinating field where apparently disordered forms become geometry. It should also be useful to researchers, physicists, and chemists, who are not yet experts in this field. This book does not claim to be an exhaustive study of all the latest research in the field, yet it does contains all the material necessary to allow the reader to tackle it. Deeper studies may be found not only in Mandelbrot’s books (Springer Verlag will publish a selection of books which bring together reprints of published articles along with many unpublished papers), but also in the very abundant, specialized existing literature, the principal references of which are located at the end of this book. The initial chapter introduces the principal mathematical concepts needed to characterize fractal structures. The next two chapters are given over to fractal geometries found in nature; the division of these two chapters is intended to xii Preface help the presentation. Chapter 2 concerns those structures which may extend to enormous sizes (galaxies, mountainous reliefs, etc.), while Chap. 3 explains those fractal structures studied by materials physicists. This classification is obviously too rigid; for example, fractures generate similar structures ranging in size from several microns to several hundreds of meters. In these two chapters devoted to fractal geometries produced by the physical world, we have introduced some very general models. Thus fractional Brownian motion is introduced to deal with reliefs, and percolation to deal with disordered media. This approach, which may seem slightly unorthodox seeing that these concepts have a much wider range of application than the examples to which they are attached, is intended to lighten the mathematical part of the subject by integrating it into a physical context. Chapter 4 concerns growth models. These display too great a diversity and richness to be dispersed in the course of the treatment of the various phenomena described. Finally, Chap. 5 introduces the dynamic aspects of transport in fractal media. Thus it completes the geometric aspects of dynamic phenomena described in the previous chapters. 