Fractals for laymen. (Q2722592)

There are two notions which are crucial in the theory of (either discrete or continuous) dynamical systems, namely ``fractal'' and ``chaos''. Both notions have mathematically rigorous definitions: a fractal is a set whose Hausdorff dimension is strictly larger than its (topological) covering dimension, and a system is called chaotic if it is sensitive with respect to initial conditions, any two points may be joined by a trajectory, and periodic orbits are dense. Of course, in view of the importance of these notions it is not surprising that nowadays there exists a vast mathematical literature, also in book form, on these topics.NEWLINENEWLINENEWLINEOn the other hand, it is hardly exaggerated to say that the notions of fractals and chaos are among the most improperly used ``vogue words'' in mass media and small-talks on parties. Among politicians, bank managers, businessmen, and stockbrokers there are lots of selfstyled ``specialists'' who see chaos and fractal structure everywhere, even where there is nothing similar. Unfortunately, the average citizen has not the slightest idea of what fractal or chaos could really mean, simply because there is a deplorable lack of corresponding literature on an elementary level. So it would be highly desirable to have some textbook which explains these concepts with the necessary mathematical rigour, on the one hand, but is accessible also to non-mathematicians with a minimal background, on the other.NEWLINENEWLINENEWLINEAs the title suggests, the book under review aims at filling this gap. It consists of 5 chapters with the following headings and subheadings: 1. The concept of fractal (The nature of fractals, classical examples, the dimension, the game of chaos, iterations); 2. The Mandelbrot set \(M\) (Generation of \(M\), anatomy of \(M\), generalizations of \(M\), mathematical software to create \(M\)); 3. Population dynamics (Introduction, the mathematical modelling, graphical analysis of the logistic equation, orbit diagrams, the Feigenbaum constant); 4. \(L\)-systems (The process, ramifications); 5. Iterated systems of functions (A historical tour d'horizon, iterated systems of functions, mathematical interlude). At the end the author adds a brief more ``folkloristic'' chapter on fractals and arts including, for example, M. C. Escher's well-known visualizations of the Poincaré model of hyperbolic geometry.NEWLINENEWLINENEWLINEThe author has done an excellent job. He guides the reader into the true heart of fractal structures, without being mathematically too presumptious, but also without trivializing the ideas. The more instructed and dauntless reader can find exact mathematical definitions and results in several ``intermezzos matemáticos'' which are scattered over the 5 chapters and distinguished from the remaining text by a different typesetting. Moreover, the interested reader may find a lot of additional material for further reading in the quite detailed list of references. There is also a collection of software sources for passionates of computer visualization. The only flaw concerns several misprints and oversights in the mathematical parts (for example, Definitions are offen denoted as Theorems, the formula on p. 4 and the diagram on p. 40 are inexact, and some other places) which should be removed in subsequent editions, if there are any.NEWLINENEWLINENEWLINESummarizing, this small book is certainly of interest not only to students in mathematics, physics, and natural sciences, but also to ``laymen'' who want to get some serious information about the fascinating concepts of fractals and chaos beyond the usual ``twaddle''. It would be nice to have an English translation available in order to provide the book with the larger readership it deserves.