Table Of Contenti “main” — 2011/10/12 — 12:06 — page i — #1 i
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Beautiful Mathematics
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c 2011bytheMathematicalAssociationofAmerica,Inc.
(cid:13)
LibraryofCongressCatalogCardNumber2011939398
PrinteditionISBN:978-0-88385-576-8
ElectroniceditionISBN:978-1-61444-509-8
PrintedintheUnitedStatesofAmerica
CurrentPrinting(lastdigit):
10987654321
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Beautiful Mathematics
Martin Erickson
Truman State University
PublishedandDistributedby
TheMathematicalAssociationofAmerica
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CouncilonPublicationsandCommunications
FrankFarris,Chair
CommitteeonBooks
GeraldBryce,Chair
SpectrumEditorialBoard
GeraldL.Alexanderson,Editor
RobertE.Bradley SusannaS.Epp
RichardK.Guy KeithM.Kendig
ShawneeL.McMurran JeffreyL.Nunemacher
KennethA.Ross FranklinF.Sheehan
JamesJ.Tattersall RobinWilson
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SPECTRUMSERIES
TheSpectrumSeriesof theMathematicalAssociationofAmericawassonamedtoreflectits pur-
pose:topublishabroadrangeofbooksincludingbiographies,accessibleexpositionsofoldornew
mathematicalideas,reprintsandrevisionsofexcellentout-of-printbooks,popularworks,andother
monographsof high interest that will appeal to a broad range of readers, including students and
teachersofmathematics,mathematicalamateurs,andresearchers.
777MathematicalConversationStarters,byJohndePillis
99PointsofIntersection:Examples—Pictures—Proofs,byHansWalser.Translatedfromtheoriginal
GermanbyPeterHiltonandJeanPedersen
AhaGotchaandAhaInsight,byMartinGardner
AlltheMathThat’sFittoPrint,byKeithDevlin
BeautifulMathematics,byMartinErickson
CalculusGems:BriefLivesandMemorableMathematics,byGeorgeF.Simmons
CarlFriedrichGauss:TitanofScience,byG.WaldoDunnington,withadditionalmaterialbyJeremy
GrayandFritz-EgbertDohse
TheChangingSpaceofGeometry,editedbyChrisPritchard
Circles:AMathematicalView,byDanPedoe
ComplexNumbersandGeometry,byLiang-shinHahn
Cryptology,byAlbrechtBeutelspacher
TheEarlyMathematicsofLeonhardEuler,byC.EdwardSandifer
TheEdgeoftheUniverse:Celebrating10YearsofMathHorizons,editedbyDeannaHaunsperger
andStephenKennedy
EulerandModernScience,editedby N. N. Bogolyubov,G. K. Mikhailov, andA. P. Yushkevich.
TranslatedfromRussianbyRobertBurns.
Eulerat300:AnAppreciation,editedbyRobertE.Bradley,LawrenceA.D’Antonio,andC.Edward
Sandifer
ExpeditionsinMathematics,editedbyTatianaShubin,DavidF.Hayes,andGeraldL.Alexanderson
FiveHundredMathematicalChallenges,byEdwardJ.Barbeau,MurrayS.Klamkin,andWilliamO.
J.Moser
TheGeniusofEuler:ReflectionsonhisLifeandWork,editedbyWilliamDunham
TheGoldenSection,byHansWalser.TranslatedfromtheoriginalGermanbyPeterHilton,withthe
assistanceofJeanPedersen.
TheHarmonyoftheWorld:75YearsofMathematicsMagazine,editedbyGeraldL.Alexanderson
withtheassistanceofPeterRoss
AHistorianLooksBack:TheCalculusasAlgebraandSelectedWritings,byJudithGrabiner
HistoryofMathematics:HighwaysandByways,byAmyDahan-Dalme´dicoandJeannePeiffer,trans-
latedbySanfordSegal
HowEulerDidIt,byC.EdwardSandifer
IsMathematicsInevitable?AMiscellany,editedbyUnderwoodDudley
IWanttoBeaMathematician,byPaulR.Halmos
JourneyintoGeometries,byMartaSved
JULIA:alifeinmathematics,byConstanceReid
TheLighterSideofMathematics:ProceedingsoftheEuge`neStrensMemorialConferenceonRecre-
ationalMathematics&ItsHistory,editedbyRichardK.GuyandRobertE.Woodrow
LureoftheIntegers,byJoeRoberts
MagicNumbersoftheProfessor,byOwenO’SheaandUnderwoodDudley
MagicTricks,CardShuffling,andDynamicComputerMemories:TheMathematicsof the Perfect
Shuffle,byS.BrentMorris
MartinGardner’sMathematicalGames:TheentirecollectionofhisScientificAmericancolumns
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TheMathChatBook,byFrankMorgan
Mathematical Adventuresfor StudentsandAmateurs,editedby DavidHayesandTatianaShubin.
WiththeassistanceofGeraldL.AlexandersonandPeterRoss
MathematicalApocrypha,byStevenG.Krantz
MathematicalApocryphaRedux,byStevenG.Krantz
MathematicalCarnival,byMartinGardner
MathematicalCirclesVolI:InMathematicalCirclesQuadrantsI,II,III,IV,byHowardW.Eves
MathematicalCirclesVolII:MathematicalCirclesRevisitedandMathematicalCirclesSquared,by
HowardW.Eves
MathematicalCirclesVolIII: MathematicalCirclesAdieuandReturntoMathematicalCircles,by
HowardW.Eves
MathematicalCircus,byMartinGardner
MathematicalCranks,byUnderwoodDudley
MathematicalEvolutions,editedbyAbeShenitzerandJohnStillwell
MathematicalFallacies,Flaws,andFlimflam,byEdwardJ.Barbeau
MathematicalMagicShow,byMartinGardner
MathematicalReminiscences,byHowardEves
MathematicalTreks:FromSurrealNumberstoMagicCircles,byIvarsPeterson
Mathematics:QueenandServantofScience,byE.T.Bell
MathematicsinHistoricalContext,,byJeffSuzuki
MemorabiliaMathematica,byRobertEdouardMoritz
MusingsoftheMasters:AnAnthologyofMathematicalReflections,editedbyRaymondG.Ayoub
NewMathematicalDiversions,byMartinGardner
Non-EuclideanGeometry,byH.S.M.Coxeter
NumericalMethodsThatWork,byFormanActon
NumerologyorWhatPythagorasWrought,byUnderwoodDudley
OutoftheMouthsofMathematicians,byRosemarySchmalz
PenroseTilestoTrapdoorCiphers...andtheReturnofDr.Matrix,byMartinGardner
Polyominoes,byGeorgeMartin
PowerPlay,byEdwardJ.Barbeau
ProofandOtherDilemmas:MathematicsandPhilosophy,editedbyBonnieGoldandRogerSimons
TheRandomWalksofGeorgePo´lya,byGeraldL.Alexanderson
RemarkableMathematicians,fromEulertovonNeumann,byIoanJames
TheSearchforE.T.Bell,alsoknownasJohnTaine,byConstanceReid
ShapingSpace,editedbyMarjorieSenechalandGeorgeFleck
SherlockHolmesinBabylonandOtherTalesofMathematicalHistory,editedbyMarlowAnderson,
VictorKatz,andRobinWilson
Student ResearchProjects in Calculus, by Marcus Cohen, Arthur Knoebel, EdwardD. Gaughan,
DouglasS.Kurtz,andDavidPengelley
Symmetry,byHansWalser.TranslatedfromtheoriginalGermanbyPeterHilton,withtheassistance
ofJeanPedersen.
TheTrisectors,byUnderwoodDudley
TwentyYearsBeforetheBlackboard,byMichaelStuebenwithDianeSandford
WhoGaveYoutheEpsilon?andOtherTalesofMathematicalHistory,editedbyMarlowAnderson,
VictorKatz,andRobinWilson
TheWordsofMathematics,byStevenSchwartzman
MAAServiceCenter
P.O.Box91112
Washington,DC20090-1112
800-331-1622 FAX301-206-9789
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To Rodman Doll,
who mentored me in mathematics
when I was a high school student
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Preface
Why are numbers beautiful? It’s likeasking why is Beethoven’s Ninth
Symphony beautiful. If you don’t see why, someone can’t tell you. I
knownumbersarebeautiful.Iftheyaren’tbeautiful,nothingis.
PAUL ERDO˝S (1913–1996)
Thisbookisaboutbeautifulmathematicalconceptsandcreations.
Some people believe that mathematics is the language of nature, others that it is an
abstractgamewithsymbolsandrules.Stillothersbelieveitisallcalculations.Platoequated
mathematicswith“thegood.”Myapproachtomathematicsisasanartform,likepainting,
sculpture,ormusic.Whiletheartistworksinatangiblemedium,themathematicianworks
in a medium of numbers, shapes, and abstract patterns. In mathematics, as in art, there
are constraints. The most stringent is that mathematical results must be true; others are
conciseness and elegance. As withotherarts, mathematical ideas have an esthetic appeal
thatcanbeappreciatedbythosewiththewillingnesstoinvestigate.
Ihopethatthisbookwillinspirereaderswiththebeautyofmathematics.Ipresentmath-
ematical topicsinthecategories ofwords,images, formulas, theorems, proofs,solutions,
and unsolved problems. We go from complex numbers to arithmetic progressions, from
Alcuin’ssequencetothezetafunction,andfromhypercubestoinfinitysquared.
Whoshouldreadthisbook?Ibelievethatthereissomethingnewinitforanymathemat-
ically-minded person. I especially recommend it to high school and college students, as
theyneedmotivationtostudymathematics, andbeautyisastrongmotivation;andtopro-
fessionalmathematicians,becausewealwaysneedfreshexamplesofmathematicalbeauty
topassalongtoothers.Withineachchapter,thetopicsrequireprogressivelymoreprerequi-
siteknowledge.Topicsthatmaybetooadvancedforabeginningreaderwillbecomemore
accessible asthereaderprogressesinmathematical study.Anappendixgivesbackground
definitionsandtheorems,whileanothergiveschallengingexercises,withsolutions,tohelp
thereaderlearnmore.
Thankstothepeoplewhohavekindlyprovidedsuggestionsconcerningthisbook:Roland
Bacher, Donald Bindner, Robert Cacioppo, Robert Dobrow, Shalom Eliahou, Ravi Fer-
nando, Suren Fernando, David Garth, Joe Hemmeter, Daniel Jordan, Ken Price, Khang
Tran,VincentVatter,andAnthonyVazzana. Thanksalsotothepeopleaffiliatedwithpub-
lishingattheMathematicalAssociationofAmerica, includingGeraldAlexanderson,Don
Albers, Carol Baxter, Rebecca Elmo, Frank Farris, Beverly Ruedi, and the anonymous
readers,fortheirhelpinmakingthisbookareality.
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