Pergamon Bullet n of Mathematical Biolooy VoL 57, No. 6, pp. 939-941, 1995

Elsevier Science Ine. © 1995 Society for Mathematical Biology

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B O O K R E V I E W

Fractals in Science, Armin Bunde and Shlomo Havlin (Editors). Springer- Verlag, Berlin, 1994. $59.00 (cloth), 298 pp.

Somewhere in cyberspace, an electronic grain of sand falls towards a virtual sandbox, which is to say a random location on a two-dimensional lattice is about to have its value incremented by one. The distribution of sand in the box is the product of many such random sandfalls and one simple rule: any square that receives a fourth grain of sand is unstable (the pile is too high) and avalanches, distributing one grain to each of its four neighbors. A disturbance at one site can therefore propagate if a neighboring site already possess three grains of sand. It is impossible to predict how far the avalanche will spread, but the frequency distribution of observed avalanche sizes is remarkable: the probability of increasing size falls offnot exponentially, but as a power law. At each new site, the chance of the disturbance propagating exactly balances the chance that it will no t - -a surprising property for randomly built piles of sand, which places this problem within the domain of fractal geometry.

The above example illustrates the concept of self-organized criticality, the "tendency for systems to drive themselves to a critical state with a wide range of length and time scales", and is presented in the book "Fractals in Science" as one possible mechanism for the dynamical origin for many of the fractal structures observed in nature. This very interesting idea is discussed by Bak and Creutz (Chapter 2) who attempt to apply it to explain the observed power law scaling of earthquakes. Millions of years of stresses have resulted in a balance within the geological structures in the crust. In this stationary state, where the build-up of stress is balanced by its release during earthquakes, there is a critical response to any initial rupture. Again, the appeal of this model is its application to many systems; for while fractal geometries in nature have been well-described, the mechanisms of their formation are less well understood.

Indeed, self-organized criticality would have been an interesting organizing theme for a book such as this. It would distinguish it in a significant way from other similar books on fractal geometry that are built as edited volumes, with papers drawn from a diversity of fields. Bak's self-organized criticality is an interesting alternative to the idea discussed by Hastings and Sugihara (1993) that fractals may reflect nature's tendency toward parsimony, with complex

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geometries built simply from the repeated application of a self-same rule (recursion). There complexity rests not in the rule, but in the diversity of building blocks that simple recursive rules can operate on.

Although organizing the volume around self-organized criticality may have made the book more provocative, it remains an extremely well done effort at an edited compendium. It is a companion volume to an earlier book "Fractals and Disordered Systems" which focused more narrowly on applications to materials science. The present book spans the gamut with contributions from geophysics, chemistry, biology, physics and computer science. The introduc- tory chapter describes the basic geometry of self-similar structures and introduces elementary concepts in calculating fractal dimension. Most subsequent articles begin with an overview of applications in each author's field, and then provide an in-depth explanation of a particular research problem. Topics include polymer formation, two-species reaction dynamics, and random walkology, with links between theory and experimental evidence consistent throughout. We believe that the explicit links within and between articles is one of the strengths of the book. The ubiquity of fractals in science is becoming well-documented, and in fact much of the research discussed in this text is not "new" in the strictest sense, having appeared in earlier publications. However, the authors pay special attention to methodological insights, greatly improving the utility of this book, both as a primer and as a guide to applications.

Some article topics are supported by computer software presented at the end of the book; however, the reader is not always made aware of the applicable demonstrations. The mostly black and white demonstrations (which are supported by both MS-DOS and Macintosh personal computers) are an excellent visual accompaniment and should not be left until the end for examination. The demonstrations mimic processes discussed in the book, such as the collapse of sand piles and the spread of forest fires, as well as illustrating simple deterministic fractals. A minor difficulty that we had is that a few of the simulations such as the "life" example run too fast to be really useful as heuristic aid (even on a MacII si). For example, when varying the rules thal govern the birth/death generation process, matrix updating is too rapid tc allow observation of the changing dynamics.

In the single chapter discussing fractal patterns in biology, well-knowr examples are used; the fractal nature of DNA sequences and long-rang~ correlations in the human heartbeat. These are hidden fractals, with som~ space-time mapping required to elucidate their self-similarity. Good explan. ation is given of the recent findings and continuing direction of such work Limited attention was given to other examples of fractals in biological systems topics such as human writing and evolution were only briefly mentioned although well-referenced.

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We strongly recommend the text for introductory graduate seminars. While the mathematical discussion does form a significant portion of the material and arguments presented, it may often be bypassed, and the required level of mathematical understanding is not overwhelming. Knowledge of calculus is critical, and at least a basic understanding of matrix algebra, ordinary and partial differential equations, and spectral analysis is helpful. Additionally, the reader is asked to find solutions individually throughout the book, which lends itself well to the discussion-style format of graduate seminars. Discussion of the model algorithms was often provided, enabling the computer literate to understand the mechanics and set up simulations to test other data. Finally, the extensive referencing at the end of each chapter makes this text an excellent starting tool for the pursuit of primary research literature regarding fractals in a wide variety of fields.

PAUL DIXON ALISTAIR HOBDAY

GEORGE SUGIHARA Scripps Institution of Oceanography 0208

University of California San Diego La Jolla, CA 92093-0208, U.S.A.

LITERATURE

Hastings, H. and G. Sugihara. 1993. Fractals: A User's Guide for the Natural Sciences, Oxford: Oxford University Press.

Bunde, A. and S. Havlin (Eds). 1991. Fractals and Disordered Systems. Berlin: Springer-Verlag.