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Book review

Author(s) Turcotte, Donald L.
Title Fractals and chaos in geology and geophysics
Publisher Cambridge University Press
Year of publication 1992
Reviewed by Hideki Takayasu

Concepts of fractals and chaos have been involved in earth sciences since their birth. Irregular fractal coastlines, first brought forward by Mandelbrot, are one of the earliest examples of fractals in nature, and deterministic chaos has been discovered by Lorenz when he was working on a weather model. "Fractals and chaos in geology and geophysics" is an elementary introduction to geophysical applications of fractals and chaos, written by a leading geophysist.

Fractals are treated in the first eight chapters; fractal dimensions and power law distributions are discussed with a variety of topics such as coastlines, landscapes, fragmentation, seismicity, volcanic eruptions, ore grade, oil fields, and so on.

Chapters from 9 to 14 are focused on chaotic temporal behavior of simple nonlinear systems. Logistic map and Lorenz equation are introduced as simplest nonlinear systems leading to deterministic chaos. The author shows how the concept of chaotic behavior can be applied to various geophysical phenomena, including earthquakes, mantle convection and polarity fluctuations of the earth's magnetic fields.

Chapter 15 is devoted to renormalization group, a powerful theoretical approach to critical phenomena. Here, problems in percolation and fragmentation are solved analytically with elementary algebraic calculations.

A recent topic of the self-organized criticality is briefly described in chapter 16 together with a direct comparison to real earthquake data. The last chapter is a one-page summary.

The overall description is intuitive using only elementary mathematics. Numerous figures effectively improve our understanding. Intuitive description is generally good for a beginner but the lack of mathematical definition sometimes causes ambiguity and misunderstanding. Throughout this book the same terminology "fractal dimension" is introduced in several different ways. Mathematically most of the fractal dimensions are different quantities, but the reader might confuse them as an identical quantity.

This book is clearly designed as a text for advanced undergraduate and introductory graduate courses in the earth sciences. At the end of each chapter problems are prepared to confirm the reader's understanding. (Some of the answers are given at the end of the book, also, there are a glossary of technical terms and tables of physical units and symbols used in the book.) References cover most of the important works in the field, so that the reader can look up the original papers if he finds a topic interesting.

The major goal in this field of study is to bridge the gaps between simple mathematical models and complicated real phenomena. Real phenomena are obviously still very far beyond, but we have a confirmation that we are approaching to the goal step by step. I believe that the reader can surely feel the attractiveness of the goal and the joy of following it.