Copyright

William Collins

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This eBook first published in Great Britain by William Collins in 2018

First published in France by Flammarion, as Le grand romans des maths in 2016

Copyright © Mickaël Launay 2016

Translation copyright © Stephen S Wilson 2018

Drawing here by Maurice Bourlon; Image here © IRSNB, Thierry HubinMap here in the public domain, courtesy of Bibliothèque nationale de France, GE BB-565 (A7,10); Image here from Wikimedia Commons; Image here by Stefan Zachow.

Mickaël Launay asserts the moral right to be identified as the author of this work

Stephen S Wilson asserts the moral right to be identified as the translator of this work in accordance with the Copyright, Designs and Patents Act 1988

A catalogue record for this book is available from the British Library

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Source ISBN: 9780008283933

Ebook Edition © October 2018 ISBN: 9780008283957

Version: 2019-09-30

Contents

Cover

Title Page

Copyright

Foreword

1. Mathematicians without knowing it

2. And then there were numbers

3. Let no one ignorant of geometry enter

4. The age of theorems

5. A little method

6. Π in the sky

7. Nothing and less than nothing

8. The power of triangles

9. Into the unknown

10. In sequence

11. Imaginary worlds

12. A language for mathematics

13. The world’s alphabet

14. The infinitely small

15. Measuring the future

16. The coming of machines

17. Maths to come

Epilogue

To go further

Footnotes

Bibliography

Index

About the Author

About the Publisher

FOREWORD

‘Oh, I’ve never been much good at maths myself!’

I’m getting a little blasé. This must be at least the tenth time I’ve heard that remark today.

But this lady has been here at my stall for a good fifteen minutes, standing with a group of other visitors, listening attentively while I describe various geometrical curiosities. That’s how the conversation started.

‘But what do you do for a living?’ she asked me.

‘I’m a mathematician.’

‘Oh, I’ve never been much good at maths myself!’

‘Really? But you seemed to be interested in what I was just talking about.’

‘Yes … but that’s not really maths … that was understandable.’

I hadn’t heard that one before. Is mathematics, by definition, a discipline that can’t be understood?

It’s the beginning of August, in the Cours Félix Faure in La Flotte-en-Ré, France. In this small summer market, I have a pop-up – there is henna tattooing and afro braids to my right, a mobile-phone accessory stall to my left, and a display of jewels and trinkets of all kinds opposite me. I’ve set up my maths stand in the middle of all this. Holidaymakers stroll peacefully by in the cool of the evening. I particularly like doing maths in unusual places. Where people aren’t expecting it. Where they are not on their guard …

‘Can’t wait to tell my parents I did some maths during the holidays!’ This from a secondary school pupil as he walks past my stall on his way back from the beach.

It’s true – I do catch them slightly unawares. But sometimes you’ve got to do what you’ve got to do. This is one of my favourite moments: observing the expression on the faces of people who thought that they had fallen out with maths for good at the instant when I tell them that they have just been doing maths for fifteen minutes. And my stall is always crowded! I present origami, magic tricks, games, riddles … there’s something for everyone.

No matter how much this amuses me, on balance I find it upsetting. How has it come about that we need to hide from people the fact that they are doing maths before they can take some pleasure in it? Why is the word so frightening? One thing is certain: had I put up a sign above my table proclaiming ‘Mathematics’ as visibly as ‘Jewels and necklaces’, ‘Phones’ or ‘Tattooing’ on the stalls around me, I would not have had a quarter of the same success. People would not have stopped. Perhaps they would even have turned away and averted their gaze.

All the same, the curiosity is there. I observe this every day. Mathematics is scary, yet even more, it is fascinating. Some may not like it, but would like to like it, or at least to be able to peep at will into its murky mysteries. Many think it is inaccessible. But this is not true. It is perfectly possible to love music without being a musician, or to like to share a nice meal without being a great cook. Then why should you have to be a mathematician, or someone exceptionally clever, in order to be open to hearing about mathematics and to enjoy having your imagination tickled by algebra or geometry? It is not necessary to delve into the technical details in order to understand the great ideas and to be able to marvel at them.

Since time out of mind, innumerable artists, creative spirits, inventors, artisans, simple dreamers, or the purely curious, have done maths without being aware of the fact: mathematicians despite themselves. They were the first to ask questions, the first to undertake research, the first to brainstorm. If we want to understand the whys and wherefores of mathematics, we have to set out on their trail, since it all began with them.

The time has come to embark on a journey. Please allow me the space of these few pages to carry you with me into the twists and turns of one of the most fascinating and most astonishing disciplines ever practised by humankind. Let us set off to meet those who have created its story through unexpected discoveries and fabulous ideas.

Let us set out on a big adventure and see how it all adds up.

1
MATHEMATICIANS WITHOUT KNOWING IT

Back in Paris, it is at the Louvre Museum, in the heart of the French capital, that I decide to begin our investigation. Am I going to do maths in the Louvre? That may seem incongruous. Nowadays the former royal residence, converted into a museum, seems to be the domain of painters, sculptors, archaeologists and historians rather than of mathematicians. However, it is here that we can now hark back to the first traces of those same mathematicians.

From the moment I arrive, the appearance of the great glass pyramid that has pride of place in the centre of the Napoleon Courtyard is already an invitation to geometry. But today I have an appointment with a much more ancient era. As I enter the museum the time machine wakes up. I pass by the Kings of France and travel back through the Renaissance and the Middle Ages before arriving in Antiquity. The rooms spool by, I come across a few Roman statues, Greek vases, Egyptian sarcophagi. Still a little further to go. Now I’m entering prehistory, and as I speed through the centuries I gradually need to forget everything: numbers, geometry, writing. In the beginning, no one knew anything. No one even knew that there was something to know.

The first stop is in Mesopotamia. We have travelled back ten thousand years.

Come to think of it, I could have gone back even further. I could have flown back another one-and-a-half million years, to the heart of the Palaeolithic, the Old Stone Age. Fire has not yet been tamed and Homo sapiens is only a distant prospect. Homo erectus reigns in Asia, and Homo ergaster in Africa, perhaps with other cousins as yet undiscovered. This is the age of flake stone tools. The biface has arrived.

Flint knappers are at work in a corner of the encampment. One of them picks up a virgin piece of flint, which is still just as it was when he chose it a few hours earlier. He sits down on the bare earth, places the stone on the ground, and holds it steady with one hand, while with the other hand he strikes its edge with a heavy stone. A first flake breaks off. He examines the result, turns his flint over and strikes it a second time on the other side. The first two flakes that have broken off in this way, opposite one another, leave a sharp edge on the flint. It only remains to repeat the operation along the whole of the contour. In some places the flint is too thick or too wide, and larger pieces have to be removed in order to give the final object the shape desired.

For the shape of the biface is not left to chance or to the whim of the moment. It is designed, worked, and transmitted from generation to generation. We find various different models, according to the period and to where they were produced. Some are shaped like a drop of water with a prominent point, other more rounded ones are egg-shaped, while others are more like an isosceles triangle with very slightly curved sides.

Biface from the Lower Palaeolithic

But they all have one thing in common: an axis of symmetry. Could there be a practical aspect to this geometry, or was it simply an aesthetic intention that drove our ancestors to adopt these shapes? The flint knapper’s blows had to be premeditated. He had to think about the shape before creating it, to construct an abstract image in his own mind of the object to be crafted. In other words, he had to do some mathematics.

When he finished the piece, the knapper inspected his new tool, held it up to the light at arm’s length to better scrutinize the contour, and altered a few cutting edges with two or three more sharp taps before he was finally satisfied. How did he feel at that moment? Did he already experience the tremendous exaltation of scientific creativity: of having been able to grasp and shape the outside world, through an abstract idea? It doesn’t matter; the finest hours of abstraction had not yet arrived. This is a time of pragmatism. The biface would be used to cut wood, to cut up meat, to pierce animal hides and to dig holes in the ground.

But no, we shall not travel so far back. Let us allow these ancient times and these few random interpretations to rest in peace, and let us return to what will become the true point of departure for our adventure: the Mesopotamian region in the eighth millennium BC.

All along the Fertile Crescent, in a zone that roughly corresponds to the area now known as Iraq, the Neolithic revolution was under way. People had been settling there for some time. In the northern plateaux, sedentarization was a success story. The region was the laboratory for all the very latest innovations. The mudbrick houses formed the first villages, and the bravest builders were even at that time adding a storey. Agriculture was a state-of-the-art technology. The temperate climate meant that the land could be cultivated without artificial irrigation. Animals and plants were gradually being domesticated. Pottery was on the verge of emerging.

Speaking of which, let’s talk about pottery. For while many testimonies to these periods have disappeared, irreparably lost in the mists of time, nevertheless archaeologists continue to gather thousands of artefacts: pots, vases, jars, plates, bowls, etc. The display cabinets around me are full of them. The earliest date from 9,000 years ago, and from room to room, like the pebbles of Hop-o’-My-Thumb, they guide us through the centuries. They come in all shapes and sizes and are variously decorated, sculpted, painted or engraved. Some have feet, others have handles. Some are intact, cracked, broken, or have been reconstituted. In some cases, only a few sparse fragments remain.

Ceramics was the first art to use fire – well before bronze, iron or glass. Artisan potters were able to use clay, that paste of malleable rock and soil material which can be harvested in abundance in these humid regions, to fashion objects as they wished. When they were satisfied with the shape, they had only to let the objects dry for a few days, before cooking them in a great fire so that they solidified. This technique had already been known for a long time. Twenty thousand years earlier people were already producing small statuettes. But the idea of producing utensils out of clay had only arisen recently with sedentarization. The new way of life required a means of storage, so pots were produced in bulk.

These earthenware receptacles rapidly came to be indispensable to everyday life and also necessary for the collective organization of the village. If there was going to be washing up to do, it might as well be beautiful – soon, pots were being decorated. Here again, there were several approaches. Some potters imprinted their patterns in the clay while it was still fresh by using a shell or a twig, before baking the pot. Some did the baking first before engraving their decorations using stone tools. Others preferred to paint on the surface with the help of natural pigments.

As I pass through the rooms of the Near Eastern Antiquities section, I am struck by the richness of the geometric patterns conceived by the Mesopotamians. Just as it was for our flint knapper’s biface, some symmetries are so ingenious that they must have been very thoughtfully designed in advance. The friezes running round the rims of these vases pique my interest.

The friezes are decorated bands with a single pattern repeated round the whole circumference of the pot. The most common ones include triangular sawtooth designs. There are also friezes showing two intertwined strings. Then there are friezes with angled rectangles, friezes with square indentations, friezes with pointed lozenges, with hatched triangles, with interlocking circles, and so on.

When one moves from one region to another or from one period to the next, trends can be seen. Certain patterns are very popular. They are reworked, transformed, improved in multiple variants. Then, several years later, they are essentially abandoned, replaced by other designs that are fashionable at the time.

My mathematician’s eyes light up as I spot symmetries, rotations and translations. Then in my mind I begin to sort and arrange all this. Several theorems from my undergraduate days come back to me. What I need is the classification of geometric transformations. I take out a notebook and a pencil and begin to scribble.

First, there are rotations. Appropriately, in front of me there is a frieze consisting of S-shaped patterns interlocking one after the other. I tilt my head to make sure I’m right. Yes, there’s no doubt, this one is invariant under a half-turn: if I were to take the earthenware jar and turn it upside down, the frieze would still look exactly the same.

Then there are symmetries. There are several types of these. I am gradually filling in my list and a treasure hunt begins. For each geometric transformation I look for the corresponding frieze. Moving from room to room, I retrace my steps. Some pieces are damaged, so I squint to try to reconstitute the patterns that ran across this clay thousands of years ago. When I find a new pattern, I tick it off. I look at the dates to try to reconstitute the chronology in which the patterns appeared.

How many patterns should I find in total? After a little thought, I finally manage to put my finger on this famous theorem. There are, in sum total, seven categories of friezes. Seven different groups of geometric transformations that can leave them invariant. Not one more, not one less.

That is not something the Mesopotamians knew. And with good reason, as the first formalization of the theory in question only began in the Renaissance. However, without knowing it, and without pretending to be doing anything other than decorating their pottery with harmonious and original designs, these prehistoric potters were actually undertaking the very first reasoning in a fantastic discipline, that would focus the efforts of a whole community of mathematicians thousands of years later.

Looking over my notes, I see almost all the patterns. Almost? I’m still missing one of the seven friezes. This doesn’t surprise me a lot, as this frieze is clearly the most complicated one on the list. I am looking for a frieze that, if you flip it over horizontally, will have the same appearance but shifted by half a pattern length. Today, we call this a glide symmetry. This would have been a real challenge for Mesopotamians.

However, I still have not visited all the rooms, so I’m not losing hope – the hunt is still on. I observe the smallest detail, the least indication. Examples of the other six categories, those I have already observed, are piling up. In my notebook, the dates, sketches and other scribbles are getting muddled. But still no sign of the mysterious seventh frieze.

All at once, a rush of adrenalin. Behind this glass panel I have just spotted a piece that looks in a bit of a sorry state. A simple fragment. From top to bottom you can clearly make out four partial friezes, one above the other, and one of these has caught my eye. The third one down – it consists of what look like fragments of inclined interlocking rectangles. My eyes are blinking. I’m peering attentively, sketching the pattern quickly in my notebook, as though I’m afraid it will vanish before my eyes. It has the right geometry. This really is the glide symmetry. The seventh frieze is unmasked.

The caption next to it says: Fragment of beaker with horizontal décor of bands and pointed lozenges. Middle of the fifth millennium BC.

I insert it mentally in my chronology. Mid-fifth millennium BC. We are still in prehistory. More than a thousand years before the invention of writing – without knowing it – the Mesopotamian potters had listed all the cases of a theorem that would only be stated and proved 6,000 years later.

A few rooms further, I come across an earthenware jar with three handles, which also falls into the seventh category: even though the pattern has become a spiral, the geometric structure remains the same. A little further on again, there is another one. I would like to continue, but suddenly the décor changes and I have arrived at the end of the Near Eastern collection. If I continue, I will reach Greece. I cast a last glance at my notes; the friezes with glide symmetry can be counted on the fingers of one hand. I had a stroke of luck.

HOW TO RECOGNIZE THE SEVEN CATEGORIES OF FRIEZES

The first category is that of friezes that have no particular geometric property, and in which a simple pattern is repeated without symmetries or centres of rotation. This is, in particular, the case for friezes that are not based on geometric figures but on figurative drawings, for instance of animals.

The second category consists of friezes in which the horizontal line dividing the figure in two is an axis of symmetry.

The third category comprises friezes that have a vertical axis of symmetry. Since a frieze consists of a pattern that is repeated horizontally, the vertical axes of symmetry are also repeated.

The fourth category is that of friezes that are invariant under a rotation by a half-turn. Whether you look at these friezes from above or below you will always see the same thing.

The fifth category is that of glide symmetries. This notable category was the last one I discovered while passing through Mesopotamia. If you reflect such a frieze about a horizontal axis of symmetry (as per the second category) the frieze obtained is similar, but translated by half a pattern length.

The sixth and seventh categories do not correspond to new geometric transformations, but combine several of the properties encountered in the previous categories. Thus, the friezes of the sixth category are those that have a horizontal symmetry, a vertical symmetry and a centre of rotation by a half-turn, all at the same time.

Finally, the seventh category consists of friezes that have a vertical symmetry, a centre of rotation and a glide symmetry.

Note that these categories only relate to the geometric structure of the friezes and do not exclude possible variations in the pattern designs. Thus, while the following friezes are all different, they all belong to the seventh category.

Therefore, all the friezes that you can possibly imagine belong to one of these seven categories. Any other combination is geometrically impossible. Curiously, the last two categories are the most common. It is spontaneously easier to draw figures with a lot of symmetries than figures that have only a few.

Bursting with pride following my successes with the Mesopotamians, I am back the very next day, ready to take ancient Greece by storm. I have barely arrived and already I am overwhelmed with choice. The hunt for friezes here is child’s play. After just a few steps, a few display cases and a few black amphoras decorated with red figures, I have already found my list of seven friezes again.

Faced with such an abundance of riches, I quickly decide not to record my statistics as I had done in Mesopotamia. I am amazed by the creativity of the artists. Ever more complex and ingenious new patterns come into view. I have to pause several times and concentrate hard in order to unravel in my mind all the intertwined visions that are swirling around me.

In one room, a loutrophoros with red figures takes my breath away. A loutrophoros is a tall slender vase with two handles; its function is to carry water for bathing – this one is almost one metre tall. The friezes come thick and fast, and I begin by ticking them off by category. One. Two. Three. Four. Five. In a few seconds I have identified five of the seven geometric structures. The vase is standing against the wall, but when I lean over I can see that it has a frieze of the sixth category on its hidden face. I’m missing just one frieze now. That would be too much to ask. Surprisingly, the missing one is not the same one as the day before. Times have changed, and fashions too; I’m no longer missing the glide symmetry, but the combination comprising vertical symmetry, rotation and glide symmetry.

I am looking frantically for it, my eyes scanning each and every feature of the object. But I can’t find it. I’m slightly disappointed and preparing to give up when my gaze alights on a detail. In the middle of the vase there is a depiction of a scene involving two people. At first sight, there does not seem to be a frieze at that point. Then, an object at the bottom right-hand corner of the scene stops me in my tracks: a vase on which the central person is resting. A vase drawn on the vase! This technique of mise en abyme, in which an image contains a smaller copy of itself, makes me smile. I look closer. The image is a little damaged, but there is no doubt about it: this image of a vase itself contains a frieze and, what a miracle, it’s the one that I’m missing!

Despite my repeated efforts I will fail to find another piece with the same property. This loutrophoros seems to be unique in the Louvre’s collections: it is the only one that bears all seven categories of friezes.

A little further on, another surprise awaits me: friezes in 3D! And I had thought perspective was an invention of the Renaissance. Bright and dark areas skilfully positioned by the artist create a chiaroscuro effect, adding volume to the geometric shapes that pursue each other around the circumference of this gigantic receptacle.

The further I proceed, the more questions arise. Some pieces are not covered by friezes but by tilings. In other words, the geometric patterns no longer simply fill a narrow band running around the object, but now also invade its whole surface, thereby reducing the possibilities for geometric combinations.

After the Greeks come the Egyptians, the Etruscans and the Romans. I discover illusions of filigree lace carved in stone. The stone threads are interwoven and pass alternately above and below one another in a perfectly regular meshing. Then, as if the works of art are not sufficient in themselves, I am soon surprised to find myself observing the Louvre itself, its ceilings, its tiled floors and its door surrounds. As I return home, it’s as though I am unable to stop myself. In the street, I look at the balconies of buildings, the patterns on the clothes worn by passers-by, the walls in the corridors of the Métro, etc.

You have only to change how you look at the world to see mathematics emerge. The search is fascinating. It has no end.

And the adventure is just beginning.

2
AND THEN THERE WERE NUMBERS

In those days, in Mesopotamia, the region was thriving. At the end of the fourth millennium BC, the small villages we visited earlier had transformed into flourishing towns. Some of them now already had several tens of thousands of inhabitants. Technology was advancing at a rate never before seen. Artisans, whether they were architects, goldsmiths, potters, weavers, carpenters or sculptors, needed to exhibit constant ingenuity to cope with the technical challenges they confronted. Metallurgy was not yet at its peak, but it was a work in progress.

Gradually, a network of routes came to criss-cross the whole region. Cultural and commercial exchanges multiplied. Increasingly complex hierarchies established themselves and Homo sapiens discovered the joys of administration. All this took extensive organization. To establish some degree of order, it was high time for our species to invent writing and enter history. Mathematics was to play a cutting-edge role in this nascent revolution.

Following the course of the Euphrates, we leave the northern plateaux that saw the birth of the first sedentary villages, and head towards the Sumer region that covers the plains of Lower Mesopotamia. It is here, in the southern steppes, that the chief population centres were then concentrated. Along the river we reach the towns of Kish, Nippur and Shuruppak. These towns were still young, but the centuries that lay ahead of them had grandeur and prosperity in store.

Then, suddenly, Uruk emerges.

The town of Uruk was a human anthill, which lit up the whole of the Near East with its prestige and power. It was built mainly of mud bricks, and its orangey hues extended over more than one hundred hectares, so that the new visitor could stray for hours in its crowded streets. At the heart of the town, several monumental temples had been constructed, dedicated to An, the father of all the gods, and above all to Inanna, the Lady of the Heavens. It was for her that the Eanna Temple was built, its largest building 80 metres long by 30 metres wide, and hugely impressive to the many travellers who were drawn here.

Summer was coming, and as in every year at this time, the whole town was beginning to bustle with a particular activity. Soon the flocks of sheep would leave for the northern pasturelands, returning only at the end of the hot season. For several months, it would be the job of the shepherds to manage their animals, to keep them fed and watered, and to return them safe and sound to their owners. The Eanna Temple itself owned several flocks, the largest numbering several tens of thousands of animals. The convoys were so impressive that some were escorted by soldiers to protect them from the dangers of the journey. However, for the owners it was out of the question to let their sheep go without a few precautions. As for the shepherds, the contract was clear: as many animals must return as had set out. There was no margin for losing part of the herd or for clandestine trading.

This then led to a problem: how do you compare the size of the flock that leaves with the size of the one that returns?

In response to this, a system of clay tokens had already been developed several centuries earlier. There were several types of token, each representing one or more objects or animals according to its shape and the patterns traced on it. For a sheep, there was a simple disc marked with a cross. At the time of leaving, a number of tokens corresponding to the size of the flock were placed in a receptacle. On return, the owners had only to compare the flock with the contents of the receptacle to check that no animal was missing. Much later on, these tokens were given the Latin name of calculi, ‘small stones’, from which the term calculus was derived.

This method was practical, but it had a disadvantage. Who looked after the tokens? For suspicion cuts both ways, and the shepherds were in turn worried that unscrupulous owners might add extra tokens to the urn during their absence. Then they could claim compensation for non-existent sheep.

After much racking of brains, a solution was found. The tokens were to be kept securely in a sealed hollow clay sphere or envelope. At the time it was sealed, both parties were to leave their signatures on the surface of the sphere to certify its authenticity. It was then impossible to modify the number of tokens without breaking the sphere. The shepherds could leave without worrying.

But then, once again, it was the owners who found drawbacks in this method. For their business requirements, they needed to know the number of animals in their flocks at any time. How could this be achieved? Could the number of sheep be committed to memory? This was not straightforward, since the Sumerian language did not yet have words to denote such large numbers. Could you have an unsealed duplicate of all the tokens contained in all the envelopes? This was not very practical.

At last a solution was found. A reed stem was cut and used to trace out on the surface of each sphere a representation of the tokens it contained. That made it possible to reference the contents of the envelope at will without having to break open the sphere.

This method now seemed to suit everyone. It was widely used, not only to count sheep, but also to set a seal on all kinds of agreements. Cereals such as barley or wheat, wool and textiles, metal, jewels, precious stones, oil and pottery also had their tokens. Even taxes were controlled by tokens. In short, at the end of the fifth millennium, in Uruk, any contract in due and proper form had to be sealed by a hollow-sphere envelope with its clay tokens.

All this worked wonderfully well, but then one day a new idea emerged, so brilliant and so simple that one wonders why no one had thought of it before. Since the number of animals was inscribed on the surface of the sphere, what was the point of continuing to put the tokens inside it? And what was the point of continuing to make spheres? You could simply draw a representation of the tokens on an arbitrary piece of clay – for example, on a flat tablet.

This came to be called writing.

I’m back in the Louvre. The collections of the department of Near Eastern Antiquities bear witness to this story. The first thing that strikes me when I see these sphere–envelopes is their size. These clay spheres that the Sumerians created simply by turning them around their thumbs are scarcely any larger than ping-pong balls. As for the tokens, they are no bigger than a centimetre.

Stepping further into the museum brings us to the first tablets. Their numbers grow and they quickly come to fill whole display slots. Over time, the writing became more precise and took on its cuneiform appearance, comprising small notched wedges in the shape of a nail, with a stem and a head. Following the disappearance of the first Mesopotamian civilizations at the beginning of the modern era, most of these pieces had lain quietly for centuries beneath the ruins of those deserted towns before they were unearthed by European archaeologists from the seventeenth century onwards. They were only gradually deciphered during the nineteenth century.

These tablets are not very large either. Some of them are the size of simple visiting cards, but covered with hundreds of tiny signs which are crammed together one above the other. Mesopotamian scribes did not waste the slightest portion of the clay when they wrote. The museum’s explanation cards placed alongside these pieces enable me to interpret these mysterious symbols. They concern livestock, jewels or cereals.

Next to me, some tourists are taking photos with their tablets – this is an amusing nod to the carousel of history where writing appears on so many different media, from clay to paper, by way of marble, wax, papyrus and parchment, and which, in a final jest, has endowed electronic tablets with the shape of their earthenware ancestors. The face-to-face encounter of these two objects has something particularly touching about it. For all we know, in five thousand years’ time these two tablets may find themselves next to each other again, but on the same side of the glass.

Time has passed. We are now at the start of the third millennium BC, and another step has been taken: numbers have been freed from the objects that they count. Previously, in the case of sphere–envelopes and the very first tablets, the counting symbols depended on the objects in question. A sheep is not a cow, so the symbol for counting a sheep was not the same as the one for counting a cow. Every object that could be counted had its own symbols, just as it had had its own tokens.

But all that was over and done. Numbers had acquired their own symbols. In other words, eight sheep were no longer counted using eight symbols each denoting a sheep, but instead the figure eight was written down followed by the symbol for a sheep. And to count eight cows, you simply had to replace the symbol for a sheep by that for a cow. The number itself stayed the same.

This stage in the history of thought was absolutely fundamental. If I had to assign an actual date to the birth of mathematics, I would be sure to choose this moment: the moment when numbers came into existence in their own right, when numbers became detached from reality to observe it from a greater height. Everything before that had simply amounted to a gestation period. Bifaces, friezes, tokens – all these amount to preludes in the scheme that led towards the birth of numbers.

From that point on, numbers became an abstract concept, and that is what gave mathematics its identity: it is the science of abstraction par excellence. The objects that mathematics studies do not have a physical existence. They are not material, they are not made of atoms, they’re purely ideas. And yet these ideas are remarkably effective when it comes to making sense of the world.

It is undoubtedly no accident that the need to be able to write down numbers acted at the time as a determining factor in the emergence of writing. While other ideas could be transmitted orally without any problem, it seemed on the other hand difficult to establish a system of numbers without moving to a written notation.

Can we, even today, dissociate the idea we have of numbers from our written representation of them? If I ask you to think of a sheep, how do you see it? You probably represent it as a bleating animal with four legs and a woolly back. You would never think of visualizing the five letters of the word ‘sheep’. Yet if I talk now about the number one hundred and twenty-eight, what do you see? Do the 1, the 2 and the 8 take shape in your brain and come together as though they are written in the intangible ink of your thoughts? Our mental representation of large numbers appears to be definitively linked to their written form.

This example came out of the blue. Although for everything else, writing was just a means of transcribing something that had previously existed in the spoken language, now, in the case of numbers, it was the writing that would dictate the language. Think about it: when you pronounce ‘One hundred and twenty-eight’, you are just reading 128: 100 + 20 + 8. Above a certain threshold, it becomes impossible to speak about numbers without the written medium. Before they were written down, there were no words for large numbers.

To this day, some indigenous peoples still only have a very limited number of words for designating numbers. For example, the members of the Pirahã tribe of hunter-gatherers who live on the banks of the Maici River in Amazonia only count to two. Above that, they use a single word signifying ‘several’ or ‘many’. Again in Amazonia, the Munduruku only have words for up to five – that is to say a hand.

In our modern societies, numbers have invaded our everyday lives. They have become so omnipresent and indispensable that we often forget just how brilliant the idea is, and that it took our ancestors centuries to provide us with evidence of this.

Throughout the ages, very many procedures for recording numbers have been invented. The simplest of these involves writing down as many signs as the desired number – for example, a sequence of bars side by side. We often still use this method, for example to count the points in a game.

The oldest known indication of the probable use of this procedure dates back to well before the invention of writing by the Sumerians. The so-called Ishango bone (actually two baboon bones) was found in the 1950s on the shores of Lake Edward in what is now the Democratic Republic of Congo and dates back some twenty thousand years. The bones are 10 and 14 centimetres long, and feature a large number of more or less regularly spaced incisions. What was the role of these incisions? It is likely that this was a first counting system. Some people think the bones are a calendar, while others extrapolate an already highly advanced knowledge of arithmetic. It is hard to know for sure. The two bones are now on display in the museum of the Royal Belgian Institute of National Sciences in Brussels.

This method of counting using a mark for each unit added rapidly reached its limits as soon as there was a need to manipulate relatively large numbers. Packets were introduced to achieve a greater speed.

The Mesopotamian tokens could already represent several units. For example, there was a particular token for representing ten sheep. This principle was retained at the time that writing came in. There are symbols to denote packets of 10, of 60, of 600, of 3,600 and of 36,000.

Here we also see the search for a logic in the construction of the symbols. For example, the 60 and the 3,600 are multiplied by 10 when a circle is included inside them. With the arrival of cuneiform writing, these first symbols were gradually transformed into the following:

Because it was so close to Mesopotamia, Egypt did not lag far behind in adopting writing, and developed its own numeration symbols from the start of the third millennium.

The system was then purely decimal: each symbol had a value ten times greater than the one before.

These additive systems, in which one just has to add the values of the written symbols, became very popular throughout the world, and numerous variants were unveiled throughout antiquity and also much of the Middle Ages. They were used, in particular, by the Greeks and the Romans, who simply employed the letters of their respective alphabets as numerical symbols.

Alongside the additive systems, a new form of notation for numbers gradually emerged: numeration by position. In these systems, the value of a symbol was defined to depend on the position it occupies within the number. Once again, the Mesopotamians were the first to come up with this.

In the second millennium BC, it was the town of Babylon that now shone brightly over the Near East. Cuneiform writing was still in vogue, but now, just two symbols were used: the simple nail that had the value ‘1’ and the chevron with the value ‘10’.

These two signs were used to provide an additive notation for all numbers up to 59. For example, the number 32 was written as three chevrons followed by two nails.

Then, from 60 onwards, groups were introduced, where the same symbols were used to denote groups of 60. Thus, in the same way as in our present-day notation, where the figures read from right to left denote the units, then the tens, then the hundreds, in this Babylonian numeration the units are read first, then the sixties, then the three-thousand-six-hundreds (that is, sixty sixties), and so on, where each rank has a value sixty times greater than its predecessor.

For example, the number 145 consists of two sixties (which make 120) to which you have to add 25 units. The Babylonians would therefore have denoted it as follows:

Based on this system, Babylonian scholars developed an extraordinary knowledge. They had a good understanding of the four basic operations of addition, subtraction, multiplication and division, and also of square roots, powers and reciprocals. They produced extremely detailed arithmetic tables and developed very good solution techniques for equations that they set themselves.

However, all this was soon forgotten. The Babylonian civilization was in decline, and a large part of its mathematical advances would be consigned to oblivion. There would be no more numeration by position and no more equations. Indeed, there was a time lag of centuries before these questions became the flavour of the day again, and it was only in the nineteenth century that the decoding of cuneiform tablets reminded us that the Mesopotamians had tackled these things before everyone else.

Following the Babylonians, the Mayans also devised a positional system, in this case with base 20. Then it was the turn of the Indians to invent a system with base 10. This last system was used by Arab scholars before it reached Europe at the end of the Middle Ages. Its symbols became known as Arabic figures, and soon spread all over the world.

0 1 2 3 4 5 6 7 8 9

With numbers, mankind gradually came to understand that it had just invented a tool for describing, analysing and understanding the world around it that surpassed any purpose it might have hoped for.

Sometimes we have been so pleased with numbers that we have even overdone it. The birth of numbers represented at the same time the birth of the practice of various forms of numerology. This involves attributing magical properties to numbers, interpreting them beyond the bounds of reason, and attempting to read into them messages from the gods and about the fate of the world.

In the sixth century BC, the Greek scholar Pythagoras made numbers the fundamental concept of his philosophy when he declared: ‘Everything is a number.’ According to him, it is numbers that produce geometric figures, which in turn give rise to the four elements of matter – fire, water, earth and air – of which all living things are made. Pythagoras thus created a whole system around numbers. The odd numbers were associated with the masculine, while the even numbers were feminine. The number 10, represented as a triangle called the tetractys, became a symbol of harmony and of the perfection of the cosmos. Pythagoreans were also behind the origin of arithmancy, which claimed to read people’s characters by associating numerical values with the letters comprising their names.

In parallel, there began to be discussions about what constitutes a number. Some authors believed that the unit – 1 – is not a number, because a number denotes a plurality and so can only be considered from 2 onwards. It was even asserted that in order for it to be able to generate all the other numbers, 1 must be simultaneously even and odd.

Later, increasingly animated discussions developed concerning the zero, the negative numbers and the imaginary numbers. In each case, the admission of these new ideas to the circle of numbers led to debate and forced mathematicians to broaden their ideas.

In short, numbers have never ceased to raise questions, and human beings still need time to learn to master these strange creatures that are their brainchildren.

3
LET NO ONE IGNORANT OF GEOMETRY ENTER

Once numbers had been invented, it did not take long for the discipline of mathematics to spread its wings. Various core branches such as arithmetic, logic and algebra gradually sprouted within it, developed to maturity and asserted themselves as disciplines in their own right.

One of these, geometry, rapidly won the popularity stakes and captivated the greatest scholars of antiquity. It was this that singled out the first celebrities of mathematics, such as Thales, Pythagoras and Archimedes, whose names still haunt the pages of our textbooks.

However, before it became a subject for great minds, geometry gained its place on the ground. Its etymology bears witness to this: it is first and foremost the science of the measurement of the Earth, and the first surveyors were hands-on mathematicians. Problems concerning the division of territory were then classics of the craft. How to divide a field into equal parts? How to determine the price of a plot of land from its area? Which of two plots is closer to the river? What route should the future canal follow to make it the shortest possible?

All these questions were paramount in ancient societies where the whole economy still revolved in a vital way around agriculture and hence around the distribution of land. In response to this, geometrical know-how was built up, enriched and transmitted from generation to generation. Anyone equipped with this know-how was certain to hold a central and indisputable place in society.

For these measurement professionals, the rope was often the primary instrument of geometry. In Egypt, ‘ropestretcher’ was a profession in its own right. When the Nile floods led to regular inundations, it was the ropestretchers who were sent for to redefine the boundaries of plots that bordered the river. Using information they recorded about the ground, they planted their stakes, stretched their long ropes across the fields, and then carried out calculations that enabled them to rediscover the boundaries erased by the floodwaters.

They were also the first port of call in constructing buildings, when they took the measurements on the ground and marked the precise location of the building based on architects’ plans. And in the case of a temple or an important monument, it was often the pharaoh in person who symbolically came to stretch the first rope.

It can be said that the rope was the all-purpose tool of geometry. Surveyors used it as a ruler, as compasses, and as a set-square.

To use it as a ruler is straightforward: if you stretch the rope between two fixed points you obtain a straight line. And if you require a graduated ruler, you just tie knots at regular intervals along your rope. For compasses, there is no magic involved either. You simply fix one of the two ends to a stake, stretch the rope and move the other end around the stake. This gives a circle. And if your rope is graduated, you can control the length of the radius exactly.

For the set-square, however, things are slightly more complicated. Let’s look at this particular problem for a few moments: what would you do to draw a right angle? With a bit of research, one can come up with several different methods. If, for example, you draw two circles that intersect each other, then the straight line that joins their centres is perpendicular to the straight line that passes through their two points of intersection. There is your right angle.

From a theoretical point of view, this construction works perfectly, but things are more complicated in practice. Imagine the surveyors, out in the fields, having to lay out two large circles precisely every time they needed to draw a right angle or, more simply, to verify that an angle which had already been constructed was actually a right angle. This was neither fast nor efficient.

The surveyors adopted a different method, which was subtler and more practical: they used their rope directly to form a triangle with a right angle (known as a right-angled triangle). The most famous one is the 3–4–5. If you take a rope divided into twelve intervals by thirteen knots, then you can form a triangle whose sides measure three, four and five intervals, respectively. And, as if by magic, the angle formed by the sides of length 3 and 4 is a perfect right angle.

Four thousand years ago, the Babylonians already had tables of numbers that could be used to construct right-angled triangles. The Plimpton 322 Tablet, which is currently in the collections of Columbia University in New York City, and dates from 1800 BC, contains a table of fifteen triples of such numbers (so-called Pythagorean triples). Apart from the 3-4-5, it has fourteen other triangles, some of which are considerably more complicated, such as the 65-72-97 or even the 1,679-2,400-2,929. Up to a few minor mistakes, such as errors in calculation or transcription, the triangles of the Plimpton Tablet are perfectly exact, and they all have a right angle.

It is difficult to know the precise period from which the Babylonian surveyors began to use their knowledge of right-angled triangles on the ground, but the use of these triangles persisted well beyond the disappearance of the Babylonian civilization. In the Middle Ages, for example, the rope with thirteen knots remained an essential tool for cathedral builders.

On our journey through the history of mathematics, it is by no means uncommon to find certain similar ideas appearing independently at removes of thousands of kilometres and in profoundly different cultural contexts. One such startling coincidence is that during the first millennium BC the Chinese civilization developed a whole mathematical know-how that corresponds remarkably to that of the Babylonian, Egyptian and Greek civilizations of the same period.

This knowledge was amassed over the centuries before being compiled under the Han dynasty, around 2,200 years ago, into one of the world’s first great mathematical works: The Nine Chapters on the Mathematical Art.

The first of these Nine Chapters is entirely devoted to the study of measurements of fields of various shapes. Rectangles, triangles, trapezia, circles, portions of circles and also rings represent geometric figures for which procedures for calculating their areas are described in minute detail. Later in the work, you discover that the ninth and final chapter deals with right-angled triangles. And guess which figure is discussed from the very first sentence of this chapter: the 3-4-5!

Good ideas are like that. They transcend cultural differences and are able to blossom spontaneously wherever human minds are ready to devise and absorb them.

SOME PROBLEMS OF THE PERIOD

Questions about fields, about architecture, or more generally about land planning led the scholars of antiquity to set themselves a great diversity of geometric problems of which the following are examples.

The following statement, taken from the Babylonian Tablet BM 85200, shows that the Babylonians were not content with plane geometry, but also thought in the space dimension.

An excavation. So much as the length, that is the depth. 1 the earth I have torn out. My ground and the earth I have accumulated, 1´10. Length and width: '50. Length and width, what?¹

As you will have gathered, the style of the mathematicians of Babylon was telegraphic in nature. If we expand it further, this same statement might look as follows:

The depth of an excavation is twelve times greater than its length.² If i dig further so that my excavation has one more unit of depth, then its volume will be equal to 7/6. If i add the length to the width i obtain 5/6.³ What are the dimensions of the excavation?

This problem was accompanied by the detailed method for solving it, ending with the solution: the length is ½, the width ⅓ and the depth 6.

Let’s now take a short trip down the Nile. As a matter of course, in the case of the Egyptians, we find problems about pyramids. The following statement is an extract from a famous papyrus, the Rhind papyrus, copied by the scribe Ahmes, dating from the first half of the sixteenth century BC.

A pyramid has a base side of 140 cubits and an inclination ⁴ of 5 palms and 1 digit, what is its altitude?’

The cubit, the palm and the digit were units of measurement of 52.5 centimetres, 7.5 centimetres and 1.88 centimetres, respectively. Ahmes also gave the solution: 93 + ⅓ cubits. In this same papyrus, the scribe also took on the geometry of the circle.

Example of the calculation of a round field with a diameter of 9 khet. What is the value of its area?

The khet is also a unit of measurement representing approximately 52.5 metres. To solve this problem, Ahmes stated that the area of this circular field is equal to that of a square field with a side of length 8 khet. The comparison is extremely useful, for it is much easier to calculate the area of a square than that of a circle. His solution was 8 × 8 = 64. However, the mathematicians who succeeded Ahmes came to discover that his result was not quite exact. The areas of the circle and the square do not quite agree. Since then, many people have attempted to answer the question: how do you construct a square with an area equal to that of a given circle? Many have worn themselves out in vain in this pursuit, but for a reason. Without knowing it, Ahmes was one of the first to tackle what would become one of history’s classic mathematical conundrums: the squaring of the circle.

In China too, people sought to calculate the area of circular fields. The following problem is taken from the first of the Nine Chapters.

Suppose one has a circular field with circumference 30 bu and diameter 10 bu. The question is how big is the field?⁵

Here, a bu is equal to about 1.4 metres. And, as in Egypt, the mathematicians kept tripping over the rug with this figure. The original statement was already known to be false, since a circle of diameter 10 has a circumference slightly greater than 30. However, that did not prevent Chinese scholars from putting an approximate value on the area (75 bu), or from complicating the task further for themselves by continuing on to questions about circular rings!

Suppose one has a field in the shape of a ring with internal circumference 92 bu, external circumference 122 bu, and transverse diameter 5 bu. The question is: how big is the field?

It seems likely that there were never any ring-shaped fields in ancient China – these latter problems suggest that the scholars of the Central Kingdom were into geometry and raised these questions as purely theoretical challenges. Research into ever more improbable and weird-looking geometric figures in order to study and understand them remains a favourite pastime of our mathematicians to this day.

Among the ranks of professional geometers, one must also include the Bematists. While it was the job of surveyors and other ropestretchers to measure fields and buildings, the Bematists had a much grander view of things. In Greece, it was the job of these men to measure long distances by counting their steps.

And sometimes their work could take them a long way from home. For example, in the fourth century BC, Alexander the Great took several Bematists with him on his campaign in Asia, which led him as far as the boundaries of what is now India. These walking geometers thus had to measure routes of several thousand kilometres in length.

Step back a little and imagine for a moment the strange spectacle of these men walking in quick time, traversing the immense landscapes of the Middle East. See them cross the plateaux of Upper Mesopotamia; walk through the arid yellow settings of the Sinai Peninsula to reach the fertile banks of the Nile Valley; then turn back, setting off to brave the mountainous massifs of the Persian Empire and the deserts of what is now Afghanistan. Can you see them, calmly walking on and on, with a dry and monotonous rhythm, and passing through the foothills of the gigantic mountains of the Hindu Kush to return via the shores of the Indian Ocean? And all the while they counted their paces tirelessly.

The image is captivating, and the outrageous challenge of their undertaking seems insane. And yet, their results were remarkably accurate: less than 5 per cent difference on average between their measurements and the true distances we know today. Alexander’s Bematists thus made it possible to describe the geography of his kingdom in a way that had never before been achieved for so vast a region.

Two centuries later, in Egypt, a scholar of Greek origin named Eratosthenes conceived an even larger project – that of measuring nothing less than the circumference of the Earth. Of course, there was no question of sending poor Bematists on a tour of the planet. However, based on skilful observations of the difference in the inclination of the sun’s rays between the towns of Syene (the present-day Aswan) and Alexandria, Eratosthenes calculated that the distance between the two towns must represent one fiftieth of the total circumference of the Earth.

Naturally, he then called upon the Bematists to undertake the measurement. Unlike their Greek counterparts, the Egyptian Bematists did not count their own steps, but those of a camel they took with them. The animal is renowned for the regularity of its walking. After long days of travelling along the Nile, the verdict was issued: the two towns were 5,000 stadia apart and the girth of our planet was therefore 250,000 stadia, or 39,375 kilometres. Once again, the result was astonishingly accurate, as we know today that this circumference measures exactly 40,008 kilometres. The error was less than 2 per cent.

Perhaps more than any other ancient people, the Greeks gave geometry pride of place in the heart of their culture. It was recognized for its rigour and its ability to form the mind. For Plato, it was an obligatory rite of passage for any budding philosopher, and legend has it that the slogan ‘Let no one ignorant of geometry enter’ was engraved at the entrance to his academy.

Geometry was so much in vogue that it eventually spilled over into other disciplines. The arithmetical properties of numbers were thus interpreted in geometric language. See, for example this definition by Euclid extracted from the seventh book of his Elements of Mathematics dating from the third century BC:

And when two numbers multiplying one another make some other number then the number so created is called plane, and its sides are the numbers which multiply one another.

If I take the product 5 × 3, the numbers 5 and 3 are thus called, according to Euclid, the ‘sides’ of the multiplication. Why is this? It is simply because a multiplication can be represented as the area of a rectangle. If the latter has a width equal to 3 and a length of 5, its area is 5 × 3. The numbers 3 and 5 are precisely the sides of the rectangle. The result of the multiplication, 15, is for its part called the ‘plane’, since it corresponds geometrically to a surface.

Similar constructions relate to other geometric figures. For example, a number is said to be triangular if it can be represented in the form of a triangle. The first triangular numbers are 1, 3, 6 and 10.

The last of the above triangles with ten points is nothing other than the famous tetractys that Pythagoras and his disciples employed as the symbol of the harmony of the cosmos. On the same principle, we also find the square numbers whose first representatives are 1, 4, 9 and 16.

And we could of course continue for a long time like that with all kinds of figures. Thus, the geometric representation of numbers makes it possible to render visual and evident properties that are otherwise seemingly incomprehensible.

By way of example, have you ever tried adding the odd numbers successively, one after another: 1 + 3 + 5 + 7 + 9 + 11 + …? No? Well, something completely surprising happens. Look:

Can you see something special about the numbers that emerge? In the order in which they occur: 1, 4, 9, 16 … These are the square numbers!

And you can continue as long as you wish; this rule will never be broken. Add the first ten odd numbers from 1 to 19, if you are brave enough, and you will reach 100, which is the tenth square:

Isn’t that surprising? But why? What kind of miracle makes this property always hold? Of course, it would be possible to give a numerical proof, but there is a much simpler way. Using the geometrical representation, you just have to slice up the square numbers and the explanation becomes plain to see.

Each slice adds an odd number of balls, and increases the side of the square by one unit. This completes the simple and lucid proof.

In the kingdom of mathematics, geometry was the queen, and no assertion could be verified without coming under its scrutiny. Its reign lasted far beyond antiquity and the Greeks. It was almost two thousand years before the scholars of the Renaissance launched a vast movement to modernize mathematics which would dethrone geometry in favour of a completely new language: that of algebra.

4
THE AGE OF THEOREMS

It is the beginning of May. At midday the sun is shining over the Parc de la Villette in northern Paris. Opposite me towers the Cité des Sciences et de l’Industrie (City of Science and Industry), the largest science museum in Europe, with its Geode in the foreground. Built in the mid-1980s, this strange cinema of some 36 metres in diameter resembles a gigantic sphere with many facets.

This is very much a stopping-off place. There are tourists clutching their cameras – they have come to see this curious building in the French capital. There are families out for their Wednesday stroll. A few lovers are sitting on the grass or walking hand in hand. Here and there, a jogger zigzags through the tide of local residents who pass through with a show of indifference, scarcely casting a glance towards the strange apparition of this shimmering sphere in the middle of their day-to-day lives. All around the sphere, children amuse themselves by observing the distorted image of the surrounding world that it reflects.

As for me, if I am here today it is because I am especially interested in its geometry. I start to walk closer, scanning it closely. Its surface comprises thousands of triangular mirrors assembled together. At first sight the assembly may appear perfectly regular, but after a few minutes’ scanning I begin to notice several irregularities. At certain well-defined points, the triangles grow distorted and are stretched as if by a malformation of the structure. Although they are grouped together six per hexagon, forming a perfectly regular meshing almost everywhere on the sphere, there are a dozen special points where these triangles are only grouped in fives.

Representation of the Geode and its thousands of triangles. The points where the triangles occur in groups of five are shown in dark grey.

These irregularities are almost invisible at first glance. The fact is that most passers-by pay no attention to them. At the same time, as a mathematician, I do not see anything surprising in them. I even have to say that I expected to find them. The architect did not make a mistake; moreover, there are numerous other buildings elsewhere in the world with a similar geometry, and all have these same dozen points where the basic pieces are grouped together in fives rather than sixes. These points are the results of inescapable geometric constraints discovered over 2,000 years ago by the Greek mathematicians.

Theaetetus of Athens was a mathematician of the fourth century BC who is generally credited with the complete description of regular polyhedra. A polyhedron, in geometry, is simply a three-dimensional figure delimited by several plane faces. For example, cubes and pyramids belong to the family of polyhedra, unlike spheres and cylinders, which have rounded faces. The geode, with its triangular faces, can also be considered as a giant polyhedron, even though its large number of faces makes it resemble a sphere from a distance.

Theaetetus was especially interested in perfectly symmetric polyhedra, those for which all the faces and all the angles are the same. And his discovery was at least disconcerting: he only found five, and proved that there are no others. Five solids and that’s all – not a single one more.

From left to right: the tetrahedron, the hexahedron, the octahedron, the dodecahedron and the icosahedron.

To this day, the polyhedra are usually named according to the number of their faces, written in Ancient Greek, followed by the suffix -hedron. For example, in geometry, the cube with its six square faces is referred to as the hexahedron. The tetrahedron, the octahedron, the dodecahedron and the icosahedron have four, eight, twelve and twenty faces, respectively. These five polyhedra subsequently became known as the Platonic solids.

Platonic? Why are they not associated in name with Theaetetus? History is sometimes unfair, and the discoverers are not always those who receive the honours of posterity. The Athenian philosopher Plato had nothing to do with the discovery of the five solids, but he made them famous through a theory that associated them with the elements of the cosmos: fire is associated with the tetrahedron, earth with the hexahedron, air with the octahedron and water with the icosahedron. As for the dodecahedron, with its pentagonal faces, Plato claimed that it was the shape of the Universe. This theory was abandoned by science a long time ago, and yet these five regular polyhedra are still conventionally associated with Plato.

As it happens, Theaetetus himself was not actually the first to discover these five solids. There exist considerably older sculpted models and written descriptions of them. For example, a collection of small sculpted stone balls reproducing the shapes of the Platonic solids came to light in Scotland and is said to date from one thousand years before Theaetetus. These pieces are currently conserved in the Ashmolean Museum in Oxford.

So is Theaetetus no more worthy than Plato? Is he also an impostor? He is not a total impostor, for while the five figures were known before his time, he was the first to demonstrate clearly that the list was complete. There is no point in looking any further, Theaetetus tells us, no one will ever find any more. This assertion has something reassuring about it. It removes a dreadful doubt. Phew – they’re all present and correct!

This step is emblematic of the way in which Greek mathematicians came to approach mathematics. For them, it was no longer a matter of simply finding solutions that worked. They wanted to exhaust the problem. They wanted to make sure that nothing escaped them. And for that reason, they developed the art of mathematical exploration to its peak.

Let us now return to our Geode. Theaetetus’ proof is final: it is impossible for a polyhedron with several hundred faces to be perfectly regular. But then what do you do when you are an architect and you want to create a building that resembles a regular sphere as closely as possible? It is technically difficult to design the building in a single piece. No, there’s nothing you can do about that, and you have to assemble a multitude of small faces. But how do you create such a structure?

We can imagine various solutions. One of these involves taking one of the Platonic solids and modifying it. Consider the icosahedron, for example. With its twenty triangular faces, it seems the roundest of the five. To smooth it out, each of its faces can be subdivided into several smaller faces. The polyhedron obtained in this way can then be deformed, as though one were inflating it by blowing into it, in order to come closer to a sphere.

Here, for example, is what happens when you subdivide each face of the icosahedron into four smaller triangles.

The icosahedron

Inflated icosahedron with subdivided faces

Icosahedron with faces subdivided into four

In geometry, such a polyhedron is called a geode. Etymologically, this means a figure that is shaped like the Earth – that is to say, it resembles a sphere. There is nothing very complicated about this principle. It is precisely this construction that was used for the Geode of la Villette! The subdivision of the faces is, however, very much finer: the basic triangles of the icosahedron are here subdivided into 400 smaller triangles, making a total of eight thousand triangular facets.

In reality, the Geode has rather fewer than 8,000 facets – only 6,433 – since it is not complete. Its base, which rests on the ground, is truncated and some triangles are missing. Nevertheless, this structure allows us to explain the presence of the twelve irregularities. The latter correspond simply to the twelve vertices of the basic icosahedron. In other words, these are the points where the original large triangles met in fives to form the points of the icosahedron. These vertices, which were originally pointed, have been flattened with the increase in the number of faces to the point of becoming almost invisible. However, their presence remains firmly fixed in the layout of the triangles, and the twelve irregularities are there to remind attentive passers-by of this.

Theaetetus probably didn’t imagine that his studies would one day permit the construction of buildings such as the Geode. And the great power of mathematics, as in the kind that the scholars of ancient Greece developed, lies in its formidable ability to engender new ideas. The Greeks gradually began to detach their questioning from concrete problems and thus to generate original and inspiring models out of simple intellectual curiosity. Although they might often have seemed to have no concrete use at the time they were conceived, these models sometimes turned out to be astonishingly useful long after their creators had disappeared.

Nowadays, we come across the five Platonic solids in different contexts. For example, they are well suited for use as dice in board games. Their regularity ensures that the dice are balanced, that is to say that all the faces have the same chance of occurring. Everyone is familiar with cubic dice with six faces, but the most inveterate players know that numerous other games also use the other four shapes to vary the enjoyment and the probabilities.

As I move away from the Geode, a little further on I come across some children who have produced a ball and are starting a makeshift game of football on the lawns of la Villette. They are not aware of it, but at this moment they too owe a very great deal to Theaetetus. Have they noticed that their ball also has its geometric patterns? Most footballs are formed on the same model: twenty hexagonal pieces (six-sided) and twelve pentagonal pieces (five-sided). On traditional balls, the hexagons are white while the pentagons are black. And even when the surface of the ball carries all kinds of printed inscriptions, you only have to look carefully at the seams between the different pieces to see once again the inevitable twenty hexagons and twelve pentagons.

A truncated icosahedron – that is the name geometers give to the football. And its structure is owed to the same constraints as the Geode: it has to be as regular and as round as possible. The only thing is that, in order to achieve this result, the creators of this model used a different method. Instead of subdividing the faces in order to be able to round off the corners, they simply chose to cut off the corners. Imagine an icosahedron in modelling clay, take a knife and simply slice off the vertices. The twenty triangles with their points sliced off become hexagons, while the twelve points removed lead to the appearance of the twelve pentagons.

The twelve pentagons on a football thus have the same origin as the twelve irregularities on the surface of the Geode: they are the original locations of the twelve vertices of the icosahedron.

As I leave the Parc de la Villette, I come across a girl with a handkerchief. What is wrong? I wonder. She doesn’t seem to be doing too well. Perhaps she has fallen victim to a nasty attack of micro-icosahedra? Certain microscopic organisms such as viruses naturally assume the form of icosahedra or dodecahedra. This is the case with the rhinoviruses, which are responsible for most colds.

If these tiny creatures adopt such forms, it is for the same reasons for which we use these forms in architecture or for our footballs, namely for the sake of symmetry and economy. Thanks to icosahedra, balls consist of only two different types of pieces. In the same way, the virus membrane consists only of a few different types of molecules (four for the rhinoviruses) that fit together by always repeating the same pattern. The genetic code needed to create such an envelope is thus very much more concise and economical than what it would have taken to describe a structure with no repeatable symmetry at all.

Once again, Theaetetus would have been very surprised to learn of all the applications in which his polyhedra are hidden.

Time to leave the Parc de la Villette and pick up the chronological flow of our history again. How did ancient mathematicians such as Theaetetus come to raise ever more general and theoretical questions? To understand this, we must go back several thousand years to the eastern rim of the Mediterranean.

While the Babylonian and Egyptian cultures were slowly declining, ancient Greece was about to enter its greatest centuries. From the sixth century BC onwards, the Greek world entered an unprecedented period of cultural and scientific effervescence. Philosophy, poetry, sculpture, architecture, theatre, medicine and even history were also disciplines that underwent a true revolution. To this day the exceptional vitality of that period retains a certain fascination and mystery, and last but not least in that vast intellectual movement, mathematics had pride of place.

When we think of ancient Greece, the first image that springs to mind is often that of the city of Athens, dominated by its Acropolis. There, we imagine citizens in white robes, who had just invented the first democracy in history, strolling among Pentelic marble temples and a few olive trees. But this vision comes nowhere near to a representation of the whole of the Greek world in all its diversity.

In the eighth and seventh centuries BC, a swarm of Greek colonies spread around the edges of the Mediterranean. Sometimes there was an integration between these colonies and the local peoples, with a partial adoption of the latter’s customs and lifestyles. It is far from the case that all Greeks shared the same form of existence. Their food, pleasures, beliefs and political systems varied widely from one region to another.

Thus, Greek mathematics did not emerge in a specific place where all the scholars knew each other and met each other daily, but over a vast geographic and cultural region. The driving forces behind the mathematical revolution included the contact with the older civilizations whose heir it became, and the mix of Greece’s own diversity. Numerous scholars undertook a pilgrimage to Egypt or the Middle East at some stage in their lives, as an obligatory rite of passage in their training. A sizeable amount of Babylonian and Egyptian mathematics was integrated and extended by Greek scholars.

The first great Greek mathematician, Thales, was born at the end of the seventh century in the town of Miletus (present-day Milet), on the south-west coast of what is now Turkey. Although he is mentioned in various sources, it is difficult today to extract reliable information about his life and his work. As with many scholars of this period, various legends spread by overzealous disciples grew up after his death, and have made it hard to unravel the true from the false. The scholars of that time were not averse to spinning yarns, and they not infrequently came to an accommodation with the truth when it did not suit them.

According to one of the many stories put about on the subject, Thales was particularly absent-minded: he is said to have been a prime example from an age-old tradition of scatter-brained scholars. One tale reports that one night he was seen to fall into a well while roaming with his nose in the air observing the stars. Another tells us that he died aged almost eighty while attending a sporting event: so wrapped up did he become in the spectacle that he forgot to eat and drink.

His scientific prowess is also the subject of singular tales. Thales is viewed as the first person to have correctly predicted a solar eclipse. This one took place in the middle of a battle between the Medes and the Lydians, on the banks of the Halys River in the west of what is now Turkey. When night intervened in broad daylight, the warriors saw it as a message from the gods, and decided at once to make peace. Nowadays, the prediction of eclipses or their reconstitution in the past has become child’s play for our astronomers. Thanks to them, we know that this eclipse took place on 28 May 584 BC, making the battle of Halys the oldest historical event that we are able to date with such accuracy.

Thales’ greatest success was achieved on a journey to Egypt. The story goes that the Pharaoh Amasis in person challenged him to measure the height of the Great Pyramid. Until then, the Egyptian scholars who had been consulted had failed to answer the question correctly. Thales not only met the challenge, but did so in an elegant way, using a particularly clever method. He planted a stick vertically in the ground and waited for the time of day when the length of its shadow was equal to its height. At that very same moment, he marked and then measured the shadow of the pyramid, which likewise must be equal to its height. That did the trick!

The story is certainly entertaining, but once again its historical truth is dubious. Moreover, as it is told, the anecdote is quite scornful of the Egyptian scholars of the period, although papyruses such as that copied by Ahmes show that these scholars knew perfectly well how to calculate the height of their pyramids more than a thousand years before Thales came on the scene. So where does the truth lie? Did Thales really measure the height of the pyramid? Was he the first to use the shadow method? And what if he had simply measured the height of an olive tree in front of his house in Miletus, and his disciples took it upon themselves to embroider the story after his death? We have to abide by the evidence, and we shall probably never have it.

Be that as it may, Thales’ geometry, on the other hand, is very real, and whether he applied it to the Great Pyramid or to an olive tree, the shadow method is no less appealing. This method represents a particular case of a property now known as Thales’ theorem. Several other mathematical results are also attributed to Thales: the circle is divided in two equal parts by any diameter (fig. 1); the angles at the base of an isosceles triangle are equal (fig. 2); the opposite angles produced at the point of intersection of two straight lines are equal (fig. 3); if the three vertices of a triangle lie on a circle and one side passes through the centre of this circle, then the triangle is a right-angled triangle (fig. 4). In fact, this last statement is sometimes also referred to as Thales’ theorem.

Fig. 1

Fig. 2

Fig. 3

Fig. 4

I want to look at this strange word that is as fascinating as it is awesome: what is a theorem? Etymologically, the word comes from the Greek thea (contemplation) and horáō (to see), so a theorem would be a sort of observation on the world of mathematics, a fact that would have been noted, examined and then recorded by mathematicians. Theorems can be transmitted orally or in writing, and are similar to one’s grandmother’s recipes or to weather lore that has been tested over the generations and is confidently believed to be true. One swallow does not make a summer, bay leaves soothe rheumatism, and the 3-4-5 triangle has a right angle. These are things we believe to be true and that we keep in mind in order to use them when they are relevant.

According to this definition, the Mesopotamians, the Egyptians and the Chinese also stated theorems. However, beginning with Thales, the Greeks gave them a new dimension. For them, a theorem not only had to state a mathematical truth, but that truth must be formulated in the most general way possible, and accompanied by a proof of its validity.

Back to one of the properties attributed to Thales: that the diameter of a circle divides the circle into two equal parts. Such an assertion may seem disappointing coming from a scholar of Thales’ calibre. It seems self-evident; why did we have to wait until the sixth century BC for such a trivial assertion to be stated? There is no doubt that Egyptian and Babylonian scholars must have known that a long time beforehand.

Make no mistake. The audacious thing about that property attributed to Thales is not so much its content as its formulation. Thales dared to speak about a circle without saying precisely which one. In order to state the same rule, the Babylonians, Egyptians and Chinese would have used an example. Draw a circle of radius 3 and draw one of its diameters, they would have said, and this circle is divided into two equal parts by this diameter. And when one example did not suffice for an understanding of the rule, they would have given a second one, a third, and a fourth if necessary. They would have given as many examples as needed to enable readers to understand that they could repeat the same procedure on every circle they might meet. But the general assertion was never formulated.

Thales reached a milestone.

‘Take a circle, any one you like, I don’t need to know which. It may be enormous or tiny. Draw it in the horizontal, in the vertical or on an inclined plane, it doesn’t matter to me. I don’t care at all about your particular circle and how you have drawn it. However, I assert that its diameter divides it into two equal parts.’

With this operation, Thales definitively assigned the status of abstract mathematical objects to geometric figures. This step in thinking was similar to that taken 2,000 years earlier by the Mesopotamians when they considered numbers independently from the objects counted. A circle was no longer a figure drawn on the ground, on a tablet or on a papyrus. The circle became a fiction, an idea, an abstract ideal all of whose real representations are merely imperfect instances.

From that point on, mathematical truths were stated in a concise and general manner, independently of the various particular cases they covered. It is these statements that the Greeks then called theorems.

Thales had several disciples in Miletus. The two most famous were Anaximenes and Anaximander. Anaximander in turn had his own disciples, who included a certain Pythagoras, who would give his name to the most famous theorem of all time.

Pythagoras was born at the start of the sixth century BC on the island of Samos, situated off the coast of what is now Turkey, and less than 40 kilometres away from the town of Miletus. As a young man he gained experience through his travels in the ancient world, and chose to settle in the town of Croton, in the south-east of present-day Italy. It was there that he founded his school in 532 BC.

Pythagoras and his disciples were not just mathematicians and scholars, but also philosophers, monks and politicians. Yet it must be said that if we were to transpose it to our time, the community begun by Pythagoras would undoubtedly be perceived to be among the most obscure and most dangerous of sects. The life of the Pythagoreans was governed by a set of precise rules. For example, anyone wishing to join the school had to go through a period of five years of silence. The Pythagoreans had no individual possessions: all their belongings were shared. They used various symbols such as the tetractys or the pentagram in the shape of a star with five branches to recognize one another. Moreover, the Pythagoreans thought of themselves as enlightened people, and thought it normal that political power should come to them. They firmly repressed the rebellions of towns that refused to accept their authority. In fact, it was in one such riot that Pythagoras died at the age of eighty-five.

The number of myths of all kinds that were invented around Pythagoras is also impressive. His disciples were scarcely lacking in imagination, as we can now see. According to them, Pythagoras was the son of the god Apollo. The name Pythagoras also means literally ‘he who was announced by the Oracle’: the Oracle of Delphi was in fact the oracle of the temple of Apollo, and it is she who is said to have told Pythagoras’ parents about the forthcoming birth of their child. According to the Oracle, Pythagoras would be the most handsome and wisest of men. After such a birth, the Greek scholar was predestined for great things. Pythagoras remembered all his previous lives. According to this, he had in particular been one of the heroes of the Trojan War called Euphorbus. In his youth, Pythagoras took part in the Olympic Games and took the laurels in all the pygmachia events (pugilism, the ancestor of our boxing). Pythagoras invented the very first musical scales. Pythagoras was able to walk in the air. Pythagoras died and was resuscitated. Pythagoras was a talented soothsayer and healer. Pythagoras had control over animals. Pythagoras had a golden thigh.

While most of these legends are so far-fetched that no one believes them, in other cases the jury is still out. Is it true, for example, that Pythagoras was the first to use the word ‘mathematics’? The facts are so sketchy that some historians have even come to speculate that Pythagoras was a purely fictional person, dreamed up by the Pythagoreans to serve as their tutelary figure.

Therefore, since it is not possible to learn more about the man, let us return to the subject for which he is still known to schoolchildren the whole world over more than 2,500 years after his death: Pythagoras’ theorem. What does this famous theorem tell us? Its statement may seem astonishing, because it establishes a link between two apparently unconnected mathematical concepts: right-angled triangles and the square numbers.

Let us return to our favourite triangle, the 3-4-5. From the lengths of the three sides we can construct three square numbers: 9, 16 and 25.

We now spot a curious coincidence: 9 + 16 = 25. The sum of the squares of sides 3 and 4 is equal to the square of side 5. This might be put down to chance, yet if we try to reproduce this calculation for a different right-angled triangle, it still works. Consider, for example, the triangle 65-72-97, which appears on the Babylonian Plimpton tablet. The three corresponding squares are 4,225, 5,184 and 9,409. And Pythagoras’ theorem does not fail: 4,225 + 5,184 = 9,409. For such large numbers, it becomes difficult to believe in a simple coincidence.

You can try with all the right-angled triangles you like, small or large, fat or thin, it always works. In a right-angled triangle, the sum of the squares of the two sides that form the right angle is always equal to the square of the third side (which we call the hypotenuse). And it also works the other way round: if, in a given triangle, the sum of the squares of the two smallest sides is equal to the square of the largest side, then the triangle is right-angled. That is Pythagoras’ Theorem.

Of course, we do not know for certain whether Pythagoras or his disciples actually contributed to this theorem. Even though the Babylonians never expressed it in the general form that we have just seen, it is highly likely that they already knew this result more than one thousand years earlier. For otherwise, how would they have been able to discover all the right-angled triangles that appear on the Plimpton tablet with such accuracy? The Egyptians and the Chinese probably also knew the theorem, which was also clearly stated in the commentaries that were added to the Nine Chapters in the centuries after it was written.

Some accounts claim that Pythagoras was the first to give a proof of the theorem. However, there is no reliable source to confirm this, and the oldest proof that has come down to us is only the one in Euclid’s Elements, which was written three centuries later.

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