The science of fractal images. With contributions by Yuval Fisher and Michael McGuire. Ed. by Heinz-Otto Peitgen and Dietmar Saupe. (Based on notes for the course fractals - introduction, basics and pespectives given as part of the SIG-GRAPH ’87, Anaheim, Calif. course program).

*(English)*Zbl 0683.58003
New York etc.: Springer-Verlag. xiii, 312 pp. DM 69.00 (1988).

This is a book on fractals and computer graphics. The principal ingredients are pictures, many of them in color, pseudo-codes of algorithms for producing them, and discussions on mathematical theories behind them. The algorithms are of varying level, some for beginners, and some more advanced. And so is the presentation of the mathematical back- ground, but as a rule it avoids technical details.

Including the appendices, there are eight writers, each handling his own section. Although these sections are related and some of them are partly based on others, they can for the most part be read independently. This is good since the style and level of difficulty varies considerably from author to author, and so the reader can choose his or her own favorite parts. But to get a good idea what is going on in fractal images, one should at least browse through the whole book.

I shall briefly explain the contents of the individual parts. The foreword titled “People and events behind the ‘Science of Fractal Images”’ is written by B. B. Mandelbrot. People range from Archimedes, Poincaré and others to the authors of the book, the events touch for example the beginning of fractals, modelling clouds, fractals in films, and the creation of the “Geometry Supercomputer Project”.

The first chapter “Fractals in nature: From characterization to simulation” is due to R. V. Voss. He introduces the concepts of self- similarity and fractal dimension, and discusses them in the context of fractal landscapes such as coastlines, mountains, and clouds. A large portion is devoted to random fractals coming from Brownian motion, and its generalizations fractional Brownian motions, which have different scaling exponents. The role of these processes in computer graphics is discussed, and some approximation and simulation methods are briefly presented. They include approximation with finitely many jumps, random midpoint displacement, and fast Fourier transform. Other topics that are considered in the connection of fractional Brownian motion are selfsimilarity and -affinity, methods for measuring fractal dimension, and relations between scaling exponent, spectral exponent, and dimension.

In the second chapter “Algorithms for random fractals” D. Saupe concentrates on a topic that was already touched by Voss: some basic algorithms for producing pictures of random fractals. Again the standard and fractional Brownian motion are the underlying processes, both in one and several dimensions. The algorithms include numerical integration from white noise, random midpoint interpolation, approximation by step functions consisting of independent jumps, and approximation by spectral synthesis. In the last method one first produces the random Fourier coefficients, and then employs some algorithm for the inverse Fourier transform. Several pseudo-codes are presented and some mathematical back- ground and details are given, in a rather pleasant way, to explain the methods.

“Fractal patterns arising in chaotic dynamical systems” is written by R. L. Devaney. He gives a very accessible introduction to simple dynamical systems and iteration for anybody knowing the basic calculation rules with numbers. For real systems he starts with the logistic equation for the population growth; \(P_{n+1}=kP_ n(1-P_ n)\), and goes on to explain the unstable orbits, chaotic sets, and strange attractors in the light of the Hénon map. For the complex systems the Julia sets are studied both for quadratic polynomials and exponential functions.

Chapter 4 “Fantastic deterministic fractals” by H.-O. Peitgen continues with complex dynamical systems. It gives several algorithms, and back- ground in more advanced mathematics. It begins by giving a detailed dynamical description of Julia sets for quadratic polynomials, and discusses algorithms for picturing them. A large part is devoted to the Mandelbrot set and very elaborate complex analytic methods behind its simulation. Also some other questions like zoom animation are discussed.

M.-F. Barnsley has written the last chapter “Fractal modelling of real world images”. The starting point is J. E. Hutchinson’s concept of an invariant set: For any contractions \(w_ 1,...,w_ N: {\mathbb R}^ n\to {\mathbb R}^ n\) there is a unique “attractor” A such that \(A=w_ 1(A)\cup...\cup w_ N(A)\). Given a geometric form, like a leave or a landscape, the idea is to find \(w_ 1,...,w_ N\) so that the attractor A provides a good approximation. In order to get more realistic pictures, probabilities \(p_ i\) are attached to the maps \(w_ i\). The maps are then assumed to be contractive only in the average; \(s_ 1^{p_ 1}...s_ N^{p_ N}<1\), where \(s_ i\) is the Lipschitz constant of \(w_ i\). Related to \(\{w_ i,p_ i\}\) there is again an attractor A, and also a measure, which is responsible for the coloring. Barnsley’s essay explains this method, gives algorithms, and illustrates with examples.

There are four appendices. The first “Fractal landscapes without creases and with rivers” by B. B. Mandelbrot discusses three defects in fractal forgeries of landscapes, and new solutions to them. The second “An eye for fractals” by M. McGuire gives a photographer’s view on fractals with brief comments and nine photographs. The third appendix by D. Saupe is called “A unified approach to fractal curves and plants”, and it introduces simple algorithms for producing von Koch and other fractal curves. They are based on Lindenmayer’s L-systems. The last appendix “Exploring the Mandelbrot set” by Y. Fisher describes an algorithm to generate black and white images of the Mandelbrot set very quickly.

Including the appendices, there are eight writers, each handling his own section. Although these sections are related and some of them are partly based on others, they can for the most part be read independently. This is good since the style and level of difficulty varies considerably from author to author, and so the reader can choose his or her own favorite parts. But to get a good idea what is going on in fractal images, one should at least browse through the whole book.

I shall briefly explain the contents of the individual parts. The foreword titled “People and events behind the ‘Science of Fractal Images”’ is written by B. B. Mandelbrot. People range from Archimedes, Poincaré and others to the authors of the book, the events touch for example the beginning of fractals, modelling clouds, fractals in films, and the creation of the “Geometry Supercomputer Project”.

The first chapter “Fractals in nature: From characterization to simulation” is due to R. V. Voss. He introduces the concepts of self- similarity and fractal dimension, and discusses them in the context of fractal landscapes such as coastlines, mountains, and clouds. A large portion is devoted to random fractals coming from Brownian motion, and its generalizations fractional Brownian motions, which have different scaling exponents. The role of these processes in computer graphics is discussed, and some approximation and simulation methods are briefly presented. They include approximation with finitely many jumps, random midpoint displacement, and fast Fourier transform. Other topics that are considered in the connection of fractional Brownian motion are selfsimilarity and -affinity, methods for measuring fractal dimension, and relations between scaling exponent, spectral exponent, and dimension.

In the second chapter “Algorithms for random fractals” D. Saupe concentrates on a topic that was already touched by Voss: some basic algorithms for producing pictures of random fractals. Again the standard and fractional Brownian motion are the underlying processes, both in one and several dimensions. The algorithms include numerical integration from white noise, random midpoint interpolation, approximation by step functions consisting of independent jumps, and approximation by spectral synthesis. In the last method one first produces the random Fourier coefficients, and then employs some algorithm for the inverse Fourier transform. Several pseudo-codes are presented and some mathematical back- ground and details are given, in a rather pleasant way, to explain the methods.

“Fractal patterns arising in chaotic dynamical systems” is written by R. L. Devaney. He gives a very accessible introduction to simple dynamical systems and iteration for anybody knowing the basic calculation rules with numbers. For real systems he starts with the logistic equation for the population growth; \(P_{n+1}=kP_ n(1-P_ n)\), and goes on to explain the unstable orbits, chaotic sets, and strange attractors in the light of the Hénon map. For the complex systems the Julia sets are studied both for quadratic polynomials and exponential functions.

Chapter 4 “Fantastic deterministic fractals” by H.-O. Peitgen continues with complex dynamical systems. It gives several algorithms, and back- ground in more advanced mathematics. It begins by giving a detailed dynamical description of Julia sets for quadratic polynomials, and discusses algorithms for picturing them. A large part is devoted to the Mandelbrot set and very elaborate complex analytic methods behind its simulation. Also some other questions like zoom animation are discussed.

M.-F. Barnsley has written the last chapter “Fractal modelling of real world images”. The starting point is J. E. Hutchinson’s concept of an invariant set: For any contractions \(w_ 1,...,w_ N: {\mathbb R}^ n\to {\mathbb R}^ n\) there is a unique “attractor” A such that \(A=w_ 1(A)\cup...\cup w_ N(A)\). Given a geometric form, like a leave or a landscape, the idea is to find \(w_ 1,...,w_ N\) so that the attractor A provides a good approximation. In order to get more realistic pictures, probabilities \(p_ i\) are attached to the maps \(w_ i\). The maps are then assumed to be contractive only in the average; \(s_ 1^{p_ 1}...s_ N^{p_ N}<1\), where \(s_ i\) is the Lipschitz constant of \(w_ i\). Related to \(\{w_ i,p_ i\}\) there is again an attractor A, and also a measure, which is responsible for the coloring. Barnsley’s essay explains this method, gives algorithms, and illustrates with examples.

There are four appendices. The first “Fractal landscapes without creases and with rivers” by B. B. Mandelbrot discusses three defects in fractal forgeries of landscapes, and new solutions to them. The second “An eye for fractals” by M. McGuire gives a photographer’s view on fractals with brief comments and nine photographs. The third appendix by D. Saupe is called “A unified approach to fractal curves and plants”, and it introduces simple algorithms for producing von Koch and other fractal curves. They are based on Lindenmayer’s L-systems. The last appendix “Exploring the Mandelbrot set” by Y. Fisher describes an algorithm to generate black and white images of the Mandelbrot set very quickly.

Reviewer: P. Mattila

##### MSC:

37-02 | Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory |

37F45 | Holomorphic families of dynamical systems; the Mandelbrot set; bifurcations (MSC2010) |

68U05 | Computer graphics; computational geometry (digital and algorithmic aspects) |

28A80 | Fractals |

00A69 | General applied mathematics |

37D45 | Strange attractors, chaotic dynamics of systems with hyperbolic behavior |

60J65 | Brownian motion |

##### Keywords:

Fractals; Anaheim, CA (USA); pseudo-code algorithm; Julia set; fractals; computer graphics; Brownian motion; Mandelbrot set
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\textit{M. F. Barnsley} et al., The science of fractal images. With contributions by Yuval Fisher and Michael McGuire. Ed. by Heinz-Otto Peitgen and Dietmar Saupe. (Based on notes for the course fractals - introduction, basics and pespectives given as part of the SIG-GRAPH '87, Anaheim, Calif. course program). New York etc.: Springer-Verlag (1988; Zbl 0683.58003)