LITMIR.BIZ

      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!