Fractals for the classroom. Part 2: Complex systems and Mandelbrot set.

*(English)*Zbl 0785.58001
Berlin etc.: Springer-Verlag. XII, 500 p. (1992).

This book contains an introduction to mathematical theories which are closely related to fractals (sets of fractional dimensions) and where fractal sets come from. This beautifully written second part of the two- volume set [for Part 1 see (1992; Zbl 0746.58005)] reveals many interconnecting links of mathematics. The reader will certainly gain from the encyclopedic knowledge of the authors expressed in a simple and attractive form.

The book contains very fine and intelligent mixture of theoretical mathematical developments, philosophical aspects, numerical results, computer experiments and physical concepts and explanations giving them practical meaning. Computer implementations are emphasized (up to discussions of round-off errors and computer arithmetics) and each chapter is accompanied by a simple BASIC program, which enlightens theoretical facts discussed there.

The results presented range from well-known old mathematical theories (like divisibility of natural numbers) to very recent papers and preprints. Many photographs, reproductions of original papers give to the reader the feeling of the participation in the process of creation of mathematical theories.

The structure and fascinating design of the book follow Part One (so this Part Two starts with Chapter 8). The organization of material is perfect and can be described as follows.

Chapter 8 deals with \(L\)-systems (which originally have been used to model purely discrete mechanisms of cells reproduction in biology). These recursive structures serve as a language for modelling growth and can be used to produce fractal sets.

Chapter 9 reveals beautiful relations between divisibility properties of elements of the Pascal triangle, cellular automata (like the famous LIFE game) and fractal sets.

Chapter 10 explores a simple quadratic iterator \(x\mapsto ax(1-x)\) to describe differences between stable, periodic and chaotic behavior of dynamical systems. Notions like Ljapunov exponent or ergodic properties are explained on simple examples. Also the relations to binary expressions of real numbers together with computer arithmetics are discussed.

Chapter 11 continues the study of the quadratic iterator. Now it is used to explain how chaos and order can be described as particular cases of a single law. This is made clear by means of so-called Feigenbaum’s diagram and discussion of the universal critical parameter (the Feigenbaum constant) which characterizes the transition from order to chaos.

Chapter 12 is devoted to strange attractors, which have been remaining a hot topic for twenty years. They are introduced through systems of differential equations. Also several classical strange attractors (like Hénon or Lorenz) and their physical meaning are discussed.

Chapter 13 brings Julia sets which appear in the study of complex dynamical systems through iterations of simple functions of complex variables. In turn, they are related to physical studies of equipotential surfaces.

Chapter 14 deals with the Mandelbrot set, which was actually the first widely known fractal set (perhaps, after the Cantor set or the Sierpiński gasket) and which was the source of the further rapid development of the subject. It is explained how a simple equation can produce the unlimited varieties of pictures.

The tools used in the book make it accessible for first-year students. At the same time it provides a rapid and inspiring introduction to the theory of dynamical systems. All in all, the reading of “Fractals for the Classroom” gives an impression of an interesting talk in a good company.

The book contains very fine and intelligent mixture of theoretical mathematical developments, philosophical aspects, numerical results, computer experiments and physical concepts and explanations giving them practical meaning. Computer implementations are emphasized (up to discussions of round-off errors and computer arithmetics) and each chapter is accompanied by a simple BASIC program, which enlightens theoretical facts discussed there.

The results presented range from well-known old mathematical theories (like divisibility of natural numbers) to very recent papers and preprints. Many photographs, reproductions of original papers give to the reader the feeling of the participation in the process of creation of mathematical theories.

The structure and fascinating design of the book follow Part One (so this Part Two starts with Chapter 8). The organization of material is perfect and can be described as follows.

Chapter 8 deals with \(L\)-systems (which originally have been used to model purely discrete mechanisms of cells reproduction in biology). These recursive structures serve as a language for modelling growth and can be used to produce fractal sets.

Chapter 9 reveals beautiful relations between divisibility properties of elements of the Pascal triangle, cellular automata (like the famous LIFE game) and fractal sets.

Chapter 10 explores a simple quadratic iterator \(x\mapsto ax(1-x)\) to describe differences between stable, periodic and chaotic behavior of dynamical systems. Notions like Ljapunov exponent or ergodic properties are explained on simple examples. Also the relations to binary expressions of real numbers together with computer arithmetics are discussed.

Chapter 11 continues the study of the quadratic iterator. Now it is used to explain how chaos and order can be described as particular cases of a single law. This is made clear by means of so-called Feigenbaum’s diagram and discussion of the universal critical parameter (the Feigenbaum constant) which characterizes the transition from order to chaos.

Chapter 12 is devoted to strange attractors, which have been remaining a hot topic for twenty years. They are introduced through systems of differential equations. Also several classical strange attractors (like Hénon or Lorenz) and their physical meaning are discussed.

Chapter 13 brings Julia sets which appear in the study of complex dynamical systems through iterations of simple functions of complex variables. In turn, they are related to physical studies of equipotential surfaces.

Chapter 14 deals with the Mandelbrot set, which was actually the first widely known fractal set (perhaps, after the Cantor set or the Sierpiński gasket) and which was the source of the further rapid development of the subject. It is explained how a simple equation can produce the unlimited varieties of pictures.

The tools used in the book make it accessible for first-year students. At the same time it provides a rapid and inspiring introduction to the theory of dynamical systems. All in all, the reading of “Fractals for the Classroom” gives an impression of an interesting talk in a good company.

Reviewer: I.S.Molchanov (Freiberg)