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Fractals everywhere
Reviews exploration because of the adverse effects of mixing deposit types and the lack of robustness for the simple linear models employed. DONALD A. SINGER U.S. Geological Survey Middlefield Road Menlo Park, CA 94025, U.S.A.
Fractais Everywhere by M. Barnsley, Academic Press, San Diego, California, 394p., ISBN 0-12-079062-9; Fraetais by J. Feders, Plenum Press, New York, 183p., ISBN 0-306-42851-2; A Random Walk Through Fractal Dimensions by B. Kaye, 1989, VCH Publishers, New York, 421p., ISBN 0-89573-888-0; The Fraetal Geometry of Nature (revised edition) by B. B. Mandelbrot, 1983, W. H. Freeman & Co., New York, 468p., ISBN 0-7167-1186-9; The Science of Fractal Images by H.-O. Peitgen and D. Saupe, Springer-Verlag, New York, 312p., ISBN 0-38796608-0. Since the publication of Benoit Mandelbrot's (1975, 1977) classic work on fractal geometry there have been numerous books written on the subject. These range from coffee table volumes that contain rich illustrations of various sets, to rigorous mathematical texts. Although the original intent of this review was to focus on one particular book, as a fractal-practitioner I decided to review several relatively new releases. The selection is not exhaustive and reflects a survey of topics covered in general texts likely to be of use to physical scientists, rather than a critique of works devoted to specialized applications. Unfortunately there is no ideal definition of a fractal, however concepts such as self-similar and scale-invariance go a long way to developing an intuitive understanding. Clearly, The Fractal Geometry of Nature (Mandelbrot, 1983) made the subject widely accessible. The book is nothing if not eclectic and inspiring. It is a compendium of mathematics, science, conjecture, history, and etymology all wrapped up into what Mandelbrot terms a "manifesto". It has some magnificent and now famous fractal landscapes mainly generated by Mandelbrot's colleague at IBM, Richard Voss. The early chapters contain a mathematical history of so-called pathological curves, that is continuous nondifferentiable functions and associated problems. The relationship between these well-known functions and the natural world is illuminated early in a discussion on Brownian motion and coastline lengths. Further chapters go on to discuss scaling relationships of many natural objects, random fractals, the relationship between fractals and complex dynamical systems, such as turbulence and numerous other topics throughout the sciences. The approach is not applied and the book requires a foundation in set-theory on behalf of the reader. Many recursive algorithms are described however code is neither
1065
listed for generating fractals nor measuring dimensionality. Although difficult to understand in places, I recommend those interested in the subject spend some time with this marvelous book. Fractals Everywhere, a recent contribution by Barnsley (1988) is a mathematics text based upon a course he gives to students at Georgia Institute of Technology. The book contains a rigorous treatment of set theory and dimensionality. Undoubtedly some of the theorems on transformations will be of great use to practicing earth scientists. Those interested in the generation of computer images will be intrigued by the chapters on iterated function systems (IFS). These show how affine transformations can be used to generate simple geometric constructs such as the Sierpinski triangle, through to complex landscapes. The book contains an explanation of the rules, algorithms, IFS codes, and BASIC programs for development of numerous images. Of interest to natural scientists is Chapter 6, "Fractal Interpolation", a section on curve fitting using fractals. The book is not concerned with random fractals and contains some annoying errors. I suspect it will be of great use to those involved with image generation, data compression, and set theory rather than the average computer-geoscientist. A book by Kaye (1989) of Laurentian University, Sudbury entitled A Random Walk Through Fractal Dimensions looks at a variety of fractal applications of particular use to material scientists. It contains a useful section on the application of fractal geometry and other techniques to the characterization of particle morphology. Chapter 4 entitled "Delinquent Coins and Staggering Drunks" is a lovely review of random processes. The remainder of the book contains interesting insights, from a fractal perspective, into a variety of subjects. Among them are, random growth processes, simulations, percolation, filters, fracture processes, and mixing. The Science of Fractal Images edited by Peitgen and Saupe has a wealth of information, explanations, computer codes, and applications. Chapter 1 by Voss is an extremely good account of the theory of fractals, and contains material on how to generate simple deterministic fractal sets, random fractal sets, and fractal landscapes. Details are given on several techniques to generate random fractals. Voss provides an unambiguous introduction to self similarity, scaling relationships, fractal dimension, self-affine, and random fractals. This chapter also contains a wonderful section on white noise, 1If noise, brown noise, and music. Not surprisingly his spectral analyses show the melody fluctuations of some modern composers' "music" is close to that of white noise! Chapter 2 by Saupe is a "how to" section that contains algorithms and code written in Pascal for generating random fractals. These can be "lifted" from the page and are useful. Chapter 3 by Devaney explores the relationship between dynamical systems, chaos, and fractals, from the H6non attractor to the Julia set. Chapter 4
1066
Reviews
by Peitgen continues the analysis of these types of sets, whereas Chapter 5 by Barnsley contains an analysis of IFS and fractal modeling similar to this text. The appendices contain more algorithms and code for both deterministic and random fractals, plus some remarkable photographs. A monograph by Feders simply entitled Fractals gives an excellent introduction to the subject and many related physical phenomena. Chapter 4 covers the viscous fingering process that results in branching patterns when one liquid is injected into another more viscous liquid. Feders reviews Hele-Shaw cell experiments, and simulations of viscous fingering using the diffusion limited aggregation (DLA) algorithm, and shows all are approximated by the Laplace equation. Other chapters discuss time series, scaling relation-
ships, and percolation from a fractal perspective. The chapter on multifractals will be useful to those working with spatially distributed data sets. REFERENCES
Mandelbrot, B. B., 1975, Les objets fractals: forme, hasard et dimension: Flammarian, Paris, 190 p. Mandelbrot, B. B., 1977, Fractals: form, chance and dimension: W. H. Freeman & Company, San Francisco, 365 p. Ottawa Carleton Geoscience Centre and Department of Geology The University of Ottawa 770 King Edward Street Ottawa, Ontario Canada KIN 6N5
A.D. FOWLER
Fractais Everywhere by M. Barnsley, Academic Press, San Diego, California, 394p., ISBN 0-12-079062-9; Fraetais by J. Feders, Plenum Press, New York, 183p., ISBN 0-306-42851-2; A Random Walk Through Fractal Dimensions by B. Kaye, 1989, VCH Publishers, New York, 421p., ISBN 0-89573-888-0; The Fraetal Geometry of Nature (revised edition) by B. B. Mandelbrot, 1983, W. H. Freeman & Co., New York, 468p., ISBN 0-7167-1186-9; The Science of Fractal Images by H.-O. Peitgen and D. Saupe, Springer-Verlag, New York, 312p., ISBN 0-38796608-0. Since the publication of Benoit Mandelbrot's (1975, 1977) classic work on fractal geometry there have been numerous books written on the subject. These range from coffee table volumes that contain rich illustrations of various sets, to rigorous mathematical texts. Although the original intent of this review was to focus on one particular book, as a fractal-practitioner I decided to review several relatively new releases. The selection is not exhaustive and reflects a survey of topics covered in general texts likely to be of use to physical scientists, rather than a critique of works devoted to specialized applications. Unfortunately there is no ideal definition of a fractal, however concepts such as self-similar and scale-invariance go a long way to developing an intuitive understanding. Clearly, The Fractal Geometry of Nature (Mandelbrot, 1983) made the subject widely accessible. The book is nothing if not eclectic and inspiring. It is a compendium of mathematics, science, conjecture, history, and etymology all wrapped up into what Mandelbrot terms a "manifesto". It has some magnificent and now famous fractal landscapes mainly generated by Mandelbrot's colleague at IBM, Richard Voss. The early chapters contain a mathematical history of so-called pathological curves, that is continuous nondifferentiable functions and associated problems. The relationship between these well-known functions and the natural world is illuminated early in a discussion on Brownian motion and coastline lengths. Further chapters go on to discuss scaling relationships of many natural objects, random fractals, the relationship between fractals and complex dynamical systems, such as turbulence and numerous other topics throughout the sciences. The approach is not applied and the book requires a foundation in set-theory on behalf of the reader. Many recursive algorithms are described however code is neither
1065
listed for generating fractals nor measuring dimensionality. Although difficult to understand in places, I recommend those interested in the subject spend some time with this marvelous book. Fractals Everywhere, a recent contribution by Barnsley (1988) is a mathematics text based upon a course he gives to students at Georgia Institute of Technology. The book contains a rigorous treatment of set theory and dimensionality. Undoubtedly some of the theorems on transformations will be of great use to practicing earth scientists. Those interested in the generation of computer images will be intrigued by the chapters on iterated function systems (IFS). These show how affine transformations can be used to generate simple geometric constructs such as the Sierpinski triangle, through to complex landscapes. The book contains an explanation of the rules, algorithms, IFS codes, and BASIC programs for development of numerous images. Of interest to natural scientists is Chapter 6, "Fractal Interpolation", a section on curve fitting using fractals. The book is not concerned with random fractals and contains some annoying errors. I suspect it will be of great use to those involved with image generation, data compression, and set theory rather than the average computer-geoscientist. A book by Kaye (1989) of Laurentian University, Sudbury entitled A Random Walk Through Fractal Dimensions looks at a variety of fractal applications of particular use to material scientists. It contains a useful section on the application of fractal geometry and other techniques to the characterization of particle morphology. Chapter 4 entitled "Delinquent Coins and Staggering Drunks" is a lovely review of random processes. The remainder of the book contains interesting insights, from a fractal perspective, into a variety of subjects. Among them are, random growth processes, simulations, percolation, filters, fracture processes, and mixing. The Science of Fractal Images edited by Peitgen and Saupe has a wealth of information, explanations, computer codes, and applications. Chapter 1 by Voss is an extremely good account of the theory of fractals, and contains material on how to generate simple deterministic fractal sets, random fractal sets, and fractal landscapes. Details are given on several techniques to generate random fractals. Voss provides an unambiguous introduction to self similarity, scaling relationships, fractal dimension, self-affine, and random fractals. This chapter also contains a wonderful section on white noise, 1If noise, brown noise, and music. Not surprisingly his spectral analyses show the melody fluctuations of some modern composers' "music" is close to that of white noise! Chapter 2 by Saupe is a "how to" section that contains algorithms and code written in Pascal for generating random fractals. These can be "lifted" from the page and are useful. Chapter 3 by Devaney explores the relationship between dynamical systems, chaos, and fractals, from the H6non attractor to the Julia set. Chapter 4
1066
Reviews
by Peitgen continues the analysis of these types of sets, whereas Chapter 5 by Barnsley contains an analysis of IFS and fractal modeling similar to this text. The appendices contain more algorithms and code for both deterministic and random fractals, plus some remarkable photographs. A monograph by Feders simply entitled Fractals gives an excellent introduction to the subject and many related physical phenomena. Chapter 4 covers the viscous fingering process that results in branching patterns when one liquid is injected into another more viscous liquid. Feders reviews Hele-Shaw cell experiments, and simulations of viscous fingering using the diffusion limited aggregation (DLA) algorithm, and shows all are approximated by the Laplace equation. Other chapters discuss time series, scaling relation-
ships, and percolation from a fractal perspective. The chapter on multifractals will be useful to those working with spatially distributed data sets. REFERENCES
Mandelbrot, B. B., 1975, Les objets fractals: forme, hasard et dimension: Flammarian, Paris, 190 p. Mandelbrot, B. B., 1977, Fractals: form, chance and dimension: W. H. Freeman & Company, San Francisco, 365 p. Ottawa Carleton Geoscience Centre and Department of Geology The University of Ottawa 770 King Edward Street Ottawa, Ontario Canada KIN 6N5
A.D. FOWLER