Towards the end of Marcus de Sautoy's mesmerising new book, one of the mathematical heroes whose quest for "monstrous moonshine" he chronicles assesses his own behaviour patterns and decides that he probably has Asperger's: "I've got a lot of the symptoms. I once read something in a newspaper and it said there are six signs of Asperger's syndrome, and I said to myself, 'Hey, I've got five of those!' " Du Sautoy, by contrast, is not your typical nerdy mathematician. If you have seen him lecture in the flesh, or perhaps watched his fabulous 2006 Royal Institution Christmas lectures on Channel 5, you will know that he looks a bit like the lead singer from Radiohead, plays the trumpet (and footy in a Sunday league) and that he is articulate, fluent, funny and personable.
He is also absolutely passionate about mathematics, with a burning desire to make the rest of us as excited as he is about its problems, its patterns and its beauty. Not for nothing was The Music of the Primes, his first popular-maths book, subtitled Why an Unsolved Problem in Mathematics Matters. As we embark with him on each fresh journey into the farthest reaches of modern pure mathematics, he is at once a stimulating and a reassuring guide.
Finding Moonshine is written with the same enthusiasm and verve as the bestselling The Music of the Primes, which took us into the tantalising world of prime numbers and some of the complex mathematical problems that are associated with their properties. This time, his subject is symmetry - a feature, we soon discover, of a mathematical world that is closely related to the one containing the primes, and rather a long way away from our ordinary notion that something is "symmetrical" simply because its left and right sides are mirror images of each other.
The mathematical world of symmetry is the world of transformations of objects such that the transformation leaves the original state of affairs apparently unchanged - like making a sixth of a turn with a hexagonal shape on the page, for example. Look away while the shape is being turned, and you would have no idea anything at all had altered. These sequences of transformations (extended to unimaginable numbers of dimensions) can be captured in mathematical objects called "simple groups". They provide a collection of building blocks for symmetry, rather as the prime numbers are the building blocks of number theory. It is, however, mathematical problems associated with 27 "sporadic groups", which do not conform to the regularities of the other members, with which du Sautoy spends his life grappling.
He introduces the general reader to mathematical symmetry via the extraordinarily rich range of patterning possibilities to be found in the tiles of the Moorish Alhambra at Granada in southern Spain. Du Sautoy and his long-suffering young son pursue an intensive quest for the 17 types of symmetrical transformation possible in two dimensions, around the walls of the Alhambra, and eventually find them all. And in two dimensions the reader can comfortably accompany them.
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But the ultimate quest for du Sautoy is the biggest of the sporadic groups, affectionately known as the "Monster" - an awesome, entirely indescribable symmetry group in 196,883 dimensions. "Moonshine" is the elusive mathematical link between the Monster and an important function in number theory, known as the j-function. Du Sautoy captures the excitement of the chase to understand the properties of the Monster, or even to prove that it really exists. But in this case, I am afraid even his narrative magic cannot take most of us with him, although he certainly gives us ample encouragement to try.
In the end, Finding Moonshine is the life story of du Sautoy himself, as he grapples with the most extreme problems of group theory and number theory. It is, indeed, structured as 12 months in his life - the watershed year when he turned 40. He captures for us with brilliant vividness the excitement of the pursuit of a solution to a difficult problem, the extraordinary optimism and patience that modern mathematics requires, when such solutions often involve years of painstaking compilation of partial solutions that might eventually contribute to a successful final outcome. We experience the thrill of a step made towards uncovering ultimate mathematical beauty and share his sense of wonder at the intricacies and patterns that the search reveals. We are drawn into the curious lives of virtuosi from the past, whose brilliant discoveries continue to underpin modern mathematics.
What du Sautoy cannot do, however, is enable us, the general readers, to understand the mathematics itself. It is telling, I think, that some reviews by mathematicians of The Music of the Primes complained that there was "not enough maths" in the book. In Finding Moonshine, too, in the midst of a gripping section of evocative presentation of the circumstances in which a particular critical insight into the mathematics of symmetry was made, we find ourselves plunged suddenly into a series of equations whose terms and relationships with other equally important fields of mathematics there is neither the time nor the space for du Sautoy completely to explain. The mathematical point that the author wants us to grasp ultimately eludes us - the equations become simply a kind of mood music, colouring our interpretation of the story.
As I was finishing reading Finding Moonshine, I happened to go to tea in Cambridge with my own mathematical guru, Bertie Bellis, who, as the young head of maths at Highgate School in London, was the person responsible for kindling my love of the subject in my teens, and giving me the confidence to venture in pursuit of some of its spellbinding problems. He viewed the book rather glumly. It looked like a good read. But he had to say that even for someone with a lifelong career in the subject, "that kind of mathematics is pretty hard to get the hang of". I felt quite a lot better.
Finding moonshine: A Mathematician's Journey Through Symmetry by Marcus du Sautoy
Fourth Estate £18.99 pp384