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S Y M M E T RY A Journey into the Patterns of Nature Marcus du Sautoy For Tomer, Magaly and Ina and my mathematical children. Contents 1 2 3 4 5 6 7 8 9 10 11 12 August: Endings and Beginnings September: The Next Roll of the Dice October: The Palace of Symmetry November: Tribal Gathering December: Connections January: Impossibilities February: Revolution March: Indivisible Shapes April: Sounding Symmetry May: Exploitation June: Sporadic July: Reflections 1 32 62 88 113 140 173 208 243 264 296 326 Further Reading Acknowledgements Picture credits Index About the Author Other Books by Marcus du Sautoy Credits Cover Copyright 355 361 363 367 About the Publisher . 1 2 1 2 4 1 1 3 August: Endings and Beginnings The universe is built on a plan the profound symmetry of which is somehow present in the inner structure of our intellect. paul vale´ ry Midday, 26 August, the Sinai Desert It’s my 40th birthday. It’s 40 degrees. I’m covered in factor 40 sun cream, hiding in the shade of a reed shack on one side of the Red Sea. Saudi Arabia shimmers across the blue water. Out to sea, waves break where the coral cliff descends to the sea floor. The mountains of Sinai tower behind me. I’m not usually terribly bothered by birthdays, but for a mathematician 40 is significant – not because of arcane and fantastical numerology, but because there is a generally held belief that by 40 you have done your best work. Mathematics, it is said, is a young man’s game. Now that I have spent 40 years roaming the mathematical gardens, is Sinai an ominous place to find myself, in a barren desert where an exiled nation wandered for 40 years? The Fields Medal, which is mathematics’ highest accolade, is awarded only to mathematicians under the age of 40. They are distributed every four years. This time next year, the latest batch will be announced in Madrid, but I am now too old to aspire to be on the list. As a child, I hadn’t wanted to be a mathematician at all. I’d decided at an early age that I was going to study languages at university. This, I realized, was the secret to fulfilling my ultimate dream: to become a spy. My mum had been in the Foreign Office before she got married. 2 Symmetry The Diplomatic Corps in the 1960s didn’t believe that motherhood was compatible with being a diplomat, so she left the Service. But according to her, they’d let her keep the little black gun that every member of the Foreign Office was required to carry. ‘You never know when you might be recalled for some secret assignment overseas,’ she said, enigmatically. The gun, she claimed, was hidden somewhere in our house. I searched high and low for the weapon, but they’d obviously been very thorough when they taught my mum the art of concealment. The only way to get my own gun was to join the Foreign Office myself and become a spy. And if I was going to look useful, I’d better be able to speak Russian. At school I signed up for every language possible: French, German and Latin. The BBC started running a Russian course on television. My French teacher, Mr Brown, tried to help me with it. But I could never get my mouth around saying ‘hello’ – zdravstvuyte – and even after eight weeks of following the course I still couldn’t pronounce it. I began to despair. I was also becoming increasingly frustrated by the fact that there was no logic behind why certain foreign verbs behaved the way they did, and why certain nouns were masculine or feminine. Latin did hold out some hope, its strict grammar appealing to my emerging desire for things which were part of some consistent, logical scheme and not just apparently random associations. Or perhaps it was because the teacher always used my name for second-declension nouns: Marcus, Marce, Marcum, . . . O