Single Digits: In Praise of Small Numbers

241 Pages·2015·7.492 MB·English

by  Marc Chamberland

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
E
TheSummitof 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