DUELLING WITH THE MONSTER: The Diary of a Conjecture

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 soundalike 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
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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.
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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 nonmathematician 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.
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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
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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.
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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 twelvesided 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.
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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
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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 topmarksman 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-
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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
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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
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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
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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.
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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 196,883dimensional 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.
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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 highlevel 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?
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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."
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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.
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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
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"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."
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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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