Fractals for the classroom. Part one: Introduction to fractals and chaos.

*(English)*Zbl 0746.58005
New York etc.: Springer-Verlag. xiv, 450 p., 289 Ill. (1992).

There are a lot of questions to ask and answer about chaos theory and fractal geometry, which seem to be two of the most interesting current areas of research. Historically, mathematical insights have never found such rapid acceptance and generated so much excitement in the public mind, as fractals and chaos did. They have literally captured the attention, enthusiasm and interest of a world-wide public.

Despite its title, the book under review is addressed to everyone who, even without much knowledge of technical mathematics, wants to understand the details of chaos theory and fractal geometry. In addition to offering the reader a broad view of the underlying notions behind fractals, chaos and dynamics, the authors also show how fractals and chaos relate both to each other and to many other aspects of mathematics as well as to natural phenomena. Special attention is given to the theory of fractal geometry, which has been proved to be a visual “language” that maps algorithms to “natural” shapes and structures, providing the teachers with a powerful and unique tool to illustrate both the dynamics of mathematics and its many interconnecting links.

It is worth to remind here that fractals and chaos have literally captivated the visual senses of a world-wide public by pictures of such power and singularity, that a collection of them became one of the most successful series of exhibitions ever sponsored by the Goethe-Institute. Alone at the venerable London Museum of Science, the exhibition “Frontiers of Chaos: Images of Complex Dynamical Systems” by H. Jürgens, H.-O. Peitgen, M. Prüfer, P. H. Richter and D. Saupe attracted more than 140,000 visitors.

The title of the present book, which comes in two volumes, reflects the authors’ intention to bring chaos theory and fractal geometry closer to an audience which is somehow involved with lessons or classes. Together with the companion materials consisting of four books on strategic classroom activities and lessons with interactive computer software, it will make an unparalleled package for educational mathematics.

From the beginning, in Chapter 1, “The Backbone of Fractals: Feedback and the Iterator”, the authors emphasize how closely related are chaos theory and fractal geometry. When we examine the development of a process over a period of time, we speak in terms used in chaos theory. When we are more interested in the structural forms which a chaotic process leaves in its wake, then we use the terminology of fractal geometry, which is really the geometry whose structures are what give order to chaos.

Although self-similarity seems to be a concept that hardly needs an explanation, and one would guess that the term has been around for centuries, it is in fact as young as only 25 years old. However, its significance for fractal geometry is so deep, that it constitutes the underlying theme of both Chapter 2, “Classical Fractals and Self- Similarity”, and Chapter 3, “Limits and Self-Similarity”.

In his famous paper on “How long is the coastline of Britain?”, published in 1967, Benoit Mandelbrot has introduced a revolutionary statement: he proved that for all practical purposes, typical coastlines do not have a meaningful length. One of the consequences — as discussed in Chapter 4, “Length, Area and Dimension: Measuring Complexity and Scaling Properties” — is that it is impossible to assign quantities such as length or surface area to natural shapes. A more appropriate characteristic would be to determine, for example, how irregular, how convoluted is a coastline, or, in other words, what is its fractal dimension?

While the elements of Euclidean geometry are basic visible forms such as lines, circles and spheres, the elements of fractal geometry do not lend themselves to direct observation, as they are in fact algorithms describing the complex forms found in nature. In Chapter 5, “Encoding Images by Simple Transformations”, the authors discuss one of the major features of the “language” of fractal geometry: it promises to provide a means to break down the forms of nature into basic elements (or “words”) which are primitive algorithms. In the mathematical terminology, these algorithms are known as “deterministic iterated function systems” (IFS).

The computational complexity of the image decoding problem when using IFS iteration is undoubtedly prohibitive. The so-called chaos game provides an alternative approach to this problem, and, as shown in Chapter 6, “The Chaos Game: How Randomness Creates Deterministic Shapes”, allows much more efficient solutions.

The problem of how to introduce randomness into the rigorously organized classical fractals as to generate realistic “natural” shapes is considered in Chapter 7, “Irregular Shapes: Randomness in Fractal Constructions”. This leads to physical, so-called percolatoin models, with applications ranging from the fragmentation of atomic nuclei to the formations of clusters of galaxies.

The book is written in a non-threatening style and each chapter can be read independently from the others. At the end of each chapter, the authors offer a short BASIC program, the “Program of the Chapter”, which is designed to highlight one of the major topics of the respective chapter.

While the present first part of the book is focusing more on fractals, the forthcoming second volume concentrates more on chaos phenomena. Explicitly, the chapters of the second part are:

8. Recursive Structures: Growing of Fractals and Plants;

9. Pascal’s Triangle, Cellular Automata and Attractors;

10. Deterministic Chaos: Sensitivity and Mixing;

11. Order and Chaos: Period Doubling and its Chaotic Mirror;

12. Julia Sets: Fractal Basin Boundaries;

13. The Mandelbrot Set: Ordering the Julia Sets.

There is no higher recognition for the achievements of the researchers from the Dynamical Systems Laboratory of the University of Bremen than the admiration that Benoit Mandelbrot himself expresses in the Foreword to the book under review.

Despite its title, the book under review is addressed to everyone who, even without much knowledge of technical mathematics, wants to understand the details of chaos theory and fractal geometry. In addition to offering the reader a broad view of the underlying notions behind fractals, chaos and dynamics, the authors also show how fractals and chaos relate both to each other and to many other aspects of mathematics as well as to natural phenomena. Special attention is given to the theory of fractal geometry, which has been proved to be a visual “language” that maps algorithms to “natural” shapes and structures, providing the teachers with a powerful and unique tool to illustrate both the dynamics of mathematics and its many interconnecting links.

It is worth to remind here that fractals and chaos have literally captivated the visual senses of a world-wide public by pictures of such power and singularity, that a collection of them became one of the most successful series of exhibitions ever sponsored by the Goethe-Institute. Alone at the venerable London Museum of Science, the exhibition “Frontiers of Chaos: Images of Complex Dynamical Systems” by H. Jürgens, H.-O. Peitgen, M. Prüfer, P. H. Richter and D. Saupe attracted more than 140,000 visitors.

The title of the present book, which comes in two volumes, reflects the authors’ intention to bring chaos theory and fractal geometry closer to an audience which is somehow involved with lessons or classes. Together with the companion materials consisting of four books on strategic classroom activities and lessons with interactive computer software, it will make an unparalleled package for educational mathematics.

From the beginning, in Chapter 1, “The Backbone of Fractals: Feedback and the Iterator”, the authors emphasize how closely related are chaos theory and fractal geometry. When we examine the development of a process over a period of time, we speak in terms used in chaos theory. When we are more interested in the structural forms which a chaotic process leaves in its wake, then we use the terminology of fractal geometry, which is really the geometry whose structures are what give order to chaos.

Although self-similarity seems to be a concept that hardly needs an explanation, and one would guess that the term has been around for centuries, it is in fact as young as only 25 years old. However, its significance for fractal geometry is so deep, that it constitutes the underlying theme of both Chapter 2, “Classical Fractals and Self- Similarity”, and Chapter 3, “Limits and Self-Similarity”.

In his famous paper on “How long is the coastline of Britain?”, published in 1967, Benoit Mandelbrot has introduced a revolutionary statement: he proved that for all practical purposes, typical coastlines do not have a meaningful length. One of the consequences — as discussed in Chapter 4, “Length, Area and Dimension: Measuring Complexity and Scaling Properties” — is that it is impossible to assign quantities such as length or surface area to natural shapes. A more appropriate characteristic would be to determine, for example, how irregular, how convoluted is a coastline, or, in other words, what is its fractal dimension?

While the elements of Euclidean geometry are basic visible forms such as lines, circles and spheres, the elements of fractal geometry do not lend themselves to direct observation, as they are in fact algorithms describing the complex forms found in nature. In Chapter 5, “Encoding Images by Simple Transformations”, the authors discuss one of the major features of the “language” of fractal geometry: it promises to provide a means to break down the forms of nature into basic elements (or “words”) which are primitive algorithms. In the mathematical terminology, these algorithms are known as “deterministic iterated function systems” (IFS).

The computational complexity of the image decoding problem when using IFS iteration is undoubtedly prohibitive. The so-called chaos game provides an alternative approach to this problem, and, as shown in Chapter 6, “The Chaos Game: How Randomness Creates Deterministic Shapes”, allows much more efficient solutions.

The problem of how to introduce randomness into the rigorously organized classical fractals as to generate realistic “natural” shapes is considered in Chapter 7, “Irregular Shapes: Randomness in Fractal Constructions”. This leads to physical, so-called percolatoin models, with applications ranging from the fragmentation of atomic nuclei to the formations of clusters of galaxies.

The book is written in a non-threatening style and each chapter can be read independently from the others. At the end of each chapter, the authors offer a short BASIC program, the “Program of the Chapter”, which is designed to highlight one of the major topics of the respective chapter.

While the present first part of the book is focusing more on fractals, the forthcoming second volume concentrates more on chaos phenomena. Explicitly, the chapters of the second part are:

8. Recursive Structures: Growing of Fractals and Plants;

9. Pascal’s Triangle, Cellular Automata and Attractors;

10. Deterministic Chaos: Sensitivity and Mixing;

11. Order and Chaos: Period Doubling and its Chaotic Mirror;

12. Julia Sets: Fractal Basin Boundaries;

13. The Mandelbrot Set: Ordering the Julia Sets.

There is no higher recognition for the achievements of the researchers from the Dynamical Systems Laboratory of the University of Bremen than the admiration that Benoit Mandelbrot himself expresses in the Foreword to the book under review.

Reviewer: D.Savin (Montreal)