The Num8er My5teries

By Marcus du Sautoy

I want you to imagine riding a train. Make it an old-fashioned and exciting steam train, all gleaming surfaces, hot metal, and pent-up power. Now imagine settling yourself comfortably back into the plush seats as the man opposite to you begins to engage you in conservation. He does so by inviting you to ponder a few everyday sorts of mysteries: the reality of climate change, the security of the internet, the stability of the solar system, beating the casino.

Such questions get you thinking. But as the train pulls away from the station, your companion begins to entertain you in the most absorbing way by drawing you in to contemplation and exploration of a bewildering variety of topics. He is so entertaining and confident that it doesn't matter whether he is sketching diagrams and doing mathematics on bits of paper in the carriage, or whether he is pointing to the view out of the window and discoursing on what he sees — you are absorbed. To illustrate his points, he tells stories, making you laugh at the antics of people he has known, or at least seems to have known. He also might pull out a game for you to play which embeds some of the truth of what he has been saying deep in your soul (games and stories are good at that).

Then you notice something curious. While it seems like he is often pulling random ideas from his head, or from what he sees in the world rushing by outside the window, in fact there might be a pattern. Indeed, all of his stories, games, mathematical exercises, and observations seem to culminate in a fascinating fact about the station into which the train is about to pull. This happens for the entire journey, at the end of which you step out of the carriage alone, but somehow changed as you look at the familiar world in an unfamiliar way.

This sense of going on a journey with a brilliant and entertaining companion, of feeling like you are never sure why the conversation is veering in this new direction, yet being confident that there is a good reason for it, of being educated and changed without suffering for it, and of finally coming to look at the familiar in a new way is the strongest sense which I got from this lovely book.

This is all the more remarkable since I have read rather a lot of "popular maths books", and after a while they all seem a little samey. Certainly different authors have different styles of writing, or different arrangements of the material, but generally speaking you know what you are going to get. The same old stories and illustrations pop up in almost every book in the genre. Yet du Sautoy has a genuine gift firstly for coming up with brand new ways of illustrating old ideas, and secondly for telling old stories in fresh ways (and correcting outmoded notions of mathematics as a linear progression of white male protagonists). I never thought I would see Mersenne primes and the Riemann hypothesis linked with dragon noodles from Mr Chang's restaurant in Taipei. The extent to which du Sautoy links mathematics not only to things in the world, but things humans do with the world, to music, to movies, to art, is exhilarating. There is never any doubt that the focus remains on the mathematics behind such phenomena: at one point he says that "biology is never as perfect as mathematics", which is surely a missed opportunity to say that therefore biology may in fact be harder than mathematics (as for example Lord Rees has intimated).

The subtitle of the book is a mathematical odyssey through every day life. This is not entirely inaccurate, but it does not reveal the whole truth. For while one of the wonderful things about the book is the way in which it reveals both the power of mathematical thinking and the ubiquity of tractable mathematical problems in everyday life, the skeleton of the book is provided by the Clay problems. The supporting structure of the book is composed of five of the seven Millennium Prize Problems proposed in 2000 by the Clay Mathematics Institute, which are really as other-worldly as you can get. Yet du Sautoy brings them into the fold of our quotidian lives.

Each of the Millenium Prize Problems carries a US$1 million prize for anyone clever enough to turn its conjecture into a theorem. (The prizes, and the Clay Mathematics Institute, is sponsored by a wealthy American business man.) This is touted as something of an attraction to the amateur, yet du Sautoy often reminds the reader that it is not the money, but the honour of solving difficult and deep problems which attracts most mathematicians. Indeed, du Sautoy discusses the example of Grigory Perelman, who recently was recognised as having solved one of the Clay problems by proving the Poincaré theorem (it was a conjecture). Yet Perelman turned down the money, saying that proving the theorem was honour enough in itself. This stance has angered and bewildered many both inside and outside the mathematical community, and I was moved to see the positive way in which du Sautoy dealt with the controversy. Indeed, he describes Perelman's decisions to reject the money (not to mention other prizes and medals), not just as "amazing", but as "something rather noble".

The five problems — the Riemann hypothesis, the Poincaré hypothesis/theorem, the P vs NP problem, the Birch & Swinnerton-Dyer conjecture, and the Navier-Stokes problem — receive one chapter each, corresponding to the stations in my over-stretched train journey metaphor above. (It bothered me that the two other problems, the Hodge and Yang-Mills conjectures, were omitted, until I read in the acknowledgements that the book grew from the Royal Institution Christmas lectures which du Sautoy gave in 2006 — the brief there being to create five one-hour, fun-packed lectures.) Each chapter builds up a welter of knowledge, via serpentine stories, puzzles, games, diagrams, and illustrations which seem to be mere detours until finally you realise that they were all absolutely essential for allowing the final "my5tery" to be proposed (and discussed) at the close of each chapter. Somehow, while this sounds like a recipe for disaster, it works!

A steam train is really the wrong metaphor entirely for this book, since in many ways this is a book which looks to the future. It is more than a book; it is a hyperbook. Distributed throughout the book are square barcodes which can be photographed by a smartphone, prompting the phone to link to a website with additional information about the topic in hand. Luddites like me — whose only phone is connected to the wall — are not left out: a simple website (hurrah for bit.ly) can be tapped into a browser which will take you to the same place. And when you get there you will find games to play, things to print out and build, apps, puzzles, movies, and more besides. I think that my favourite was the site which automatically generated artwork in the style of Coldplay's X&Y album art. This was related to the discussion on codes — and the pitfalls of using them on album art covers.

By the way, don't worry if you have never heard of Coldplay — there is album art from the Beatles, too. Not heard of them either? Well, at least du Sautoy tried to cover several age groups. Indeed, determining the intended audience of the book is maddening. The Royal Institution Christmas lectures are aimed at children (including teenagers), and certainly many of the games and puzzles (with solutions) are aimed squarely at that category, not to mention the hyperlinked nature of the book. But what would such an audience make of references to "the head on your Guinness" (in a discussion on foam, bubbles and the Ponicaré Theorem) and the drug ecstasy (in the build-up to the P vs NP problem)? The book cover claims that the book is aimed at "ages 1-101", which might be a tad optimistic, particularly at the lower end, but is broadly true: a book for everyone!

Well, at least everyone who can cope with multiple references to football. Du Sautoy makes no secret of his passion for the beautiful game, and there are not many parts of the book which don't have an explanation or illustration based on soccer. I can't stand the sport myself, but somehow even this aspect didn't jar with me because, frankly, the football-based illustrations were good ones!

If I have to find one area for improvement it would be in the figures. On the plus side, although all black and white, they are very varied, with little purpose-made quirky works of art, photos, diagrams, sketches. But it seems that the publisher did not want to pay for reproductions of many images, so for example chapter three has to describe, rather than show, Dürer's Melancholia, a Roman 20-sided die, and the Sagrada Familia cathedral in Barcelona. In fact, I could only count three images in the book not produced by one of the team. But this is a small quibble.

Telling old stories with new twists (just how big is that pile of rice on the chessboard?), bringing new illustrations to bear on old problems (what is the history of the tetrahedral tea bag?), asking insightful questions (why can't you blow a cubical bubble?): in all of these du Sautoy excels. The book contains some of the clearest and most remarkable explanations I have ever read of some of the deepest questions in mathematics. It illustrates like no other the beauty, power, fun, ubiquity, and compulsive nature of mathematics.

Book details:

The Num8er My5teries

Marcus du Sautoy

hardback — 320 pages (2010)

Fourth Estate

ISBN: 978-0007278626

About the author

Phil Wilson is a lecturer in mathematics at the University of Canterbury, New Zealand. He applies mathematics to the biological, medical, industrial, and natural worlds.