Finding Moonshine, By Marcus Du Sautoy

How many symmetries can you find in the letter that begins this sentence? You will probably count the symmetries to do with reflection – swapping the "H" left-to-right, up-to-down and across diagonals – before you think to add a rotational symmetry to the list, turning the letter through 180 degrees. If you could pick the letter up, you could also flip it over: another rotational symmetry. If you turned it around and then flipped it over, would this be the same as flipping it over and turning it around?

How would you find out? The whole point about symmetry is that it allows an object to change its orientation without altering its appearance. This can be a headache even when handling two-dimensional objects. Ever tried feeding letterhead paper into an unfamiliar printer? Now imagine handling a 196,883-dimensional object.

Reading takes time, and time destroys symmetry. Nonetheless, Du Sautoy has invested a lot in creating a very symmetrical book, one whose ends are thinner than its middle. We begin and end gently. Symmetry appears in the real world, metastasizes through multidimensional space, then returns to earth for a long, gentle decompression.

To start with, we are told that symmetry has prestige. It's expensive to do: only the fittest and healthiest plants and animals can afford to devote resources to it. So symmetry carries meaning: it is a sign of success. Symmetry has cultural prestige for the same reason: without access to mass-manufacture, symmetrical objects are hard to craft. Near the end of Du Sautoy's account we hear symmetry in music, discover a connection between symmetry and empathy, and we see why, although symmetrical shapes are hard for living things of any size to achieve, they are easy for things that are very small. Some viruses are Platonic solids – objects that are symmetrical in three dimensions. Death, more often than not, has 20 faces: herpes, rubella and HIV are all icosahedrons.

The real work of the book is its middle. Some objects are more symmetrical than others, and this is as true of hypothetical, impossible-to-visualise multidimensional objects as it is of dice and bathroom tiles. How do we find symmetrical objects we can't see – and how do we work out how their symmetries relate? In the middle of the 20th century, a branch of mathematics called group theory devoted 30 years to an exhaustive hunt for new species of multidimen-sional symmetry. This hunt shared some qualities with astronomy. Just when you think your account of the cosmos is complete, someone working on another continent rings up to report something inexplicable.

To bring us up to speed with group theory, Du Sautoy rattles through a lot of history. There is not a lot else he can do. The best way of explaining maths is through the history of maths. The history is not really the point – and this is as well, since he unapologetically describes in geometrical terms much work that acquired its relevance for symmetry only much later. Du Sautoy's strategy places clarity a long way above precision. His hope is that the reader can visualise, more than understand. "A child starting out on an instrument will have no idea... how to improvise a blues lick, yet they can still get a kick out of hearing someone else do it." Finding Moonshine is a superlative mathematical entertainment; not pretty to the purist eye, but oh, so effective.

Du Sautoy adds whig memoir to whig history when he harnesses himself and his family to his account. He neatly captures his spiky relations with his son Tomer; at one point he tells him off for using his Nintendo, only to discover the poor kid was using it to study the very symmetries his father lives for. Attempts to personalise a subject in this way are not to everyone's taste, but his account of a family visit to the Alhambra (pictured left), home to all 17 two-dimensional symmetries, has won him at least one convert.

Most of symmetry's big game hunters used mathematics to hide from life. They developed techniques "almost like the craft skills of the medieval stone-masons". Now, for all their awards, they are yesterday's men. The mathematical experience is a stark one, lived out in a world where superhuman flights of analytical thinking shade seemlessly into autistic compulsions; where trust is tricky, and success can make you redundant; and no-one, not even your your family, will ever understand why you are smiling. Du Sautoy tells us that when he was a child, he wanted to grow up to be a secret agent.

In a funny way, he got his wish.