Fractals are very popular, even outside the mathematical community, mostly because of their aesthetic qualities and their relations with chaos theory. Accordingly, there are many expository books on fractals addressed to a non-mathematical audience. Edgar’s book is aimed instead at providing a rigorous introduction to the subject. Prerequisites are a basic understanding of topology and, I think, of measure theory (even if there is a whole chapter devoted to the fundamentals of the Lebesgue measure, it is utilitarian and it is probably more useful as a quick reference than as a ‘crash course’).
What is a fractal? Actually, there are several definitions of fractals in the literature. The founder of the theory, the French mathematician Benoit Mandelbrot, originally defined a fractal as ‘a set whose Hausdorff dimension exceeds the topological dimension’. However, later he expressed some reserves about this definition and it is not the one chosen by the author (to be found at page 177). The term fractal was coined in 1975 by Mandelbrot, who published his ideas in Les objets fractals, forme, hasard et dimension. In 1982 Mandelbrot expanded and updated his ideas in The Fractal Geometry of Nature, a very influential work which popularized the theory.
Both Hausdorff and topological dimensions are explained in depth by the author. There are many ways to define the dimension of a set, and this book provides a thorough introduction to dimension theory. The topological dimension of a topological space is defined to be the minimum value of n, such that every open cover has a refinement in which no point is included in more than n+1 elements. The Hausdorff dimension gives another way to define dimension, which takes the metric into account. Fractals often are spaces whose Hausdorff dimension strictly exceeds the topological dimension. For example, the famous Cantor set has topological dimension 0, but has Hausdorff dimension equal to log2/log3.
The fractal illustrated below is known as the Sierpinski triangle.
A triangle is two-dimensional, but the ‘Sierpinski sieve’ (i.e. deleting an infinite sequence of sub-triangles according to a recursive rule, as shown in the picture) produces an ‘infinitely porous’ object, and this suggests that it should have a dimension between 1 and 2 (in fact, its Hausdorff dimension is equal to log3/log2). The pictures shows another essential geometrical feature of the fractals: the self-similarity (i.e. at any magnification there is a smaller piece of the object that is similar to the whole).
The first chapter of the book introduces several classical fractal sets in a direct and constructive way, making its reading accessible to everyone; technical matters are postponed to later chapters. A special emphasis is put on the self-similarity of fractals. The second chapter contains classical material on the topology of metric spaces and most mathematicians will be able to skip it. The fourth chapter, on self-similarity, can be read right after the second chapter, since it does not require any result from dimension theory, which is the object of the third chapter. The fifth chapter provides a short introduction to measure theory, a necessary background in order to understand the next chapter on Hausdorff dimension.
For some of the fractal sets presented in the book, the author includes the Logo programs that can be used to draw them (Logo is a programming language created in 1967 for educational use).
Each chapter concludes with a section of remarks, containing additional references, historical facts and suggestions for course instructors. Also, several exercises are proposed, at various levels of difficulty.
Summing up: as a non-specialist, I found this book very helpful. It gave me a better understanding of the nature of fractals, and of the technical issues involved in the theory. I think it will be valuable as a textbook for undergraduate students in mathematics, and also for researchers wanting to learn fractal geometry from scratch. The material is well-organized and the proofs are clear; the abundance of examples is an asset for a book on measure theory and topology.
Fabio Mainardi earned a PhD in Mathematics at the University of Paris 13. His research interests are Iwasawa theory, p-adic L-functions and the arithmetic of automorphic forms. He may be reached at firstname.lastname@example.org.