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.
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