Hidden Harmonies: The Lives and Times of the Pythagorean Theorem PDF
Preview Hidden Harmonies: The Lives and Times of the Pythagorean Theorem
HIDDEN HARMONIES THE LIVES AND TIMES OF THE PYTHAGOREAN THEOREM ROBERT KAPLAN AND ELLEN KAPLAN Illustrations by Ellen Kaplan L’arc et la fleche Mesure du carré de l’hypotenuse Mont-St-Michel 12th century FOR BARRY MAZUR A hidden connection is stronger than an apparent one. —HERACLITUS Contents An Outlook on Insights CHAPTER ONE The Mathematician as Demigod CHAPTER TWO Desert Virtuosi CHAPTER THREE Through the Veil CHAPTER FOUR Rebuilding the Cosmos CHAPTER FIVE Touching the Bronze Sky CHAPTER SIX Exuberant Life CHAPTER SEVEN Number Emerges from Shape CHAPTER EIGHT Living at the Limit CHAPTER NINE The Deep Point of the Dream CHAPTER TEN Magic Casements AFTERWORD Reaching Through—or Past—History? Acknowledgments Notes Selected Bibliography An Outlook on Insights Manisthemeasureofallthings.WhileProtagorashadthemoraldimensioninmind,itisalsotruethat weareuncannilygoodasphysicalmeasurers.Wejudgedistances—fromthrowingapunchtofittinga beam—with easy accuracy, seeing at a glance, for instance, that these two lines add up to this one: What’s odd is that we are much less accurate at estimating areas. How do the combined areas of these two squares: compare to the area of this? It’s hard to believe that they are the same—which is why the Pythagorean Theorem is so startling: Wheredidthisinsightcomefrom—andhowdomathematicalinsightsingeneralsurface?Howdid thetraditionevolveofmeasuringtheworthofourinsightsbyproofs,andwhatdoesthistellusabout people and times far distant from ours—yet with whom we share the measures that matter? Wesharewiththemtoothefreedomofthisshiningcityofmathematics,lookingoutwherewewill from its highest buildings or in astonishment at the imaginative proofs that support them, made by minds for minds. You can safely no more than glance at these proofs in passing, knowing they wait, accessibly, for when you choose to explore their ingenious engineering. Come and see. CHAPTER ONE The Mathematician as Demigod TheEnglishmanlookeddownfromthebalconyofhisvillaoutsideFlorence.Guido,thepeasant’ssix- year- old son, was scratching something on the paving- stones with a burnt stick. He was inventing a 2 2 2 proofofthePythagoreanTheorem;whatwerememberasthealgebraicabstractiona +b =c hesaw as real squares on the sides of a real triangle— “ ‘Do just look at this. Do.’ He coaxed and cajoled. ‘It’s so beautiful. It’s so easy.’ ” And Guido showedtheEnglishman’ssonhowthesamesquarecouldbefilledwithfourcopiesofarighttriangle and the squares on its sides, or the same four triangles and the square on the hypotenuse— so that the two squares of his first diagram must equal in area the one square in the second.
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