The Science of Fractal Images

Computers & Geosciences Vol. 17, No. 3, pp. 469-475, 1991 Pergamon Press plc. Printed in Great Britain

REVIEWS The Science of Fractai Images by Heinz-Otto Peitgen and Dietmar Saupe, editors, 1988, Springer Verlag, New York, xiii + 312 p., 142 illustrations, 39 color plates, $39.95 (US); Chaos by Arun V. Holden, editor, 1986, Princeton University Press, Princeton, New Jersey, vi + 324 p., ISBN 0691-084-246 (paperback), $19.95 (US). With the two books edited by Holden and by Peitgen and Saupe we have two typical aspects of chaos theory or fractal geometry: (a) the geometrical approach and (b) the mathematical-graphicaldynamical systems approach. The geometrical approach in Peitgen's and Saupe's book focusses on the description of forms, shapes, and patterns; the fractal is presented as something fascinating, and as something that resembles nature, and last but not least is beautiful. The next issue then is the computation of fractal images. Holden's book also presents a "zoo" of fractals, but under the perspective of bifurcations and chaotic behavior of dynamic systems. The emphasis is in the application in biology and physiology. While The Science of Fractal Images naturally contains a lot of pictures, uses simple mathematics mostly for demonstrations, and is easy readable, more mathematics is used and needed throughout Chaos, but most of the text can be followed without going through all the maths. The many figures in this book are illustrative, whereas in the first book computer graphics play the major role. For a reader to whom fractals are completely new, certainly a picture book is far more motivating and fascinating. If one is interested in learning something about the nature of fractals, and possibly applying them, then a science application oriented book such as Holden's can be recommended. In the following, I give a short account of each of the books.

The Science of Fractal Images This book is another beautiful picture book on fractals, but it is more than a follow-up of H.-O. Peitgen's and P. H. Richter's The Beauty of Fractals (1986, Springer Verlag, Berlin). Both books place themselves in the tradition of B. B. Mandelbrot's understanding of The Fractal Geometry of Nature (title of his famous book, W.H. Freeman, New York, 1982). The main emphasis in The Science of Fractal Images is the visualization of fractals by computer graphics, and created by one of the leading groups in the field (University of Bremen, Germany), the pictures are truly fascinating. The center of the book,

by any understanding, is formed by color plates (p. 114-125, 36 plates on 8 p. with captions), which are the present culmination of computer visualization of fractals, showing not so much the scientific viewpoint but a creative play with forms and shapes-adding for example colors and clouds with an artistic intention. More than leaving the reader in admiration of the pictures generated with today's increased computer power, the book also provides some mathematical background and an introduction into coding algorithms (so to enable the reader to make "his own" fractals), termed "amateur mathematics" (p. vi). From the geoscience point of view the most interesting aspects are the application of fractals in the natural sciences and the fractal modeling of real world objects. An extended foreword is written by B. B. Mandelbrot, entitled "People and events behind the 'Science of Fractal Images' ", it gives a history of fractals from Poincar6 to supercomputer graphics (p. 1-19). For my taste, this account is rather self-centered, concerned with the recent history of people involved in this work and the book. The book consists of five chapters and four long appendices. Chapter l "Fractals in nature: from characterization to simulation" by R. F. Voss introduces random fractals via the most well-known examples (coastline, Koch snowflake, landscapes, Brownian motion, clouds) and gives some mathematical concepts (self-similarity, self-affinity) and algorithms (FFT, random fractal functions, spectral density) associated with random fractais (p. 21-70). "Algorithms for random fractals" is the title of Chapter 2 by D. Saupe (p. 71-113, p. 126--136). The chapter is easy to understand with little previous mathematical knowledge, for instance, even basic definitions from probability theory are given (probability, distribution, variance, random function). While Chapters l and 2 dealt with random fractals, Chapters 3-5 are devoted to deterministic fractals, starting from dynamical systems theory. Chapter 3 "Fractals arising in chaotic dynamical systems" by R. L. Devancy is meant as an introduction to Chapters 4 and 5, giving basic ideas of dynamical systems and links to chaos theory, leading to the Julia set (p. 137-167) (with a mathematical gap between the introductory maths and the Julia set). The title of Chapter 4 "Fantastic deterministic fractals" (by H.-O. Peitgen, p. 169-218) reflects the issue of the book: the use of mathematics to create pictures. A lot of numerical and programming details are included. Chapter 5 "Fractal modelling of real world images" by M. F. Barnsley (p. 219-242)

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finally appeals to the geoscientist with modeling of clouds, landscapes, and vegetation, but left me rather disappointed because the modeled landscape was mainly chimneys and not geomorphology (as the Brownian landscape published earlier in Mandelbrot's books). The nicest example is the Black Spleenworth fern. The geomorphologist's hopes get better fulfilled in Appendix A "Fractal landscapes without creases and with rivers" by B. B. Mandelbrot (p. 243-260), which is an extended description of the possibilities of Brownian landscapes already presented in Mandelbrot's earlier books (e.g. as referenced previously). A new aspect is the simulation built on prescribed river networks, watersheds, and drainage basins. In Appendix B "An eye for fractals" by M. McGuire (p. 261-271) somewhat of a "proof" of the fractal geometry being the geometry of nature is attained by photographs of objects which occur in nature that look similar to some of the fractals "created" on the computer: self-similarity of rocksand mountains-landscapes, of trees and branches, of lava streams, and of sediments. Appendix C "A unified approach to fractal curves and plants" by D. Saupe (p. 273-286) gives an introduction into the programming of "rewriting systems". "Rewriting systems" provides a way to generate classic fractals (Koch snowflake, Peano curve) and to model the geometry of plants (branches of trees and bushes). Appendix D "Exploring the Mandelbrot set" by Y. Fisher (p. 273-286) describes an algorithm that is efficient enough to be used to create black-and-white images of the Mandelbrot set on a microcomputer. Overall, this book assumes a lower mathematical level than The Fractal Geometry of Nature, and also contains less information per page (because of typesetting and style). This is certainly a consequence of the popularization of fractals, with the aim to sell to a broader readership. But the book is certainly to be recommended, even if just for one's own enjoyment and fascination.

Chaos A. V. Holden's book contains a list of useful introductory essays on chaos theory. Although the science primarily addressed is biology and physiology, it can be recommended as a reading for geologists interested in constructively using chaos theory. The fifteen single contributions are organized into six parts: I--Prologue, II--Iterations, III--Endogeneous Chaos, IV--Forced Chaos, VmMeasuring Chaos, and VI--Epilogue. The Prologue consists of a philosophical introduction by M. Conrad to chaos in environmental and biological systems, with special emphasis on adaptability. A. V. Holden and M. A. Muhammad present "A graphical zoo of strange and peculiar attractors". More definitions from the theory of differential equations are given in Part II (one- and two-dimensional iterative maps, by H A. Lauwerier), those stay basic enough not to require

prior knowledge of the mathematics of dynamic systems, for instance, "attractor", "orbit", "fixed point", "chaos" vs "hyperchaos" are explained. Part III is concerned with "Chaos in feedback systems" (A. Mees), "The Lorenz equations" (C. Sparrow), and several applications. While feedback is normally used to organize and stabilize a system, both technically and environmentally, chaos or noise also may be employed for optimization in systems with several limit states, for instance in neurological computation. The application chapters cover laser optics ("Instabilities and chaos in lasers and optical resonators", by W. J. Firth), ¢c,ology and epidemology ("Differential systems in ecology and epidemology", by W. M. Schaffer and M. Kot), and cell physiology ("Oscillations and chaos in cellular metabolism and physiological systems", by P. E. Rapp). The examples under Part IV, "Forced Chaos" are of lesser interest for the geologically oriented reader and fairly specialized ("Periodically forced nonlinear oscillators" by K. Tomita, "Chaotic cardiac rhythms, by L. Glass, A. Shrier, and J. Belair, "Chaotic oscillations and bifurcations in squid giant axons" by K. Aihara and G. Matsumoto: squids have long nerve cells which can be used for study of information transfer speed). Part V is important for geologic applications, because it relates to questions such as: How can I tell whether the system I study might be a fractal? Is this process governed by chaos, or can it be predicted? Such questions seem to be relevant in exploration geology, tectonics, and sedimentary geology. Two approaches are Lyapunov exponents (loosely speaking, eigenvalues in a decomposition of limit orbits) and the fractal dimension of attractors ("Quantifying chaos with Lyapunov exponents" by A. Wolf, "Estimating the fractal dimensions and entropies of strange attractors" by P. Grassberger). The Epilogue by O. E. Roessler addresses the question "How chaotic is the universe?". The essays generally are balanced successfully between descriptive text and consistent mathematics. Some experience in differential equations is helpful. Even in mathematically more demanding parts the text is interesting enough to keep the reader going from one example to the next. Various examples are given (chemical field, Lorenz equations, logistic maps, Hopf bifurcation, Feigenbaum number, to name only a few), so one can get a feeling of "what may happen". The figures in the book have a limited resolution, which is adequate for the introductory purpose. In most situations a significant list of references is given. The book has a short general index.

Scripps Institution of UTE CHRISTINAHERZFELD Oceanography Geological Research Division University of California~San Diego La Jolla, CA 92093 U.S.A.