The geometry of fractal sets is a cross-disciplinary direction of scientific research which owes most of its tremendous development in recent years to the creative ideas of B. Mandelbrot; his inspiring work, much of which was synthesized in three consecutive versions of his monumental work [Les objets fractals: forme, hasard et dimension (Flammarion, Paris 1975); Fractals: form, chance, and dimension (1977; Zbl 0376.28020); The fractal geometry of nature (1982; Zbl 0504.28001)], has stirred up a widespread interest among both mathematicians and people working in various fields of the physical sciences or of related disciplines.
The theory of fractals, as a link between geometries on different scales, appears now to be an unavoidable tool in the understanding of the relationship between microscopic behaviour in various natural phenomena and observation at microscopic scale. The present book discusses various aspects of this general problem; it is a result of the author’s own research on phase-transitions, on the aggregation of immunoglobulins, and on the viscous fingering of fluid displacement in porous media.
After a short introduction in Chapter 1, the author defines in Chapter 2 the Hausdorff-Besicovitch fractal dimension and the related concept of the similarity dimension; he also presents some simple examples of fractal sets. In Chapter 3 he introduces the cluster fractal dimension and uses it in the description of the results of experiments on aggregation, gelation, and sedimentation of particles. Chapter 4 presents the theoretical background for the study of the displacement of fluids in porous media; a discussion of experimental results provides evidence for the fractal nature of the phenomenon of viscous fingering. After a short presentation of Cantor sets in Chapter 5, a detailed treatment is included in Chapter 6 of the ideas of fractal measures and multifractals; these are used in the discussion of some recent experimental results on thermal convection and viscous fingering. Chapter 7 contains a lengthy discussion of percolation processes, which provide well understood examples of random fractals and which appear as an essential ingredient of natural phenomena such as fluid-fluid displacement. Chapter 8 discusses Hurst’s analysis of records in time and the use of the empirical R/S analysis in the study of fractal statistics. The related problem of fractional Brownian motion is treated in detail in Chapter 9. Chapter 10 is devoted to self-affine fractals and discusses the various fractal dimensions associated with them. As an application of the R/S statistics to a self-affine record in time, the wave-height statistics of ocean waves are analyzed in Chapter 11. Chapter 12 deals with the perimeter-area relation for fractal sets; the discussion forms the basis for understanding the fractal nature of surfaces of clouds and of river systems. Chapter 13 presents some simple fractal surfaces and some experiments in computer generation of landscapes. The recent experimental evidence for the fractal structure of surfaces, based on topographic measurements, is presented in Chapter 14. Also included are a set of computer generated full-color representations of fractal landscapes and an extensive bibliography.
The book is certainly a most valuable contribution towards the understanding of some of the most exciting scientific applications of contemporary mathematics.