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Indra鈥檚 Pearls by David Mumford, Caroline Series and David Wright, Cambridge, 拢29.95, ISBN 0521352533

FRACTALS. 鈥淭hat damn hippy graphic again?鈥� Thus artists of my acquaintance react, looking over my shoulder at a laptop screen proudly displaying the paisley-patterned extravaganzas that are the Mandelbrot set. Some mathematicians may concur: it seems there鈥檚 been an awful lot more random exploration of the infinite detail of these pictures than production of deep results.

Indra鈥檚 Pearls promises a lot more than this. David Mumford, Caroline Series and David Wright鈥攑rofessors of mathematics at Brown, Warwick and Oklahama State universities respectively鈥攐pen their consideration of a quite different family of fractals with the fundamental theory of symmetry groups. In passing they give the reader a sound introduction to the sort of topological questions that now crop up in cosmology.

And, though it鈥檚 always hard to be sure in mathematics, it seems likely that they鈥檙e on to something really interesting about automata, those logical machines that generate complexity using rules more fundamental than arithmetic. They may even pique your imagination on space-time.

This is a strange beast of a book: a popularisation of their mathematics aimed squarely at other mathematicians, or at least those who share a steely determination to become mathematicians. I recommend you acquire that determination, whether to explore the theory or to write the code to produce stunning images.