Strange mathematical objects – Mickaël Launay (Amphitheatre, 11:45 a.m.) =================================================================================================================================================================================================================================================================================
Having machines does not mean we can dispense with thinking. Sometimes their limitations even force us to be more ingenious. Take the Caroline calculator, marketed in the mid-20th century. Each wheel corresponds to a decimal place (units, tens...), and to add the desired numbers, one simply turns the wheels by the required number of notches. There is just one catch: the wheels turn in only one direction. There is no way to go backwards, and therefore no way to subtract. Or so it seems at first, because if you enter 9,999,999 + 1, the machine, whose capacity is limited to seven digits, will display... 0. This tells us that 9,999,999 = -1. In short, we are calculating modulo 10,000,000, and likewise -2 = 9,999,998 and -42 = 9,999,958. We can now subtract again! Although today's computers have no trouble with subtraction, they, like the Caroline, have their limits, and it is still up to humans to use them with ingenuity and creativity.
Choosing sounds and their ratios in music – Daniel Justens (Amphitheatre, 1 p.m.) ===============================================================================================================================================================================================================================================================================================================================================
Music is a universal language. In this respect, it has much in common with mathematics. But the connection goes further: music is mathematical in its very conception. When a string vibrates, it produces a sound. Cut the string into two equal parts and set one half vibrating. The sound resembles the first, but we describe it as "higher-pitched." Musically, we give the two sounds the same name—the same note, with one an octave above the other—because our brains perceive them as similar. In fact, the half-string vibrates faster than the whole string: twice as fast. Now divide the original string into three. This time, the sound produced by one third of the string seems different from the original sound, but their combination—the chord they form—sounds harmonious to us. The third of the string vibrates three times as fast. We have therefore established a system for constructing pairs of sounds that are pleasing to the ear. Nothing prevents us from continuing. But when should we stop? We stop when we come approximately back to one of the octaves of the original sound. That is why all musical systems in the world rely on 5, 12 or 43 sounds, whose corresponding powers of three are close to powers of two!
Where does mathematical vocabulary come from? – Bertrand Hauchecorne (Amphitheatre, 2:45 p.m.) =======================================================================================================================================================
The Greeks were not the first to use mathematics, but they were certainly pioneers in asking what constitutes a proof. They therefore created the words they needed to develop their reasoning. In their language, mathemata was the plural of a term meaning both the act of learning and its result: knowledge, and then science, primarily science based on reasoning. The need to survey land led to questions about the simplest shapes—circles, rectangles and so on—and geometry occupied a very important place at the time. In Greek antiquity, the goddess Gê personified the Earth; geometry was thus the measuring of the earth. Proofs require statements. To contemplate a theorem is to watch a spectacle: that is what the word's etymology whispers to us, for the root thea, meaning spectacle, also appears in theatre and theory. The Greeks forged the concept of number, independent of the objects being counted, and called it arithmos. Thus arithmetic, the science of numbers, was born. These words, twenty-five centuries old, offer us a highly poetic vision of mathematics and remind us of the bond the ancients shared with the Queen of Sciences.
Chicago, an Oulipian game – Olivier Salon and Hervé Le Tellier (Amphitheatre, 3:45 p.m.) ========================================================================================================================================================================================================
Chicago is an invention by Paul Fournel in which players guess a word, rather like in a charade, except that the charade's definitions are replaced by four words from the same semantic field, which logically point to the syllable to be guessed. For example, the clues:
• Bridge → He does me a favor
• Vault → He is useful to me
• Arcade → He serves me
• Portico → He helps me out
legitimately lead to the answer: Arch—he aids me.
Hervé Le Tellier and Olivier Salon, partners in mischief, Oulipo members and no strangers to mathematics, get the audience playing Chicagos based on mathematicians' names for Tangente's 30th anniversary. A playful, fast-paced, funny and exuberant session!
John Conway's Game of Life – Jean-Paul Delahaye (Amphitheatre, 4:30 p.m.) ================================================================================================================================================================================================
Conceived by mathematician John Conway in 1970, the Game of Life cellular automaton defines an imaginary world. The physics of this universe has been carefully studied and continues to be explored by several thousand enthusiasts. Constructing initial configurations with particular properties—for example, generating the prime numbers one by one or computing the number π—has given us a remarkable repertoire of dynamic forms. They are the inhabitants of a parallel, self-contained and mysterious universe. Their combinations and the movements they produce appear to us as manifestations of a form of life different from the one we know on Earth. Perhaps this is the dream of the forms in our computers?
The image shows three stages, at different scales, in the development of a Game of Life configuration that computes the number π. The population at generation n is asymptotic to (π - 2)n2/720. In other words, if m is the number of living cells at stage n, then 720m/n2 + 2 is an approximation of π. The configuration was created by Dean Hickerson.