The Number Mysteries: A Mathematical Odyssey through Everyday Life. Marcus Sautoy du
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Название: The Number Mysteries: A Mathematical Odyssey through Everyday Life
Автор: Marcus Sautoy du
Издательство: HarperCollins
Жанр: Математика
isbn: 9780007362561
isbn:
FIGURE 1.02 The interaction over 100 years between populations of cicadas with a 7-year life cycle and predators with a 6-year life cycle.
For example, suppose that the predators appear every 6 years. Cicadas that appear every 7 years will coincide with the predators only every 42 years. In contrast, cicadas that appear every 8 years will coincide with the predators every 24 years; cicadas appearing every 9 years will coincide even more frequently: every 18 years.
FIGURE 1.03 The interaction over 100 years between populations of cicadas with a 9-year life cycle and predators with a 6-year life cycle.
Across the forests of North America there seems to have been real competition to find the biggest prime. The cicadas have been so successful that the predators have either starved or moved out, leaving the cicadas with their strange prime number life cycle. But as we shall see, cicadas are not the only ones to have exploited the syncopated rhythm of the primes.
Cicadas v predators
Cut out the predators and the two cicada families. Place predators on the numbers in the six times table. Each player takes a family of cicadas. Take three standard six-sided dice. The roll of the dice will determine how often your family of cicadas appears. For example, if you roll an 8, then place cicadas on each number in the 8 times table. But if there is a predator already on a number, you can’t place a cicada—for example, you can’t place a cicada on 24 because it’s already occupied by a predator. The winner is the person with the most cicadas on the board. You can vary the game by changing the period of the predator, from 6 to some other number.
How are the primes 17 and 29 the key to the end of time?
During the Second World War, the French composer Olivier Messiaen was incarcerated as a prisoner of war in Stalag VIII-A, where he discovered a clarinettist, a cellist and a violinist among his fellow inmates. He decided to compose a quartet for these three musicians and himself on piano. The result was one of the great works of twentieth-century music: Quatuor pour la fin du temps—‘The Quartet for the End of Time’. It was first performed to inmates and prison officers inside Stalag VIII-A, with Messiaen playing a rickety upright piano they found in the camp.
In the first movement, called ‘Liturgie de Crystal’, Messiaen wanted to create a sense of never-ending time, and the primes 17 and 29 turned out to be the key. While the violin and clarinet exchange themes representing birdsong, the cello and piano provide the rhythmic structure. In the piano part there is a 17-note rhythmic sequence repeated over and over, and the chord sequence that is played on top of this rhythm consists of 29 chords. So as the 17-note rhythm starts for the second time, the chord sequence is just coming up to about two-thirds of the way through. The effect of the choice of prime numbers 17 and 29 is that the rhythmic and chordal sequences wouldn’t repeat themselves until 17×29 notes through the piece.
It is this continually shifting music that creates the sense of timelessness that Messiaen was keen to establish—and he is using the same trick as the cicadas with their predators. Think of the cicadas as the rhythm and the predators as the chords. The different primes 17 and 29 keep the two out of sync, so that the piece finishes before you ever hear the music repeat itself.
FIGURE 1.04 Messiaen’s ‘Liturgie de Crystal’ from the Quatuor pour la fin du temps. The first vertical line indicates where the 17-note rhythm sequence ends. The second line indicates the end of the 29-note harmonic sequence.
Messiaen wasn’t the only composer to have utilized prime numbers in music. Alban Berg also used a prime number as a signature in his music. Just like David Beckham, Berg sported the number 23—in fact he was obsessed by it. For example, in his Lyric Suite, 23-bar sequences make up the structure of the piece. But embedded inside the piece is a representation of a love affair that Berg was having with a rich married woman. His lover was denoted by a 10-bar sequence which he entwined with his own signature 23, using the combination of mathematics and music to bring alive his affair.
Like Messiaen’s use of primes in the ‘Quartet for the End of Time’, mathematics has recently been used to create a piece that although not timeless, nevertheless won’t repeat itself for a thousand years. To mark the turn of the new millennium, Jem Finer, a founding member of The Pogues, decided to create a music installation in the East End of London that would repeat itself for the first time at the turn of the next millennium, in 3000. It’s called, appropriately, Longplayer.
Finer started with a piece of music created with Tibetan singing bowls and gongs of different sizes. The original source music is 20 minutes and 20 seconds long, and by using some mathematics similar to the tricks employed by Messiaen he expanded it into a piece which is 1,000 years long. Six copies of the original source music are played simultaneously but at different speeds. In addition, every 20 seconds each track is restarted a set distance from the original playback, but the amount by which each track is shifted is different. It is in the decision of how much to shift each track that the mathematics is used to guarantee that the tracks won’t align perfectly again until 1,000 years later.
You can listen to Longplayer at http://longplayer.org or by using your smartphone to scan this code.
It’s not just musicians who are obsessed with prime numbers: they seem to strike a chord with practitioners in many different fields of the arts. The author Mark Haddon only used prime number chapters in his best-selling book The Curious Incident of the Dog in the Night-time. The narrator of the story is a boy with Asperger’s syndrome called Christopher who likes the mathematical world because he can understand how it will behave—the logic of this world means there are no surprises. Human interactions, though, are full of the uncertainties and illogical twists that Christopher can’t cope with. As Christopher explains, ‘I like prime numbers … I think prime numbers are like life. They are very logical but you could never work out the rules, even if you spent all your lifetime thinking about them.’
Prime numbers have even had an outing in the movies. In the futuristic thriller Cube, seven characters are trapped in a maze of rooms which resembles a complex Rubik’s cube. Each room in the maze is cube-shaped with six doors leading through to more rooms in the maze. The film begins when the characters wake up to find themselves inside this maze. They have no idea how they got there, but they have to find a way out. The trouble is that some of the rooms are booby-trapped. The characters need to find some way of telling whether a room is safe before they enter it, for a whole array of horrific deaths await them, including being incinerated, covered in acid and being cheese-wired into tiny cubes—as they discover when one of them is killed.
One of the characters, Joan, is a mathematical whiz, and she suddenly sees that the numbers at the entrance to each room hold the key to revealing whether a trap lies ahead. It seems that if any of the numbers at the entrance are prime, then the room contains a trap. ‘You beautiful brain,’ declares the leader of the group at this piece of mathematical deduction. It turns out that they also have to watch out for prime powers, but this proves beyond the clever Joan. Instead they have to rely on one of their number who is an autistic savant, and he turns out to be the only one to make it out of the prime number maze alive.
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