Single digits: in praise of small numbers PDF
Preview Single digits: in praise of small numbers
s i n g l e d i g i t s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . s i n g l e d i g i t s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . In Praise of Small Numbers MARC CHAMBERLAND PrincetonUniversityPress Princeton&Oxford Copyright(cid:1)c 2015byPrincetonUniversityPress PublishedbyPrincetonUniversityPress,41WilliamStreet,Princeton, NewJersey08540 IntheUnitedKingdom:PrincetonUniversityPress,6OxfordStreet, Woodstock,OxfordshireOX201TW press.princeton.edu AllRightsReserved ThesecondepigraphbyPaulMcCartneyonpage111istakenfrom TheBeatlesandisreproducedwithpermissionofCurtisBrownGroupLtd., LondononbehalfofTheBeneficiariesoftheEstateofHunterDavies. Copyright(cid:1)c HunterDavies2009. Theepigraphonpage170istakenfromHarryPotterandtheHalfBlood Prince:Copyright(cid:1)c J.K.Rowling2005 Theepigraphsonpage205arereprintedwihtthepermissionofthe FreePress,aDivisionofSimon&Schuster,Inc.,fromBornonaBlueDay: InsidetheExtraordinaryMindofanAustisticSavantbyDanielTammet. Copyright(cid:1)c 2006byDanielTammet.Originallypublishedin GreatBritainin2006byHodder&Stoughton.Allrightsreserved. LibraryofCongressCataloging-in-PublicationData Chamberland,Marc,1964– Singledigits:inpraiseofsmallnumbers/MarcChamberland. pagescm Includesbibliographicalreferencesandindex. ISBN978-0-691-16114-3(hardcover:alk.paper)1.Mathematical analysis. 2.Sequences(Mathematics) 3.Combinatorial analysis. 4.Mathematics–Miscellanea. I.Title. QA300.C44122015 510—dc23 2014047680 BritishLibraryCataloging-in-PublicationDataisavailable ThisbookhasbeencomposedinAdobeCaslonProandMyriadPro Printedonacid-freepaper.∞ TypesetbySRNovaPvtLtd,Bangalore,India PrintedintheUnitedStatesofAmerica 1 3 5 7 9 10 8 6 4 2 c o n t e n t s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . preface xi Chapter1 TheNumberOne 1 SlicedOrigami 1 FibonacciNumbersandtheGoldenRatio 2 RepresentingNumbersUniquely 5 FactoringKnots 6 CountingandtheSternSequence 8 Fractals 10 Gilbreath’sConjecture 13 Benford’sLaw 13 TheBrouwerFixed-PointTheorem 16 InverseProblems 17 PerfectSquares 19 TheBohr–MollerupTheorem 19 ThePicardTheorems 21 Chapter2 TheNumberTwo 24 TheJordanCurveTheoremandParityArguments 24 AspectRatio 26 HowSymmetricAreYou? 27 vi contents ThePythagoreanTheorem 29 BeattySequences 32 Euler’sFormula 33 MattersofPrimeImportance 34 TheHamSandwichTheorem 38 PowerSetsandPowersofTwo 39 TheSylvester–GallaiTheorem 42 (cid:1) Formulasfor 43 Multiplication 44 TheThue–MorseSequence 45 Duals 48 ApollonianCirclePackings 51 PerfectNumbersandMersennePrimes 53 PythagoreanTuningandtheSquareRootof2 54 InverseSquareLaws 56 TheArithmetic-GeometricMeanInequality 57 PositivePolynomials 59 Newton’sMethodforRootFinding 60 MoreDivisionviaMultiplication 63 TheAllureof(cid:1)2/6 64 JacobianConjectures 66 Chapter3 TheNumberThree 69 x + The3 1Problem 69 TriangularNumbersandBulgarianSolitaire 71 Rock-Paper-ScissorsandBorromeanRings 73 RandomWalks 74 TrisectinganAngle 77 TheThree-BodyProblem 78 TheLorenzAttractorandChaos 81 PeriodThreeImpliesChaos 83 PatternsamongtheStars 85 Fermat’sLastTheorem 86 LeftoversAnyone? 89 EgyptianFractions 90 Arrow’sImpossibilityTheorem 93 contents vii MappingSurfaces 95 GuardinganArtGallery 96 ThePoincaréConjecture 97 Monge’sThree-CircleTheorem 100 Marden’sTheorem 100 TheReuleauxTriangle 103 TheThirdCriticalPoint 106 SumsofCubes 108 ApproximatingDecay 109 Chapter4 TheNumberFour 111 TheFourColorTheorem 111 TheTennisBallTheorem 114 SumofSquaresIdentities 114 RearrangingFourPieces 115 DucciSequences 116 Euler’sSumofPowersConjecture 119 VillarceauCircles 122 TheInscribedSquareProblem 123 RegularPolygonsonaComputerScreen 124 TheFourTravelersProblem 125 TheFourExponentialsConjecture 127 ConcentricQuadrilaterals 129 TheFourHatsProblem 131 Chapter5 TheNumberFive 132 TheMiquelFiveCirclesTheorem 132 ThePlatonicSolids 133 SolvingPolynomialEquations 134 DiophantineApproximation 137 ThePetersenGraph 138 TheHappyEndingProblem 140 Tessellations 141 OfBallsandSausages 143 Knight’sToursonRectangularBoards 145 MagicwithFiveCards 145 viii contents SoccerBallsandDomes 148 RecyclingadInfinitum 148 TheRogers–RamanujanIdentities 150 Chapter6 TheNumberSix 156 OptimalPacking 156 OfFriendsandStrangers 161 SixDegreesofSeparation 161 ANecklaceofSpheres 163 HexagonsinPascal’sTriangle 164 TheGameofHex 165 TheWendtDeterminant 167 SixLengthsinGeometry 168 Chapter7 TheNumberSeven 170 TheSevenCirclesTheorem 170 / Digitsof1 7andEllipses 171 Strassen’sMatrixMultiplication 173 TheFanoPlane 175 BorderPatterns 177 TheSzilassiPolyhedronandtheHeawoodGraph 178 TheKuratowskiClosure–ComplementTheorem 179 CanYouHeartheShapeofaDrum? 182 BarkerCodes 184 RecreationalMathematics 186 ExperimentswithIntegrals 188 Chapter8 TheNumberEight 191 ThePizzaTheorem 191 ShufflingCards 192 TheGameofLife 193 RepetitioninPascal’sTriangle 196 TheSierpin´skiCarpet 197 QuaternionsandOctonions 198 TheSummitofE 202 8 contents ix Chapter9 TheNumberNine 205 NinePointsandCollinearity 205 CastingOutNines 207 PrimesandNines 208 TheFifteenTheorem 209 CirclePackingswithTwoSizes 210 Catalan’sConjecture 211 TheHeegnerNumbers 213 Chapter10 Solutions 216 RearrangingFourPieces(Chapter4) 216 TheFourHatsProblem(Chapter4) 216 TheKuratowskiClosure–ComplementTheorem (Chapter7) 217 RecreationalMathematics(Chapter7) 217 further reading 219 credits for illustrations 223 index 225
The list of books you might like
