Duelling with the Monster By Marcus du Sautoy

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DUELLING WITH THE MONSTER

Marcus du Sautoy

DUELLING WITH THE MONSTER will combine a personal insight into the mind of a working mathematician together with the story of one of the biggest adventures in mathematics: the search for symmetry.

1. some background

My intention with this second book is to build on the strengths of THE MUSIC OF THE PRIMES – where a meaty mathematical story was told through the lives of a fascinating dramatis personae of mathematicians. In commercial (and publicity) terms, THE MUSIC OF THE PRIMES had the advantage over its competition because it was written by a working mathematician. I now want to extend this advantage and to create something unique within the genre of popular science books, through a book that still has a strong mathematical story at its heart, but which goes one step further.

I am constantly striving to push my own boundaries in finding ways to share the excitement of mathematics with a broader audience. THE MUSIC OF THE PRIMES was a first step for me – beyond my journalism – into the world of popular science writing. I want to work on a project that will be as stimulating for me to write as that first book. But I have a strong sense that the market is looking for a fresh approach to telling a scientific story. I was a judge of the Aventis Science Book Prize last year, and I read 94 popular science books from the year’s crop. That experience certainly made me realise that the market is flooded by sound- alike books.

But where to go?

The reception of THE MUSIC OF THE PRIMES both in the media and through the huge amount of correspondence I have received suggests to me that the reading public has a huge appetite to find out about the world of the mathematician as much as about the subject of mathematics. Readers are keen to get under the skin of the mathematician and to get some inkling into the sensation of doing mathematics. This was

2 particularly reinforced by the surprising response to a small academic writing project I undertook for the Gulbenkian Foundation at the time that THE MUSIC OF THE PRIMES was being put to bed.

The idea was to collect into one volume the diaries of scientists over a short period in their academic lives. The book is called SCIENCE, NOT ART. Since publication, there has been fantastic media interest in the record I kept. The level of interest is especially striking given the academic origins and resources of the project. The Guardian chose my diary alone as a serialisation. Radio 4 also featured my diary on “Front Row” in the form of an interview and reading. My diary was also read as Radio 4’s “Book of the Week” in February 2004.

I was flattered and initially amazed that my diary won out over the diaries of scientists in such obviously glamorous subjects as the ecologist studying the Amazon, the physiologist investigating medicine in space or the marine biologist in his submarine. But the mysteries of the mathematical world hold a great fascination for readers. They yearn to be transported into this alien world that seems far removed from the physical space around them yet seems to be filled with such universal truth and beauty.

Given the success of both THE MUSIC OF THE PRIMES and my diary, I have begun to explore the exciting possibility of combining the two approaches.

The central thrust of DUELLING WITH THE MONSTER will be to tell the story of how humankind has come to its understanding of the bizarre world of symmetry – a subject of fundamental significance to the way we interpret the world around us. Our eyes and minds are drawn to symmetrical objects, from the sphere to the swastika, from the pyramid to the pentagon. “Symmetry” indicates a dynamic relationship or connection between objects, and it is all-pervasive: in chemistry and physics the concept of symmetry explains the structure of crystals or the theory of fundamental particles; in evolutionary biology, the natural world exploits symmetry in the fight for survival; symmetry and the breaking of symmetry are central to ideas in art, architecture and music; the mathematics of symmetry is even exploited in industry, for example to find efficient ways to store more music on a CD or to keep your mobile phone conversation from cracking up through interference.

Mathematics has provided us with the language to master and to articulate this rich world. It is only through mathematics that we could have discovered some of the extraordinary symmetrical objects that Nature has concocted. And the most fearful and exciting of all the symmetries discovered by mathematicians is the Monster: a huge snowflake that lives in 196,883-dimensional space with more symmetries than there are atoms in the sun. The discovery of the Monster represents the summit of mathematicians’ mastery of symmetry, and it will also be the pinnacle in DUELLING WITH THE MONSTER.

3 The mathematical/historical threads in this story are rich enough, but what will make this book unique is the personal story that will be woven through it. It is here that I hope to explain what it is really like to be a mathematician. DUELLING WITH THE MONSTER will balance the grand historical exploration with my own personal battles – one with a particular deep conjecture about symmetry (my day job, as it were), and one relating specifically to the Monster mentioned in the above paragraph. The stories will interweave very naturally and will illustrate the sense of transformation and discovery in both the personal and historical narratives.

The aim in the personal narrative will be to capture the dynamic tension that made the diary for the Gulbenkian Foundation so successful. There are moments of deep frustration, flashes of revelation, fears of competition, tales of friendship and collaboration. These personal episodes in my own battle will echo, and be echoed by, similar themes that run through the historical account.

The format of the book will attempt to create a direct dialogue with the reader. In all my work, both academically and through the media, I try to make people understand the way my mind creates, and at the same time to lead them on the path that I have cleared. It is too bold a claim that I can give the non- mathematician an access-all-areas back-stage pass to the mathematical mind, but the personal journey in this new book offers the tantalizing possibility of bringing people extremely close to it.

There are no other mathematicians out there who would contemplate such exposure. Even the short diary I did the Gulbenkian Foundation has drawn comments from the mathematical community. Although they recognized in the diary all the emotions that rage in their own mathematical lives, exposing their souls just isn’t what mathematicians do. . .

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2. structure of the book

There are three distinct threads:

The first is the broad mathematical story, a classic “narrative non-fiction” recounting the history of man’s understanding of symmetry, and culminating in the uncovering of the Monster, and what we have learned from it.

The second and third threads are personal challenges:

One relates specifically to the Monster, and it is simple: can I understand it, and how much of my understanding can I communicate to my reader?

The other relates to my ongoing efforts to crack the PORC conjecture, one of the outstanding mysteries about symmetry.

My intention is to divide the historical narrative into four distinct sections -- or “seasons”. Each section will contain three chapters -- or “months”. The personal narrative will frame each chapter of the historical story. The sense of time built into the historical narrative will be mirrored in the development of the personal story.

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3. brief outline of the historical narrative.

Spring: The symmetry we can see. The ancient Greeks are the fathers of our mathematical world. It was the likes of Plato and Aristotle who identified the importance of shapes with many symmetries. The five Platonic solids are the beginning of the science of shape and symmetry. Similarly, the iconoclastic belief of Islam meant that the Arabs were particularly fond of expressing their religious belief through designs that included much symmetry. The beginning of the story explores the tangible first shoots of a theory of symmetry.

Summer: From pictures to language. The French revolution represented a key turning point in the story of symmetry. The French revolutionary and mathematician Galois single-handedly forged a new language that allowed mathematicians to articulate the symmetry of the unseen. The importance of language will be a big theme in the book, representing a similar strand to the “music” that ran through The Music of the Primes. The power of this language reveals that there is symmetry at work in an unexpected variety of mathematical and scientific settings. Not only does the 3-dimensional physical world like symmetry, but the theory underpins the behaviour of other mathematical objects. Although killed in a duel aged 21, Galois’ mathematical language immortalized the young mathematician. For the first time it allowed us to see symmetries beyond the physical world around us.

Autumn: Mapping the world of symmetry. Following Galois’s breakthrough, mathematicians embarked on an epic journey of discovery. Their aim was to produce a complete list of building blocks from which all symmetries could be constructed. In the same way that numbers are built by multiplying together primes, the symmetry seekers wanted a Periodic Table containing a list of all the different basic atomic symmetries. At times the task looked impossible, so wild and varied was the list becoming. Until in the early 1980s a group based in Cambridge produced an Atlas that, they claimed, represented a complete map of the world of symmetry.

Winter: The mysteries of the Monster. The end of one journey marks the beginning of the next venture. Amongst the list of building blocks recorded in the Atlas, one stuck out as a complete anomaly. Christened the Monster, this symmetrical object only appeared once mathematicians investigated 196,883-dimensional space. Despite its exceptional character, mathematicians spotted strange connections between the monster and other areas of mathematics and physics. The last two decades have been spent grappling with this mysterious beast and understanding why it is so important.

6 4. brief outline of the personal narrative

To keep a clear structure to the book, the personal narrative will top and tail each chapter. The historical story gives the book its sense of time progressing between each entry in the personal story, plus there will be room within the historical account to interweave some of the personal perspective as and when it seems relevant.

I will be using the seasons of the historical story as a framework for telling my story. As the book proceeds I want to create a dynamic, living narrative that pulls the reader along, with a sense of tension and expectation similar to my Gulbenkian diary. An important point, though, is that DUELLING WITH THE MONSTER won’t contain diary entries as such; I will be aiming for a fluid narrative.

I have spent my mathematical life journeying alongside the modern day navigators of the world of symmetry. One element of the personal narrative will tell the story of my attempts to crack one of the big questions still outstanding in the theory of symmetry. The Atlas provides the building blocks of symmetry. These are the atoms – but what are the chemicals you can build from these atoms? The chemists can make water from hydrogen and oxygen. What symmetries can be built from our Periodic Table of Symmetry? This is what drives my mathematical explorations. In particular I have been battling for years to solve the PORC Conjecture. Formulated in 1959 by Graham Higman, one of the grandfathers of the modern band of symmetry searchers, the PORC conjecture tries to count how many different symmetrical objects you might be able to build using the atoms from the Atlas. It doesn't go as far as to say what the objects look like but it is a first step in our attempts to understand quite how complicated the world of symmetry might be beyond the symmetries in the Atlas. For years this conjecture seemed impenetrable; but my recent work has shown a way in. Using techniques originally developed to study primes not symmetry, there now exists the chance to crack this enigma.

The live element here will mean that the reader will not be sure until the end whether the mystery, the conjecture, is solved or not. In essence, the personal narrative will recount the drama of being a mathematician. Neither the reader nor the mathematician will be sure when the next twist will arrive.

The other thread concerns my relationship with the Monster. Ever since I first heard about this object, it has held a deep fascination and challenge for me. I have battled on and off for years to get my head around it. Now I will attempt to do so in earnest. Can I? Or – and this will be humbling – can’t I? Or will I be able to realise my credo that “if someone can understand it, then I can understand it; and if I can, everyone can.” This personal duel mirrors the historical challenge to master the world of symmetry. There is a constant fear when trying to crack a mathematical problem that perhaps this time things will just be too complicated

7 for the human mind to conceive and understand. Perhaps there are just some things that will remain beyond the abilities of humankind to comprehend.

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5. more detail on the mathematical story

The eyesight of a bee is extremely limited. As it flies through the air in search of food, it has to find some way to make sense of the onslaught of images it is bombarded with. Evolution has tuned the bee to recognize shapes full of symmetry because this is where it will find the sustenance that will keep it alive. The flower is equally dependent on the bee for its survival. It has evolved to form a symmetrical shape in the hope of attracting the bee. Nature has made symmetry an important part of the evolutionary language. Symmetry marks out the intentional, something with design, something with meaning or a message against the background noise.

Nature enjoys hiding mysterious symmetries at the heart of many parts of the natural world – fundamental physics, biology and chemistry all depend on a complex variety of symmetrical objects. The six-sided symmetry of the snow-flake, the eight-sided symmetry of the medusa, the simple reflection symmetry in the human face are some of the obvious manifestations of how much Nature loves symmetry. The symmetry in the natural world serves a function for each object and isn’t simply a thing of beauty. The honeycomb built by the bee is built from hexagons only because this six-sided figure is perfectly adapted to packing things efficiently.

9 Of all the animals in Nature’s kingdom, the mathematician likes to think he is the most sensitive to and in tune with the language of symmetry. Mathematics is often called the quest for patterns. But is it possible to classify all the possible patterns that could be found in Nature? Is there a limit to what patterns we might find? Could we even make a list of all these possible symmetries? For the mathematician, the pattern searcher, understanding symmetry is one of the principal themes in charting the mathematical world.

The word symmetry conjures to mind objects that are well-balanced, with perfect proportions. They capture a sense of beauty and form. The human mind is constantly drawn to concepts that realise some aspect of symmetry. Our brain seems programmed to notice and search for things with order and structure. Artwork, architecture and music from ancient times to the present day play on the idea of things which mirror each other in interesting ways. Symmetry is about connections between different parts of the same object. Symmetry is often a sign of meaning and intent and can therefore be interpreted as a very basic, almost primeval, form of communication.

In the natural world symmetry is an important theme in the way our environment has been fashioned – on a biological, chemical and atomic level. Evolution has marked out the animals that can achieve perfect symmetry as those fittest for survival. Beauty in the human face is strongly correlated with perfect symmetry. In the chemical world symmetry provides efficiency and strength. In physics, scientists have discovered connections between seemingly unrelated bits of the physical world by understanding how these parts are simply two different sides of some common symmetrical object.

Artists and musicians have for centuries played with the ideas of symmetry in constructing buildings and symphonies, frescos and paintings. Bach’s Goldberg variations are a mathematical exercise in the different ways you can rotate, invert, reflect a musical theme and create a new variation. Egyptian ornaments, Roman mosaics, Arabic tiles have all stretched the possibilities of symmetry to produce fresh and original designs. These ancient artists are some of the first explorers of the mathematics of symmetry. Today the business world has cottoned on to the commercial potential that symmetry provides in the world of telecommunications. Strange symmetrical objects are exploited in manufacturers attempts to squeeze as much data onto a CD, just as the bee exploited the hexagon to pack the most into the honeycomb. Different symmetries are used to preserve the quality of data as it transmitted from one destination to another. For example the pictures that the Mars space probes are able to send back to Earth are so clear and aren’t disrupted by interference because the data is encoded using important symmetrical objects discovered by mathematicians. Even the clarity of a mobile phone conversation depends on the same idea of symmetry.

Such technological developments are a by-product of the mathematician’s search for the deepest understanding of symmetry, and they will naturally form part of the story in DUELLING WITH THE MONSTER.

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Over time mathematicians have successfully created a language to allow us to navigate the world of symmetry. The idea at the centre of this language is quite simple. To understand the different symmetries of an object, like a twenty pence piece, draw an outline of the object. The different symmetries of the coin are then represented by the different ways I can pick up the coin and place it back inside its outline. For the seven-sided coin there are actually 14 different moves I can make. These moves capture how much and what sort of symmetry the coin has.

DUELLING WITH THE MONSTER will follow the attempts of mathematicians to list all the possible symmetrical objects that Nature has scattered round the mathematical world. The Greek and Arab mathematicians discovered all the symmetrical objects that exist in our three-dimensional visual world. The Platonic solids, things like the pyramid shaped tetrahedron or the twelve-sided dodecahedron, held deep mystical significance for the ancient Greeks. They were at the heart of the Greek view of the chemical world and even their astronomical theories. Plato believed there were deep connections between the Tetrahedron and fire, the Cube and earth, the Icosahedron and water, the Octahedron and air. The twelve- sided Dodecahedron he believed related to the stuff of which the constellations and heavens were made of. Although the Greek ideas are quaint from our twenty-first century perspective, there are nonetheless deep connections between symmetry and the chemical and astronomical world.

The Arabs too were fascinated by the visual appeal of symmetry. Embedded in the artwork of the Alhambra are all of the seventeen different sorts of patterns that can be drawn in two-dimensional space.

11 Even if the pictures are different there are actually only seventeen essentially different patterns that artists can use to wall-paper a wall or tile a floor so that the pattern repeats itself horizontally and vertically.

17 different sorts of wallpaper

Although the Arabs and the Greeks were extremely creative and inventive in coming up with so many of the basic forms of symmetry in the three dimensional world, it would not be until the inspirational mathematical vision of the nineteenth century French revolutionary Evariste Galois that mathematicians would finally have the tools to prove that there wasn’t an eighteenth wallpaper pattern that the Arabs had missed.

While trying to win the Grand Prix du Paris in 1830, Galois invented a whole new language that allowed mathematicians to articulate subtleties about symmetry that a mere geometrical viewpoint could only obscure. Galois’s contribution is central to our ability in the last two centuries to navigate the complexities of this world and to allow us to “see” new and bizarre symmetries in 4, 5 and higher dimensional space. The symmetries of a 7 sided coin can be understood as the different ways you can move the coin so that is can sit back inside its original outline. Galois saw that this symmetry can be captured by a language to describe how to permute the seven points of the coin. For Galois a rotation is described by a language which says that vertex 1 moves to vertex 2, vertex 2 to vertex 3 and so on. Although it looks a very innocent translation, the power of turning symmetry into a language of permutations was inspirational. Suddenly the shuffles of a pack of cards can be seen as the different symmetries of an object in 52 dimensional space. For the mathematician, each card can be thought of as living in its own independent

12 direction. Shuffling the pack of cards then looks like spinning this 52 dimensional object so that cards move in different directions. We can illustrate this in our limited three dimensional world by taking a pack of only 3 cards, say the ace, king and queen of diamonds. By putting the 3 cards on a cube, all the shuffles of the cards can be realised by different symmetries in the cube. For example, imagine the cube aligned so that two sides face east-west, two sides face north-south and the last two sides are facing up and down. Place the ace on the east facing side, the king on the north face and the queen on the top. If I want to shuffle the cards so that the ace, king and queen cycle round, I can achieve this shuffle by spinning the cube at the point where the east, north and top faces meet. Other permutations of the three cards then correspond to other symmetries of the cube.

Galois’s perspective revolutionised the theory of symmetry. Combined with the excitement of his mathematics, Galois’s personal story is also one filled with romance and intrigue. Ignored by his mathematical superiors and feared by the political establishment he was killed aged 20 in a duel by the top- marksman in Paris. The night before that fateful duel, he worked till dawn recording his ideas for understanding symmetry lest they be lost for eternity, blown away by the bullet of his political adversary.

Galois’s life has been described in popular literature by a number of authors. But very recent research has revealed a new and extremely fascinating perspective on why and how Galois died. In contrast to previous accounts, Galois was not the victim of a plot by the authorities to remove this tiresome revolutionary. Instead it appears that Galois offered to sacrifice his own life in the cause of the revolution hoping that by spreading the rumour that he had be assassinated, it would stir the masses into public revolt. He had become so disillusioned with his failure to get his ideas to be accepted by the mathematical establishment and had been rejected by the woman he loved that the 20 year old decided his life was worth nothing. He believed his death, however, could prove a powerful spark to ignite the powder-keg of revolution. But the sacrifice of his life in the cause of the revolution turned out to be wasted. His revolutionary friends were handed a bigger political firework with the death of one of Napoleon’s generals. Instead of initiating the planned riot that was to accompany Galois’s funeral, the revolution was delayed. It appears that Galois’s personal sacrifice was pointless, a new perspective that to my knowledge is yet to be published in a book aimed at the general reader.

When later generations realised the mathematical legacy that Galois had left them, the floodgates were opened. Mathematicians began to build and see new symmetries everywhere, which brought new levels of understanding in countless fields of human endeavour. The chaotic jumble of fundamental particles suddenly made sense once mathematicians discovered a special symmetrical shape in 8 dimensions. The Herpes virus, we discovered, is so virulent thanks to the symmetrical dodecahedron shape it assumes. The strength of a diamond relies on carbon atoms exploiting the power of symmetry. Symmetries undreamt of by the Arabs or Greeks were accessed once mathematicians explored other dimensions. While 3-

13 dimensional grocers continued to pack 3-dimensional oranges in the most efficient hexagon arrangement, mathematicians were getting busy with more exotic 24-dimensional oranges, and making discoveries with practical implications.

The Herpes virus

Dimensions are used to keep track of anything we might be interested in. So for example in physics we talk about three dimensions for space and fourth dimension of time. Economists on the other hand might want to investigate the relationship between interest rates, inflation, unemployment and the national debt. The economy can therefore be thought of as a landscape in four dimensions. Although we cannot physically represent this landscape in our limited 3-dimensional world, it is possible to use the equations of mathematics to explore the shape of the economy. Mathematical language can therefore describe an orange in 24 dimensions. Instead of using the visual language that most people employ when they imagine a sphere, the mathematician translates the visual into a language of numbers. This language is how a mathematician can talk so easily about higher dimensional oranges or spheres. We are all used to looking up in the index of an atlas and finding the location of Antigua translated into a set of numbers. It is this same translation which is at the heart of the mathematician's ability to conceive of a 24-dimensional sphere.

Draw a circle of radius one centimetre on a piece of paper. Each point on the circle can be identified by two numbers detailing the coordinates of that point on the map. A three-dimensional sphere consists of points which need three coordinates to pinpoint. A point is on the surface of the sphere if its distance from the centre is one centimetre. As physicists, engineers and economists began to bang at the mathematician's door demanding tools to be able to keep track of more than just three spatial dimensions but also concepts like time, temperature or the national debt, mathematicians saw how objects like their beloved sphere also had analogues in these higher dimensional worlds. Mathematicians realised that you can take a leap into the unknown and talk about a four-dimensional sphere or even a 24-dimensional sphere. We can't build it physically or see it with our eyes but we can describe it using our mathematical eyes. The 4-dimensional sphere is described by using four numbers or coordinates. To identify which points with four coordinates make up the four dimensional sphere, the concept of the ruler which makes sense in three dimensions is replaced by a mathematical equation which tells us that, say, a point is one centimetre from the centre of

14 the sphere. The conceptual leap to leave a concrete three dimensional picture behind and rely on the language of mathematics to do the “seeing” is an important theme which will also be picked up in the personal narrative.

As time went on mathematicians realised that it might be possible to make a complete list of objects that were the building blocks for all symmetrical objects – a Periodic Table of symmetry. But as more and more strange symmetries were discovered mathematicians became pessimistic that the process would ever end. Perhaps Nature had cooked up a never-ending mess of symmetrical objects that defy any attempt by mathematicians to comprehend. But then a group of mathematicians in Cambridge became much more optimistic that their exploration would eventually end in a complete understanding. Their journey culminated in one of the most incredible feats of intellectual history. In the 1980s a definitive list was made of these building blocks. Mathematicians had spent the last two millennia searching for new symmetries. At the end of the twentieth century, these explorers returned with an “Atlas”. Inside the “Atlas” they charted the discovery of what they claimed was a complete list of symmetries from which every symmetry in Nature could be built.

At the head of this adventurous band of mathematical explorers was Professor John Conway, one of the most colourful characters in the modern world of mathematics. When I first visited him in the eighties in Cambridge the common room was buzzing with his ideas, or more likely one of the many mathematical games he loves to invent. Conway's passion for games infects much of his serious mathematics. Many people are drawn initially to the subject by its playful nature but many soon forget this element, and get caught up in the serious day-to-day rigours of being a professional mathematician. It is always refreshing therefore to listen to someone like Conway.

His mathematical and personal charisma have given him almost cult status. Conway's performances when he presents the spoils of his mathematical journey into symmetry are almost magical in quality. He weaves together what at first sight look like mathematical curios or tricks but by the end of the lecture has arrived via these games at answers to very deep questions of mathematics. These deep insights are preceded by his characteristic laugh as if he too is surprised at where he has arrived. At the same time he has reduced a room of serious academics to playful children. They rush up at the end of the lecture to play with the mathematical toys he produces from a suitcase of tricks that he often carries with him.

Conway has always been obsessed with patterns and symmetry. I remember when I asked whether it would be possible for me to join his band of explorers, he explained that only on condition I changed my name. I was rather perplexed. But then he explained that a sixth name could only be added to the list of authors of the Atlas if, like the other five authors, the new name had six letters with the vowels in exactly the same position - Conway, Curtis, Norton, Parker and Wilson.

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He once described to me his fascination with the designs of Escher. ''I have a book of Escher's pictures on my piano. I try to ration myself to an Escher picture a day. Often I can't resist cheating and turning the page early but I always insist on at least going out of the room first before I can turn the next page.'' One of Conway’s favourite Escher designs is a beautiful 12-sided chocolate box that Escher made for the fiftieth anniversary of a Dutch chocolate manufacturer. The design on the box consists of starfish and shells. But the interesting thing for Conway is that the symmetries in Escher’s chocolate box are also the first building blocks of symmetry to be found in Conway’s Atlas. These symmetries are simple enough, and were understood by the ancient Greeks, but they are just the beginning of a list of symmetrical shapes that turned out to be more bizarre than any would have imagined.

Amongst all these objects there is one symmetrical object that held the mathematical community in awe - a very strange figure, a complex snowflake that lives in 196,883 dimensional space. This object was a complete anomaly amongst all the other objects recorded. Conway christened this strange symmetrical object the Monster not least because it has more symmetries than there are atoms in the sun. Understanding this seeming freak of Nature began to obsess the symmetry searchers. The duel with the Monster had begun.

It is unique minds like that of Conway’s that can conceive of such a monstrous object. On my visit to Cambridge in the mid-80s I remember sitting in awe as Conway and his group played with this object in their minds like it was some simple Rubik’s cube. Conway sat there in his sandals wearing a tee-shirt with 200 decimal places of pi on his back – and he wouldn’t have to take off his shirt to tell you what the 200 digits were. Remembering 200 digits is one thing. Manipulating an object with 808017424794512875886459904961710757005754368000000000 different symmetries is feat beyond most mortals. But it was finding a hidden code that explained how this object is put together which enabled Conway and his crew to get to grips with the Monster. Called the Golay Code after its founder Marcel Golay, it is made out of strings of 24 numbers. There is a special rule for saying when a string of 24 numbers is an admissible word in the code. It is certain symmetries in the collection of code words that provide the secrets of how to construct the Monster. Although one can think of the code purely arithmetically there is actually a beautiful geometrical meaning hiding behind these code words.

The code provides a remarkable way to pack 24-dimensional oranges. Your grocer probably arranges his oranges in the shape of a hexagon when he packs them as this is considered to be the packing that wastes least space. (It is striking that it took mathematicians till the year 2000 to prove that this 3-dimensional packing is the best.) Mathematicians though are not only interested in packing 3 dimensional oranges or spheres. Although the mathematician can't see or draw a 4 -dimensional sphere, he can describe it using 4 numbers or coordinates to locate each point on the sphere. If we take a collection of four dimensional spheres, how can they be arranged so that they fill 4 dimensional space most efficiently. It turns out that an

16 analogue of the hexagonal packing still provides the best way to arrange the centres of the spheres so that fill most space but don't overlap. But, as we increase the dimensions, mathematicians discovered that the 24-dimensional grocer has a special packing which allows him to stack his oranges in a much more efficient manner than the analogue of the hexagonal packing. And it is the Golay code at the heart of Monster which tells the mathematician how to arrange the spheres. Each code word is a string of 24 numbers. These 24 numbers provide the coordinates in 24 dimensional space for where to place the centre of each sphere to achieve this amazingly efficient packing. The symmetry of this packing explains many of the facets of the Monster. Although this may seem interesting only to mathematicians and 24-dimensional grocers, this packing has been a breakthrough in the design of very efficient codes that are used today by the 3-dimensional military. As these beautiful properties became apparent and people became better acquainted with the Monster, Griess, the first mathematician to construct the Monster, decided to rechristen it the Friendly Giant.

Conway holding the 3-dimensional hexagonal packing

The proof of the “Atlas” is so immense that it was claimed by some that it had holes in but because no one had managed to single-handedly (or –mindedly) hold the proof in their heads, the gaps had not been revealed. As with any atlas, this Atlas is the culmination of decades of exploring – in this case by over 100 mathematicians. Their quest, which covers over 15,000 journal pages laid the groundwork for Conway and his colleagues to complete and collate the description of the building blocks of symmetry. But concerns still pervade the mathematical community about how complete the project truly is. Some make sure to alert

17 others to the fact they are using the Atlas as if it is almost an article of faith. “I am a believer in the Gospel according to the Atlas”. Is there actually the possibility that there are other islands out there that the survey of symmetry had overlooked in their desire for resolution and closure? Nevertheless the existing list contains many challenges for the mathematicians of symmetry. In particular the Monster turned out to have a number of surprises in store.

It was while playing around with this fearsome Rubik’s cube that Conway noticed something rather strange and remarkable about the Monster: certain strange numbers kept appearing which he felt contained some sort of hidden message. At some point whilst Conway was browsing through the library he suddenly noticed these numbers appearing in a seemingly unrelated area of mathematics called modular functions. These functions are essential to the proof of Fermat's Last Theorem. This incredible numerological coincidence seems to hint at a deep connection between these two areas but at the time the connection remained somewhat elusive.

Conway describes this discovery as one of the most exciting moments of his life. It is rather like an archaeologist uncovering designs in tombs in Egypt which have only ever been seen before in Mayan tombs in South America. Mathematics is full of such connections and many mathematicians will admit that they are often the greatest incentive in mathematical discovery.

Conway talked with his collaborator Simon Norton. Norton is the anti-thesis of Conway's ebullient extravert character, happier at the end of a station platform spotting trains than entertaining a lecture hall full of people. But Norton and Conway make a great team. They christened the strange numerological phenomenon “Monstrous Moonshine” since these numbers seemed to be reflecting beams from the unrelated world of modular functions. This unexpected connection with fundamental objects in number theory made people begin to realise that the Monster was more than just a freak in a mathematical side show. Understanding what it was that was producing this strange moonlight became the holy grail of the world of symmetry. Like Bottom in A Midsummer Night's Dream, mathematicians couldn’t resist the mathematical weaver John Conway's call to "Find out Moonshine".

It was John Conway’s student Richard Borcherds who finally revealed in the late 90s the source of this numerological coincidence. Borcherds’ work uncovered strange new connections emerging between the Monster and fundamental questions of physics. Speculation became rife following his proof of Monstrous moonshine. Could Nature really have chosen this bizarre exceptional object as the building block for the fundamental framework of the universe? Or were mathematicians reading too much into the numerical coincidence they had spotted? People talked mystically about it being the symmetry group of the universe and it began to take on an almost religious status. Although mathematicians have failed so far to find God

18 in the Monster, Borcherds work revealed there was an intimate relation between the Monster and what physicists call string theory.

Richard Borcherds

His groundbreaking discoveries earned him a Fields medal in 1998, the mathematician's Nobel prize. But Borcherds is a strange character. Lost outside the world of mathematics, he is cast from the same mould that made the likes of the obsessive train-spotter Simon Norton. In an interview to the Guardian, Borcherds admitted to suffering six out of the seven key indicators for asperger's. He was a contemporary of mine during my time in Cambridge and conversation with him was near impossible outside of the comfort of the equations of mathematics. The psychologist Baron-Cohen dedicates a whole chapter to Borcherds in his recent book about autism and the extreme male brain. Is it these qualities that equipped Borcherds with the mind to fight so successfully with the Monster? Like the innocent fool of mythology, Borcherds knew no fear in the face of something so fearsome. His duel revealed how important the Monster is to physics and mathematics.

19 6. Extended outline of the personal narrative

The narrative will not only record a historical duel but also a personal one. My life has been spent journeying alongside other mathematicians in the wild regions of the mathematical world. We each have our own individual quarry that we are tracking, and the book will follow me on one of my hunts.

The aim of the personal journey will be to give people the insight into what it is a mathematician does all day. Whilst most people can understand the research world of the marine biologist, the ecologist, the chemist, what are the objects that the mathematicians are investigating? The diary of scientists who explored the bottom of the sea, the depths of the rain forest, even medicine in outer space have an obvious glamour. Yet for me, the exploration of the mathematical world is as exciting and visceral as diving to the bottom of the sea. It may not be a world you can easily see around you, but these mathematical journeys of the mind are for me as adventurous and risky. I want to try to take people into this world. The innocent looking room at the top of my house is the magic wardrobe through which I step each morning to this world with extremes as challenging as the ice-caped landscape of Antarctica or the bare deserts of the Gobi desert.

Do mathematicians really look at the world in a different way from the rest of the world? How does one get to the level of meditative understanding to be able to talk about symmetrical objects in 196,883- dimensional space? Is there something unique about the brains of those that do mathematics? Is the mathematical mind hard-wired? Or can everyone learn and speak this language?

These questions will be approached through the actual practice of doing mathematics. By producing a live real-time narrative of my mathematical battles, readers can experience the highs and lows, the depression and exhilaration, the jealousy and joy of doing mathematics. My own personal story will inevitably mirror the historical narrative that tells the story of our exploration of symmetry over the last two millennia. My personal narrative will provide the vehicle for an insight into the mathematical world. It is as much a piece of creative writing as a factual account and I intend that it should be a gripping read with a strong and fluid narrative drive.

There are two projects that I will be working on during the writing period. The first concerns my attempts to prove a 40-year-old conjecture about how the building blocks of symmetry may be put together. Although we have a list of these building blocks, it is still a deep mystery how they can be put together. The conjecture I am working on attempts to count how many symmetrical objects there are with a given number of symmetries even if we can’t identify the objects. The tools that I bring to bear on this problem are extremely varied drawing on many of the major mathematical breakthroughs in the last two centuries.

20 The second project is to see how close I can come to an understanding of our current knowledge of the Monster. Very few mathematicians around the world can truly admit to feeling at home with such a creature. In 1986 I made a visit to Cambridge to discuss doing my research there. Sitting around the common room were the five of the creators of the “Atlas”. I sat there in awe as these mathematicians were able to swap huge, complex symmetries in this 196,883 dimensional space. At the time I scurried back to Oxford daunted by the feat these mathematicians were able to perform. Now with the experience of my research years I am intrigued to come back to this Monster and to explore whether it is possible to master it. My own personal duel to understand this beast will expose how the mathematician can perceive what at first sight seems beyond the grasp of the human mind.

This second personal duel with the Monster will provide me with the vehicle to go and talk with the key characters in the modern mathematical story. These characters will then be brought to life in the narrative. The masters of the Monster are an extremely strange selection of personalities representing some of the classic stereotypes of the mad mathematician. In my attempts to explore the workings of the Monster I shall meet: John Conway, mathematical magician and Friendly Giant whose abilities to remember pi to thousands of decimal places were the perfect skill for manipulating a snowflake with symmetries to match the number of atoms in the sun; Simon Norton, called the Baby Monster by his friends whose appearance resembles tramp rather than world-class mathematician, the bags of train timetables he carries round in plastic bags are as obsessively studied as the symmetries of the Monster he carries round in head. Richard Borcherds, Fields Medal winner for his mastery of the Monster and self-diagnosed aspergic mathematician, his character illustrates how the Monster can become your friend when all around the pressures of engaging in the social world of humans can become too overwhelming.

Through talking with experts in the field I want to see whether it is possible for my mind to hold this object in its grasp or whether there is mathematics that some can perceive that is too distant for me. Will my journey reveal why others can’t see my work or will I be able to realise my credo that “if someone can understand it, then I can understand it; and if I can, everyone can.” I want to get as close to the Monster as my skills will support, and I want to take the reader with me.

The discussions with the mathematicians of symmetry will not only contribute to the historical narrative but will provide interesting anecdotal material in the personal story. It will give me the chance to compare my own feelings of being a mathematician with others in the subject.

The other theme that will run through the personal narrative is a selection of fun mathematical puzzles that emerge out of every day life. Each chapter will have its own puzzle which will be described during the personal narrative at the head of each chapter. The solution will then be explained at the end of the chapter. These puzzles are a potent way to empower the reader. The ability to either solve the problem or at least to

21 understand the solution will give the reader a sense of mathematical development and achievement. All the puzzles will be vary basic but will illustrate different traits of being a mathematician. They will be chosen as to be integral to the story. Many of them will also depend on ideas of symmetry for their solution. This will also lend to the thesis of the book that symmetry pervades many aspects of our world.

For example, I have been responsible for home tutoring Tomer, my eight year old son, whilst we lived in Guatemala. One of the problems we tried to solve concerned how many different ways there were to get home from the supermarket. Antigua, the town we are living in, was the first to be constructed in the classic Manhattan grid shape with 7 avenues and 7 streets. The supermarket is in the bottom left corner of the grid, our house is in the top right. How many different ways are there through the network of avenues and streets to get home? We eventually found a beautiful way to solve the problem by changing it into a question about how many necklaces you can make with 7 red beads and 7 yellow beads. The answer to this depended on the different symmetries you can make of the necklace. Just after we’d cracked the problem Tomer and I made a trip up a volcano with a bus full of 25 year old travellers. We told them about the problem whilst driving to the volcano. For hours everyone was busily trying to crack the problem ignoring the lava bubbling around us. It is this fascination with puzzles like these that I want to tap into. Other examples will include: How many presents did my true love give at Christmas; Mayan mathematics based on numbers base 20; the probability that Turkey, Belgium, Russia and Holland all avoided each other in the Euro 2004 play-offs; how mathematics can tell you the best time to get married. Such puzzles are a powerful way to illustrate important characteristics of being a mathematician.

There are a great many junctions between my story and the historical narrative. Sometimes an event in my life will also be the moment to introduce a new concept in the historical story. Or, for example, a trip to Israel or to Japan to meet one of my collaborators might provide the springboard for a passage about cultural differences in mathematics; news of somebody else’s breakthrough in a given subject might lead to an update on how my own research is working, or not working, as the case may be.

My aim will be to build a kaleidoscopic narrative that will pull the reader further and further into the otherworld that is the mathematician’s life. The historical mathematical narrative has a very clear resolution with the complete understanding of the building blocks of symmetry. The personal element is obviously much more open-ended. Mathematics is a journey where the excitement is not knowing the exact final destination. As the famous mathematician Gauss once said “It is not knowledge, but learning, not possessing, but production, not being there, but travelling there, which provides the greatest pleasure.” I hope my journey will culminate in cracking both the conjecture and the mysteries of the Monster. Whatever the outcome I will at least have an entertaining and rewarding book at the end of it.

22 The following includes some of the major themes in the personal story and how they relate to the historical narrative.

Beginnings - what made me into a mathematician? This mirrors the history of why did people start creating mathematics to understand symmetry. It comes down to a human desire for pattern and order, and to the maths teacher at my comprehensive school who showed me, one break time, some of the beauty in the mathematical world.

The Mathematician’s Laboratory – what is it I am doing all day? Despite common perceptions of the mathematician, I am not doing long division to a lot of decimal places. I sit playing in my mind with little toy examples of symmetry in the hope that some pattern or structure will emerge in these small examples that will work for every example I might choose. The process often feels like playing with a Rubik’s cube waiting for all the colours to match up. (Indeed, solving the Rubik’s cube depends on the mathematics of symmetry). This element of my work mirrors the first explorations by the Ancient Greeks and the Arabs of what constitutes symmetry, where experiment and artistic representation began to reveal what was possible. The computer is not a tool I use often. Instead I spend my time manipulating the language of mathematics and symmetry in an attempt to grasp what its internal logic and grammar will enforce on our understanding of symmetry. Of course there are formulas and geometrical shapes that I play with, but the surprising thing for many lay people is that it is the linguistic side of mathematics that is of paramount importance in my explorations.

The physical place of the mathematical laboratory can be anywhere – the beach in Rio, the garden in Guatemala, the BA 53 to Tokyo or most often just the office at the top of my flat in Stoke Newington. The hardest part is finding one’s way into the mathematical world -- both on a daily basis and more generally. It requires an act of meditation that rivals any Buddhist trance. I sometimes feel there is a secret door in the office at the top of my flat that grants me access to the mystical mathematical world.

Frustration – the land can remain covered in an impenetrable mist. This mist is often due to the lack of language to express things. These are the pre-conscious moments where you feel you see something but it remains beyond expression. You can sit there feeling like nothing is happening. Mathematics is actually a very physical activity – at these moments you are constantly squirming away in your chair desperate for progress. The mathematicians of the early nineteenth century experienced this frustration in their inability to articulate properties of symmetry. Equally there was deep despair amongst the symmetry searchers of the middle of the twentieth century as they contemplated the possibility that the building blocks of symmetry might just be too wild and unwieldy for a classification ever to be possible.

23 Playing – often to counter the enormity of exploring vast wastes and tundra of the mathematical world, it is important to play around with small toy examples. The personal narrative will be threaded through with small playful challenges whose solutions will empower the reader: how many paths are there from the supermarket to home, how many presents does “my true love” give at Christmas, what are the chances of football teams avoiding each other in the draw for the Euro 2004 playoffs. Each example will illustrate the power of simple examples to reveal new ways of thinking. The same playfulness is essential in doing high- level research. I shall be talking with John Conway one of the master navigators of the world of symmetry who is also inventor of the Game of Life and many other games. His playful perspective was key to his discovery of what makes the Monster tick.

Breakthrough – the drug of the mathematician. Like bashing away on a piano crashing out notes, then suddenly your mind throws up a chord full of harmony. These moments of revelation often occur away from the desk. The subconscious carries on working away, throwing up an idea on a train, playing football, sitting in a concert. The exhilarating rush of emotions that happens at these moments is what you spend your whole life trying to recreate. Many mathematicians have talked about how such moments can bring tears to their eyes. Just as some pieces of music can make the hair on the back of your neck stand on end, these mathematical moments can be equally physical. The historical narrative is littered with these mathematical epiphanies each contributing a new piece to the grand mathematical puzzle.

Unexpected doorways – these breakthroughs often are like opening a door onto another world. These wonderful surprises are what make exploration so fun. The story of the Monster tells of how strange connections were suddenly discovered between this strange symmetrical object and patterns in the seemingly unrelated worlds of number theory and physics. These tunnels between worlds are what I personally find the most exciting part of doing mathematics revealing how interconnected and perfectly constructed my world is.

Creation or discovery – one of the most exciting moments of my mathematical life was the creation of a new object with very weird symmetrical properties. But that act of creation melted into a feeling that the thing was always out there waiting for me to discover. The discoverers of the Monster went through a very similar experience. This feeling of the external reality of these symmetries goes to the heart of the Platonist view mathematicians have of their subject. Yet are these just ways of seeing the world that are dependent on the human psyche for their existence?

Competition – the constant threat of being beaten to the discovery. Mathematicians still have a great schoolboy mentality of wanting be top of the class. Having your name on a theorem is the payback for all the hard work. The discovery that it might not be your name on the theorem can be devastating. Jealousy, fear and loathing are common emotions in the fight to be the first to reach the final destination. There is

24 still great controversy in the mathematical world over whether it was Robert Griess or Bernd Fisher who really discovered the Monster.

Mathematics: a young man’s game? – The slips and mistakes along the way will give me the chance to explore the fears that beset any creative mind. Is mathematics really a subject that you can’t do after 40? At 38, is my time running out to make the big contribution I hanker after? Have I missed the chance for a Fields Medal? There is news on the grapevine that the Poincaré Conjecture has been proved by an obscure Russian. Are these people on a different plane or is it somewhere we can all aspire to? Will I be ousted by the students who are hot on my heels? The young Galois found his new language of symmetry rejected by the establishment. Were they jealous of this new kid on the block who had ideas for solving the problem they'd dedicated years to? Conway too has made way for his student Borcherds who won the Fields medal ahead of his mentor for mastering the Monster.

Collaboration – some of the most exciting moments of being a mathematician are created in collaboration. Working with my mathematical friends can often feel like a jazz jam session where you play together with themes that have begun with one player and then are taken up and developed by another instrument. The strange thing is how this common language of mathematics has forged connections with a strange array of characters spread across the globe: the West Bank orthodox settler, the French gastronome, the Japanese Go player.

Autism – going out for a drink after a mathematics seminar can be deadly. Mathematicians are not the best socially when forced out of the comfort of their mathematical world. There is a strong strain of aspergic qualities that run through the character of the mathematician that I recognize in myself and others in my world. Mathematics is the cupboard under the stairs where we go to retreat from the world around us. It is a world that is easy to negotiate emotionally. No one is making irrational moves. Anything which looks confusing will eventually yield to the logical eye and will no longer be threatening. Mathematics provides a home for people like Simon Norton, train spotter extraordinaire and the man who helped complete the Atlas describing the world of symmetry. Christened the Baby Monster by colleagues, Norton will be one of the eccentric characters that I will bring alive both in the personal narrative through my meetings with him over the years and the historical story.

Too beautiful – the perfection of the mathematical world can sometimes prove too hypnotic. Life is messy and not always as reliable as the axioms of maths. Symmetry is a symbol of death for writers like Thomas Mann. Objects that are too perfect lack life. Their icy precision has an almost life-denying quality. Artists and architects have felt compelled often to introduce imperfection in their work to reassert the human aspect of their creativity. Too much time in the mathematical world can have a destructive effect on one’s

25 personal life. Conway admits to having several disastrous marriages and tried to commit suicide eight years ago. Galois on the other hand succeeded where Conway failed, sacrificing his life in the cause of revolution.

Never ending story – one of the excitements of doing mathematics is that, even with the breakthroughs, there are still uncharted waters out there beckoning us on. I can’t resist the mathematical sirens call. Each year is different, bringing new stories, surprises, new faces, old conjectures solved. I measure the passing of time against what I did or did not know about the world of mathematics when I was 15, 25, 35… Mathematics gives me my sense of growth, development and maturity. The development of human civilization too is mirrored in the mathematician's duel to master the complexities of symmetry.

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7. the background to my working life

My diary for the Gulbenkian Foundation was inevitably flecked with mentions of my non-mathematical life. I think this was one of the elements that drew people in, that indicated somehow that I was “real” and by extension that the world of mathematics is “real”. So often mathematics is presented as a clinical finished project where the mathematician is just a name on a theorem or a formula, and I think people clicked with the idea that I was human. Therefore DUELLING WITH THE MONSTER will also portray the human side of being a mathematician. I should however stress that at no point will the book subside into autobiography for the sake of autobiography. There are, though, events that are very likely to occur, or to be in my mind, during the writing period that will inevitably give a distinctive flavour to my mathematics. The following gives a sense of some of those elements although the personal narrative in the book will only draw on those strands which become directly relevant to the narrative:

-- My family’s attempts to adopt twin baby girls from Guatemala; this will mean a trip to Guatemala during the writing. Guatemala has a fascinating mathematical heritage. The Mayan calendar was built on strange numerical cycles. How a natural and an adopted child develop mathematically in our family is of great interest to me in my attempts to understand whether mathematics is a matter of nature or nurture.

-- I am involved in a dialogue with a composer who is interested in natural structures like the Fibonacci sequence as a framework for musical composition. Symmetry is very important in musical composition and I am curious to know whether there are contributions that a deeper understanding of symmetry beyond simple reflections and inversions of themes can make to the compositional experience. This plays to my fantasy that if I hadn’t have been a mathematician, I wanted to be a composer.

-- Football is a major part of the cycle of my year – both playing and watching. The highs and lows of our Hackney team in the Super Sunday League Division Two as we play our hearts out on Hackney Marshes is likely to have an influence. Will we be promoted or suffer one defeat after another at the hands of London’s footballing elite?

-- The possibility that we might move to Israel constantly looms large in our family life. This is interesting in that it plays to the interesting question of how differently countries and nationalities do their mathematics. Israel has played a major role in shaping me as a mathematician thanks to the ideas of one of my collaborators. Alex is an orthodox religious Jew and a west-bank settler. My politics are completely polar to his yet our mathematics has united us in a common dialogue. The issue of travel and the links across the world will also provide an interesting perspective on the historical narrative. For example, I have

27 been invited to go to Japan next year after a mathematician there was inspired by reading ideas of mine. There are very few mathematical contacts between the Far East and West. Is there some difference in our mathematical perspectives that explains this?

28 8. other characters

These are some of the people that I will be meeting during the project:

John Conway. Conway, who christened the Monster, is one of the most colourful characters on the mathematical circuit. Professor at Princeton, he was the principal navigator in the mathematical exploration of symmetry, the Columbus of the world of mathematics. He approaches the world as one great game, a perspective that lead to his invention of the seminal “Game of Life”.

It was his discovery of strange Monstrous Moonshine, as he called it, that lead to the discovery that there was more to the Monster than just abstract mathematics. I can still recall my first meeting with Conway in Cambridge when I was trying to decide where to do my research in the mid 80s when Monster fever was raging.

[See an article I wrote on Conway and the Monster for the Times on my webpage: Patterns that hold secrets to the universe.]

Richard Borcherds. In 1998, Borcherds won a Fields Medal for understanding the source of the Light that was producing Conway’s Monstrous Moonshine. In an interview, following the award of the prize, Borcherds admitted to thinking he was actually aspergic. His story provides an interesting spring board into trying to get to grips with the unique qualities of the mathematical mind. Is there something about the extreme male brain that draws it to the barren unpopulated mathematical world? Why are there so few women who do mathematics?

Simon Norton Conway’s first mate on his navigations. Norton is an obsessive train-spotter, making tortuous journeys across Britain to test the network, trailing plastic bags of timetables and sporting a dirty anorak and wild black tangled hair. The train-spotter in him is what gave him the skills to be able to put in the last piece in the classification of the building blocks of symmetry. Yet his eccentricities were too much for the mathematical establishment and after placing the last piece in the jigsaw he has been abandoned, left to live off private money from a grand inheritance.

Richard Parker One of Conway’s crew on the charting of the Atlas. Would not be out of place as an officer in the Imperial Raj. Full of good stories and an example of how un-aspergic mathematicians can be.

29 Bern Fischer The man who created the Monster. He almost lost his job in Bielefeld when the sixties Maoist student movement decided they were against any more mathematics of symmetry being taught in the University. My German collaborator Fritz Grunewald remembers demonstrating outside the maths department with a placard declaring “Down with symmetry!”

Graham Higman He is the father of the conjecture I have been working on. Now in his 80s I am curious to explore his perspective on the importance of his conjecture to him and the importance of mathematics to someone at the end of their life.

Alex Lubotzky Orthodox Jew, West-Bank settler, member of the Israeli parliament, Lubotzky is one of my mathematical collaborators. The strange bond that mathematics has provided has bridged what appeared to be a social and political gulf.

Nobushige Kurokawa Japanese collaborator and champion Go player, an ancient game that depends on ideas of symmetry for a complete mastery. The differences between mathematics in the East and West.

Francois Loeser French collaborator and gastronome extraordinaire. The French are a strangely snobby lot mathematically and have always looked down on the search for the building blocks of symmetry as a rather Anglo-Saxon messy affair. They have never understood the fascination with the Monster preferring grand theories to quirky exceptions. The recent discoveries of the fundamental nature of the Monster has however forced them to rethink their viewpoint. The French angle will be interesting in relation to Galois’s life and problems in getting his ideas recognized.

30 9. some personal details about the author

Marcus du Sautoy is author of the best-selling book “The Music of the Primes”. Published by Fourth Estate in the UK in August 2003 and HarperCollins in the USA in April 2003, it was received with great critical acclaim. Quotations from major newspapers are attached. The book is due to be translated into German, Italian, Hebrew, Japanese, Korean and Greek. On publication, du Sautoy was interviewed by The Observer, The Sunday Times, The Financial Times, the Village Voice, the New Scientist and Esquire Magazine. “The Music of the Primes” was picked by five different newspapers as one of their books of the year 2003.

Marcus du Sautoy is Professor of Mathematics at the University of Oxford and a Fellow of All Souls College. He is currently a Research Fellow at the Royal Society, the premier independent scientific academy of the UK dedicated to promoting excellence in science. He has been named by the Independent on Sunday as one of the UK's leading scientists. In 2001 he won the prestigious Berwick Prize of the London Mathematical Society awarded every two years to reward the best mathematical research made by a mathematician under 40. In 2004 Esquire Magazine chose him as one of the 100 most influential people under 40 in Britain alongside David Beckham and Thierry Henry. He is author of numerous academic articles and books on mathematics. He has been a visiting Professor at the École Normale Supérieure in Paris, the Max Planck Institute in Bonn, the Hebrew University in Jerusalem and the Australian National University in Canberra.

Marcus du Sautoy writes regularly for the Times, Daily Telegraph and the Guardian and is frequently asked for comment on BBC radio and television. He will present his own five part programme on BBC Radio 4 this autumn called Five Shapes and is currently talking to Channel 4 and Windfall Films about presenting three one-hour programmes on mathematics called The Million Dollar Number Hunt. He has given many presentations to the banking community - in Europe, Asia and the Americas - on the mathematics of internet security. He is a member of the Royal Society's Science and Society Committee which seeks to establish a dialogue between the public and the scientific community. His presentations on mathematics, which include “Why Beckham chose the 23 shirt”, have played to a wide range of audiences: from theatre directors to school children, from diplomats to academics.

Marcus du Sautoy plays the trumpet and football. Like Beckham he also plays in a prime number shirt, no 17, for Recreativo FC based in the Hackney Marshes. Born in 1965, he lives in London with his wife, son and cat Freddie Ljungberg.

More details can be found at the websites www.maths.ox.ac.uk/~dusautoy www.musicoftheprimes.com

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10. critical acclaim for “The Music of the Primes”

"Written with incisive clarity, Marcus du Sautoy's The Music of the Primes tells an enthralling story." "the saga is also one of profoundly human passions and griefs, of rivalries and collaborative labours. In what are today somewhat tawdry times, the history of this great hunt is quite simply one of rare human dignity. Du Sautoy brings it to passionate life even for the layman. A book not to be put down." George Steiner’s Book of the Year in the Times Literary Supplement, December 5 2003.

“I was gripped by Marcus du Sautoy’s The Music of the Primes, an exploration of the mystery of prime numbers – which has driven some mathematicians mad. I am innumerate, but this book is so well written, and tells its story so vividly and with such interesting human detail, that even I could follow much of it. I read every page, even those with lots of numbers on them.” Margaret Drabble’s Book of the Year in the Guardian, December 6 2003.

“The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics by Marcus du Sautoy (Fourth Estate, £18.99; offer £15.19) is my pick of a wonderful crop of popular science books this year. We have entered a new age of eloquent, informed scientific communicators, so that even the most armchair of enthusiasts can get to grips with those key areas of knowledge which shape the world around us.” Lisa Jardine’s Book of the Year in the Times, December 6 2003.

Equally mind bending, Marcus du Sautoy’s The Music of the Primes: a devotee writing about a mathematical mystery results in the best kind of popular science book. Observer Christmas Choice 21 December 2003.

It was also picked by the Economist as one of their books of the year.

An amazing book! Hugely enjoyable. I could not put it down once I had started. Du Sautoy provides a stunning journey into the world of primes, a journey made human and even more enthralling because he presents the personalities and lives of some of history's greatest mathematicians with the same vividness and brilliance as he presents their ideas." Oliver Sacks, Author of The Man Who Mistook His Wife For a Hat.

"Du Sautoy provides a panoramic history of prime-number crunching, rich with anecdote and unfailingly patient with the mathematical fine points."

32 Village Voice

"This fascinating account is written like the purest poetry. Marcus du Sautoy’s enthusiasm shines through every line of this hymn to the joy of high intelligence, illuminating as it does so even the darkest corners of his most arcane universe." Simon Winchester, author of The Professor and the Madman

"Du Sautoy brings out well the character of mathematicians and their world. ...as in Simon Singh's book Fermat's Last Theorem, the exhilaration of the chase comes over even when you don't know exactly what the quarry looks like" Financial Times August 9 2003.

"he brings hugely enjoyable writing, full of zest and passion, to the most fundamental questions in the pursuit of true knowledge." Marina Warner, The Sunday Times 10 August 2003.

"du Sautoy provides an engaging and accessible history of work on prime numbers and the Riemann Hypothesis. He also has an eye for modern applications" The Economist July 12 2003.

"The connection between music and the primes is not trivial, but it is cleverly made plausible to the mathematically terrified in this delightfully entertaining book." "He has certainly been successful in setting up a compelling dramatis personae of mathematicians, with every character vividly illuminated with anecdotes and felicitous comment." Graham Farmelo, The Guardian 6 September 2003.

"Du Sautoy laces the ideas with history, anecdote and personalia - an entertaining mix that renders an austere subject palatable." "Even those with a mathematical allergy can enjoy du Sautoy's depictions of his cast of characters, just as a vividly-written book about composers could hold the attention of someone with no musical ear nor musical knowledge." Martin Rees, Astronomer Royal, The Times 13 August 2003.

"Marcus du Sautoy whose combination of brains and charm should soften up even the most wilfully innumerate of readers." Jonathan Heawood, The Observer 24 August 2003.

"this is a gripping, entertaining and thought-provoking book. Du Sautoy is certainly a brilliant storyteller.

33 Even if you don't understand the maths, this is still a fascinating book." Scarlett Thomas, The Sunday Independent

"Both the man and the book make a fascinating case for the importance of the primes." "The book is full of neat cameos and clever metaphors. Du Sautoy has uncovered a wealth of intriguing anecdotes that he has woven into a compelling narrative." Andrew Antony, The Observer 3 August 2003.

"Marcus du Sautoy's The Music of the Primes is a mesmerising journey into the world of mathematics. The subject - what are prime numbers and what are their secrets? - is daunting but du Sautoy writes with admirable clarity and verve" Matthew d'Ancona, Daily Mail July 25 2003.

"The story of that quest is told in an engrossing new book called THE MUSIC OF THE PRIMES." Godfrey Smith, The Sunday Times, September 28 2003

"Look swotty by carrying around maths book THE MUSIC OF THE PRIMES." In Vogue November.

"Marcus du Sautoy's entertaining book THE MUSIC OF THE PRIMES is aimed at the more popular end of the market and looks certain to be a great success." Timothy Gowers, Fields Medal winner 2002, in Nature vol 425 9 October 2003

"The Music of the Primes (Fourth Estate, £18.99), Prof Marcus du Sautoy's fascinating new book on a mystery in the theory of numbers" "Prof du Sautoy's book is an engrossing account of the struggle of some of the world's most brilliant mathematicians to find the order amid this chaos, and to hear the "music" of the primes." The Daily Telegraph 3 September 2003.

"du Sautoy offers an engaging account of those -- including John Nash of A Beautiful Mind fame -- who have tried and failed to make "the primes sing." Maclean's May 19 2003.

"Engaging. . . . [Du Sautoy] is a fluent expositor of more tractable mathematics, and his portraits of math notables are quite vivid." Publishers Weekly

34 "Fascinating." Washington Post Book World

"A highly engaging and entertaining account of the problem that most mathematicians put at the top of their most wanted list. No matter what your mathematical IQ, you will enjoy reading The Music of the Primes." Keith Devlin, Stanford University, author of The Math Gene and The Millennium Problems: The Seven Greatest Unsolved Mathematical Puzzles of Our Time

"Du Sautoy shows how computers are used to discover reams of detail about the primes and how this detail is important to Web commerce. His account of current work takes us as close to the frontier as we can get without a passport." Los Angeles Times Book Review

"[T]his account is fascinating, filled with odd twists. . . . Marcus du Sautoy attempts to explain some of the efforts that have been made on this Everest of Mathematics." Christian Science Monitor

"This is a wonderful book about one of the greatest remaining mysteries in mathematics. Marcus du Sautoy has done an excellent job exploring this topic and explaining the significance of prime numbers and the zeta function." Amir Aczel, author of Fermat's Last Theorem and The Riddle of the Compass

"[A] lively history. . . . Du Sautoy keeps the story moving and gives a clear sense of the way number theory is played in his accessible text. A must for math buffs." Kirkus Reviews

"Exceptional. . . . A book that will draw readers normally indifferent to the subject deep into the adventure of mathematics." Booklist

"Du Sautoyˆs narrative conjures up the characters and their profound ideas with wonderful verve and a poetic gift for explanation. It is enormously entertaining."

35 New Scientist

“The Music of the Primes gives the poetry, the imagery, the tantalising ramifications. Marcus du Sautoy is a mathematician and reading him you will hear the thinking and dreaming of real live mathematicians today.” Reuben Hirsch, The Times Higher Educational Supplement

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