Magic numbers

Finding Moonshine: A Mathematician's Journey Through Symmetry by Marcus du Sautoy 376pp, Fourth Estate, £18.99

This is a marvellous account of a 4,000-year obsession with symmetry and the secret language of nature: the pentagonal structure of the honeysuckle, the hexagons of the honeycomb, the perfectly spherical symmetry of the raindrop. Underlying the beauty and efficiency of nature, for mathematicians, are perplexing riddles. What is symmetry? How many symmetries can there be? Are there underlying laws that govern their kind and number? Are they visible manifestations of much deeper mathematical truth? Does symmetry hold a mirror up to the universe, and if so, how could you ever prove this to anyone's satisfaction?

After covering the Babylonians, Greeks and Romans, Marcus du Sautoy introduces a cast of characters who seem to fulfil the layman's expectations of mathematicians: madmen with cruel afflictions, odd personal habits and unhappy destinies, such as Niccolo Fontana of Brescia, known as Tartaglia. Having confected a formula for solving cubic equations, he was pursued, bribed and tormented for the answer by a mathematically-obsessed Renaissance gambler who let slip a hint to a youth called Ferrari. Tartaglia was then humiliated by Ferrari in a public mathematical contest and died in wretched poverty. The victor picked up a plum job and was subsequently poisoned for his inheritance by his sister. The gambler committed suicide.

Such stories read like Italian romances, but the lesson of Finding Moonshine is that mathematicians are driven men, and often driven to despair. Even after Renaissance mathematicians cracked the secret of cubic and quartic equations, quintic equations resisted solution for 250 years. Niels Abel, the hungry son of Norwegian alcoholics, picked up the challenges left by Tartaglia and Ferrari: he established something different, that there was no magic formula for quintic equations. He was too poor to pay for more than six printed pages of proof, and banked all on sending his abbreviated solution to Cauchy, a French academician. Cauchy, alas, had his own problems, emotional as well as mathematical. The starving Abel went back to Norway and died at 26, on the very day someone wrote to tell him his name was made, and worth a professorship in Berlin.

This rich, readable, sometimes demanding book is in part a personal journey for Du Sautoy and his family, in part an exposition of the passion for patterns that characterises all mathematicians. It confirms the suspicion that they really are a different race: Du Sautoy's colleagues are caught up in an eccentric pursuit, and - to put it delicately - are inclined to conduct themselves eccentrically.

But these exasperating people with unkempt beards, untidy clothes, impossible manners and extraordinary obsessions, become, in this narrative, ultimately enviable. Most of us cannot comprehend the world as it is: Du Sautoy's crazed calculators and driven autodidacts have gone far beyond the three dimensions of physical geography and now inhabit a world that has no physical existence at all, a world that nevertheless includes all possible universes, and firmly and forever eliminates the impossible ones. They may do things the rest of us would not do - memorise rail timetables, scribble their reasoning on rolls of wallpaper, calculate to more places than there are atoms in the solar system - but they have made a one-way trip to eternity, or at least to some eternal truths.