In The Computational Beauty of Nature Flake has produced a guide to computational modeling which doesn't really fit either the textbook or the popular science mould, but which combines some of the best elements of both genres. It isn't a replacement for more technical works, but it could be used as the skeleton for an introductory undergraduate course; it will also be a great starting place for the amateur, fascinated perhaps by Mandelbrot sets and after something more substantial.

As the subtitle suggests, The Computational Beauty of Nature is built around the five pillars of Computation, Fractals, Chaos, Complex Systems, and Adaptation. An amazing amount of material is packed in, though it surprisingly doesn't result in clutter. The section on computation introduces Cantor diagonalisation, Turing machines, a cut-down Lisp (Stutter), recursively enumerable functions, and Gödel's theorem; that on fractals, self-similarity and fractal geometry, L-systems and affine transformation fractals, and Mandelbrot and Julia sets; that on chaos, simple non-linear dynamics, strange attractors, prey-predator systems, and chaos control; that on complex systems, cellular automata (including the Game of Life), autonomous agents, game theory and the Prisoner's Dilemma, and neural networks; that on adaptation, genetic algorithms, classifier systems, and more neural networks (perceptrons and back-propagation).

Flake also connects his core pillars: on them rest rafters forming a pentagon (Figure 1.1). These are labeled recursion, incomputability, strange attractors, phase transitions, cellular automata, growth models, self-organization, emergence, hierarchical models, and coadaptation. Each of the five areas has its own roof (a more philosophical postscript), but Flake doesn't really get embroiled in overarching philosophical issues until it comes to the lightweight central roof (the epilogue), which is made up of largely ornamental analogies.

While he hasn't really got the space to fit in more than an outline of all these topics, Flake still manages to treat them at a level that combines accessibility with real substance — enough to be interesting to me, though I found little that was actually new. He has a knack for succinct but clear explanations that should be accessible to those with no more than school mathematics. The mathematics is explained as it is needed, but rather concisely, with just a page or two for each of calculus, linear algebra and affine transformations, complex numbers, vector calculus, matrix algebra, and so forth. A really gifted novice might enjoy this, but for most it will be best used as a refresher. Flake says that he wrote The Computational Beauty of Nature for himself at a younger age, and I would certainly have been ecstatic over a copy when I was at school.

The Computational Beauty of Nature is not as austere as this might suggest. It is fundamentally about simulation and comes with an array of almost forty programs which illustrate its contents; instructions on using these are scattered throughout. The Computational Beauty of Nature is perfectly usable without these programs, but they make some things much clearer and are also a lot of fun to play with. They are available, along with source code, on the web.

All told, in The Computational Beauty of Nature Flake has created a truly splendid edifice, a genuine "labour of love". And only a few proof-reading slip-ups mar otherwise attractive construction by the MIT Press.

February 1999

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