
This is a review of the book The Colours of Infinity (ed. Nigel LesmoirGordon), written for the online reviews of the Mathematical Association of America.
This is a strikingly oldfashioned book, full of swirling colour graphics of the Mandelbrot set, demonstrations that fractals can model mountains, and hyperbole such as the description of the Mandelbrot set as "the thumbprint of God". It is an outgrowth of a British television documentary from 1994 (which goes some way towards explaining the retro feel), presented by the science fiction author Arthur C. Clarke. Each chapter is an exposition of some aspect of fractals. The list of contributors is described by Clarke, with spectacular inaccuracy, as "all the leading names in the fractal geometry field". There are sober chapters and there are feverish chapters, the most feverish being by Benoît Mandelbrot and by Clarke himself, who calls the Mandelbrot set "the most extraordinary discovery in the history of mathematics". It is not a book for mathematicians. I would even suggest that, in a small way, it could be actively harmful to mathematicians who have had no professional involvement with fractal geometry: the lavish graphics and wild overstatements add weight to the unfortunate perception that fractals are little more than a gimmick, that the mathematics behind them is not profound. This perception is absolutely false. But many mathematicians seem to believe it, in an unthinking way, as a result of the way in which fractals have been popularized. For example, every mathematician knows that the Mandelbrot set has something to do with iterating the map that sends z to z^{2}+c. But how many know that an appropriate general setting is the iteration of a holomorphic selfmap of a Riemann surface? (An important special case is the Riemann sphere C∪{∞}, when the map is a rational function over the complex numbers. A special case of that is when the map is a polynomial, and a special case of that is when it is a quadratic z^{2}+c.) The case that everyone knows about is actually just one small part of a substantial theory, deeply rooted in complex analysis and hyperbolic geometry. Or take complex dynamics, the branch of mathematics to which the Mandelbrot set belongs. Everyone knows about Mandelbrot's contribution — but how many know that some of the most important contributions to the subject have been made by John Milnor and Bill Thurston? Even the weight of those eminent names does not seem to have counterbalanced the negative perception. Indeed, some researchers whose work involves fractals deliberately avoid using that tainted fword. If you can ignore all that, there are some things to enjoy. Ian Stewart is reliably excellent on applications. He tells us, for instance, that just before a rockfall in a mine, the pattern of tiny sounds in the rock undergoes a telltale change in fractal dimension. (But even Stewart lets himself get carried away by the general giddy tone: having carefully explained that just as there are no perfect spheres in nature, there are no perfect fractals, he spoils it by declaring that "the universe is full of fractals. Indeed, it may even be one.") Will Rood's chapter says the most about complex dynamics. Gary Flake and David Pennock do a good job of explaining the selfsimilarity that has arisen spontaneously in the structure of the internet. Their chapter, like Stewart's, is both pleasant to read and genuinely informative. The only reason I can think of for a mathematician to obtain this book is if they were preparing a talk on fractals for the general public. Even then, I would question the speaker's choice of subject, which has already been popularized far out of proportion to its importance in contemporary mathematics. Continuing to tread this very wellworn path may not be doing any favours to mathematics, and it is a mixed blessing, at best, for those whose work involves the serious and sober study of fractal spaces.
Tom Leinster is a mathematician at the University of Glasgow, Scotland. He works on category theory and its many applications: most recently, to selfsimilar and recursivelydefined objects, to metric geometry, and to the measurement of biodiversity. He is a host of The nCategory Café.
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