Fractals and chaos. The Mandelbrot set and beyond, by Benoit B. Mandelbrot. Pp. 308. £38.50. 2004, ISBN 0 387 20158 0 (Springer-Verlag).

Fractals and chaos. The Mandelbrot set and beyond, by Benoit B. Mandelbrot. Pp. 308. £38.50. 2004, ISBN 0 387 20158 0 (Springer-Verlag). Fractals, fractal geometry or chaos theory have been a hot topic in scientific research. It may come as a surprise that much of the theory as we know it was initiated during the last 30 years and by the vision of one man: Benoit Mandelbrot. The story starts in 1975 with Mandelbrot's small booklet Les objets fractals. Forme, hasard et dimension (published by Flammarion, Paris). Already in 1977 it was translated and expanded into Fractals: Form, Chance and Dimension (W. H. Freeman, San Francisco), but the breakthrough came in 1980 with the first picture of the Mandelbrot set in 'Fractal aspects of the iteration of z >-» Xz (1 z) for complex X and z', Ann. New York Acad. Sci. and reprinted on pp. 37—51 in the present volume. It was, however, Mandelbrot's 1982 masterpiece The Fractal Geometry of Nature (W. H. Freeman, New York) that popularized the subject. Mandelbrot's book is a scientific, philosophic and pictorial treatise at the same time and it is one of the rare specimen of serious mathematics books that can be read and re-read at many different levels. The volume under review is 'Selecta C of Mandelbrot's oeuvre. It is a selection of papers which appeared between 1980 and 2003, dealing with (non-)quadratic rational dynamics, iterated (nonlinear) function systems and multifractal measures. Alongside some important and very technical original papers, there is a highly readable (also for the non-specialist) introduction and survey-type original contributions, extracts from his 1982 monograph as well as unpublished material. The last chapter is devoted to a brief historical account of the subject's early heroes: Pierre Fatou and Gaston Julia. Rather than being a juxtaposition of papers, Mandelbrot succeeded in creating a readable selection of material which contains new original contributions. The papers featured in the book are sometimes corrected and annotated; that in this process the original pagination was lost is somewhat unfortunate. The style is what one could call 'truly Mandelbrotian', a mixture of hard science, often with a personal touch, some (sometimes quite lop-sided) personal notes and recollections and always the urge to convey a message.