Finding Moonshine, by Marcus du Sautoy

A monster hit for the maths world
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Although authors of popular books on mathematics often resort to the sublime in order to seduce the un-mathematical hoi polloi, the poetic metaphor in Marcus du Sautoy's title is not his own. "Moonshine" is a technical term first employed by mathematicians three decades ago to refer to suspicions of a profound relationship between the totally unrelated, as they were thought until then, fields of group theory and the theory of numbers.

Moonshine developed from suspicion to a precise conjecture, whose spectacular proof won its perpetrator, British mathematician Richard Borcherds, a Fields Medal: the mathematician's Nobel. Like many discoveries, it began with a chance encounter. One day in 1978, group theorist John McKay was flipping through a paper in number theory, when a particular number attracted his attention, as it was but one unit over 196,883; ie, the smallest amount of dimensions in which his his own field's newest star, a humongous collection of transformations affectionately called "the Monster", can exist. Non-mathematicians, who need to add, à la Einstein, the notion of time to the Euclidean triad of length, width and height, in order to get some intuitive sense of four-dimensional space, can have fun figuring out names for the 196,879 additional dimensions.

McKay was intrigued to learn that 196,884 is a coefficient of something of crucial importance to number theorists, called the "modular function". He wrote to John Thompson, a giant of group theory, who investigated further and discovered to his amazement that the next coefficients of the modular function, the ungainly numbers 21,493,760 and 864,299,970 (not Finding Moonshine's 864,229,970) could be derived by adding up a few of the higher dimensions in which the Monster appeared. In a sense, this was like digging up an 8th-century BC site in Mexico and finding a clay tablet with the first hundred lines of the Iliad, written in proto-Yucatecan. There's no way this could be ascribed to coincidence.

Du Sautoy's book begins and ends with moonshine. The rest is a guided tour through group theory and the concept of symmetry, which motivated its birth and drives many of its advances. Two chapters are given to the prehistory of the field in Greek and Arabic culture, including a symmetry-hunting visit to the palace of Alhambra; two more explore symmetry in music and modern science, providing non-mathematical readers with a context for the harder stuff.

At the book's core is the mathematical story, which du Sautoy tells with the narrative flair and storyteller's sense of detail, development and suspense also exhibited in his first book, The Music of the Primes. The basic dilemma an author of a book of mathematical popularisation faces is whether to opt for making the subject more appealing to the non-expert, or to cater to the appetites of the mathematician manqué. The dilemma usually boils down to more or less narrative or, inversely, less or more mathematics. Du Sautoy has moved closer to the hard side of this dichotomy since his last book. Especially in the earlier parts of Finding Moonshine, he gets into considerable detail, explaining the concepts, working through examples. Readers looking for the storytelling of a book like Simon Singh's Fermat's Last Theorem may become slightly frustrated unless they skip the more technical pages.

But those who read the book to find out more about mathematics will be amply recompensed. Avoiding the abstract notion of "group" for the intuitively clearer notion of symmetry, du Sautoy gives an excellent account of the early triumphs of group theory, centering on the discoveries of Évariste Galois. This preparation makes the reader more capable of enjoying the central action: the hunt for the theory's hidden protagonists, the "simple" groups.

The struggle for their classification involved almost a hundred mathematicians. Its complete form is renowned as the Leviathan of mathematical proofs, covering approximately 10,000 pages spread over a few dozen specialist journals. During the last stages of this mega-proof the Monster appeared. It's the biggest simple group of a sui generis variety measuring 808017424794512875886459904961710757005754368000000000 elements. That big!

Though different versions of this story have been well told in books by Mark Ronan and Ian Stewart, Finding Moonshine is unique. Du Sautoy, a professor at Oxford, is himself an important group theorist, active in groundbreaking research, and writes his book hopping back and forth from history and popularisation to autobiography and research diary. This makes his book an attractive two-in-one. Finding Moonshine is one of the few popular first-person accounts - I'd be hard-pressed to name a single other - from the frontiers of modern mathematics. Moving from the highs of illumination leading to new discoveries, to the lows of professional rivalries and peregrinations in intellectual labyrinths, it gives an inspiring testimony of what it is like to be a research mathematician.

The only blemish is an occasional sloppiness with non-mathematical material. Plato's great dialogue Timaeus deserves a neater reference than "his text Timeus"; any informed theatre-goer will cringe at reading that Peter Brook supports the elitist view of theatre, especially when substantiated by a quote from The Empty Space obviously referring to what Brook calls "holy theatre" – distinguished from his model.

Finally, it's a pity that du Sautoy goes for a giggle at the expense of St Augustine, when it is well known that his denigrating comments on mathematici refer to astrologers. If mathematicians want to be listened at more attentively by people on the other side of the cultural rift, they should accord their idols the same respect they show their own.

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Apostolos Doxiadis's graphic novel 'Logicomix' will be published by Bloomsbury next year