Project1_MMV 7x10 12/6/11 1:54 PM Page 1 THE MATHEMATICAL ASSOCIATION OF AMERICA Claudi Alsina & Roger B. Nelsen ® ® MAA MATH MADE VISUAL M A T MATH MADE VISUAL H Creating Images for Understanding Mathematics M Creating Images for Understanding Mathematics A D Claudi Alsina & Roger B. Nelsen E V I S U Is it possibleto make mathematical drawings that help to understand mathematical ideas, A proofs and arguments? The authors of this book are convinced that the answer is yes and L the objective of this book is to show how some visualization techniques may be employed C r e to produce pictures that have both mathematical and pedagogical interest. a t i n g I m Mathematical drawings related to proofs have been produced since antiquity in China, a g Arabia, Greece and India, but only in the last thirty years has there been a growing e s f interest in so-called “proofs without words.” Hundreds of these have been published in o r U Mathematics Magazineand The College Mathematics Journal, as well as in other journals, n d e books and on the World Wide Web. r s t a n d Oftentimes, a person encountering a “proof without words” may have the feeling that in g the pictures involved are the result of a serendipitous discovery or the consequence M a of exceptional ingenuity on the part of the picture’s creator. In this book the authors show th e m that behind most of the pictures “proving” mathematical relations are some a t i well-understood methods. As the reader shall see, a given mathematical idea or relation c s may have many different images that justify it, so that depending on the teaching level or the objectives for producing the pictures, one can choose the best alternative. C la u d i A ls in a & R o Classroom Resource Materials g e r B . N e Classroom Resource Materials ls e n Math Made Visual Creating Images for Understanding Mathematics (cid:13)c 2006by TheMathematicalAssociationofAmerica(Incorporated) LibraryofCongressControlNumber2005937269 PrintISBN978-0-88385-746-5 ElectronicISBN978-1-61444-100-7 PrintedintheUnitedStatesofAmerica CurrentPrinting(lastdigit): 1098765432 Math Made Visual Creating Images for Understanding Mathematics Claudi Alsina UniversitatPolite`cnicadeCatalunya and RogerB. Nelsen Lewis&ClarkCollege PublishedandDistributedby TheMathematicalAssociationofAmerica CouncilonPublications RogerNelsen,Chair ClassroomResourceMaterialsEditorialBoard ZavenA.Karian,Editor WilliamC.Bauldry DouglasMeade GeraldBryce JudithA.Palagallo GeorgeExner WayneRoberts WilliamJ.Higgins KayB.Somers StanleyE.Seltzer CLASSROOMRESOURCEMATERIALS Classroom Resource Materials is intended to provide supplementary classroom material forstudents—laboratoryexercises,projects,historicalinformation,textbookswithunusual approachesforpresentingmathematicalideas,careerinformation,etc. 101CareersinMathematics,2ndeditioneditedbyAndrewSterrett Archimedes:WhatDidHeDoBesidesCryEureka?,ShermanStein CalculusMysteriesandThrillers,R.GrantWoods Combinatorics:AProblemOrientedApproach,DanielA.Marcus ConjectureandProof,Miklo´sLaczkovich ACourseinMathematicalModeling,DouglasMooneyandRandallSwift CryptologicalMathematics,RobertEdwardLewand ElementaryMathematicalModels,DanKalman EnvironmentalMathematicsintheClassroom,editedbyB.A.FusaroandP.C.Kenschaft EssentialsofMathematics,MargieHale ExploratoryExamplesforRealAnalysis,JoanneE.SnowandKirkE.Weller FourierSeries,RajendraBhatia GeometryFromAfrica:MathematicalandEducationalExplorations,PaulusGerdes HistoricalModulesfortheTeachingandLearningofMathematics(CD),editedbyVictor KatzandKarenDeeMichalowicz IdentificationNumbersandCheckDigitSchemes,JosephKirtland InterdisciplinaryLivelyApplicationProjects,editedbyChrisArney InverseProblems:ActivitiesforUndergraduates,CharlesW.Groetsch LaboratoryExperiencesinGroupTheory,EllenMaycockParker Learn from the Masters, Frank Swetz, John Fauvel, OttoBekken, Bengt Johansson, and VictorKatz MathematicalConnections:ACompanionforTeachersandOthers,AlCuoco MathematicalEvolutions,editedbyAbeShenitzerandJohnStillwell MathematicalModelingintheEnvironment,CharlesHadlock Mathematicsfor Business Decisions Part 1: ProbabilityandSimulation(electronic text- book),RichardB.ThompsonandChristopherG.Lamoureux Mathematicsfor Business Decisions Part 2: Calculusand Optimization(electronic text- book),RichardB.ThompsonandChristopherG.Lamoureux MathMade Visual:CreatingImagesforUnderstandingMathematics,ClaudiAlsinaand RogerB.Nelsen OrdinaryDifferentialEquations:ABriefEclecticTour,DavidA.Sa´nchez OvalTrackandOtherPermutationPuzzles,JohnO.Kiltinen APrimerofAbstractMathematics,RobertB.Ash ProofsWithoutWords,RogerB.Nelsen ProofsWithoutWordsII,RogerB.Nelsen ARadicalApproachtoRealAnalysis,DavidM.Bressoud RealInfiniteSeries,DanielD.BonarandMichaelKhoury,Jr. SheDoesMath!,editedbyMarlaParker SolveThis:MathActivitiesforStudentsandClubs,JamesS.Tanton StudentManualforMathematicsforBusiness DecisionsPart1:ProbabilityandSimula- tion,DavidWilliamson,MarilouMendel,JulieTarr,andDeborahYoklic StudentManualforMathematicsforBusiness DecisionsPart 2:CalculusandOptimiza- tion,DavidWilliamson,MarilouMendel,JulieTarr,andDeborahYoklic TeachingStatisticsUsingBaseball,JimAlbert TopologyNow!,RobertMesserandPhilipStraffin UnderstandingourQuantitativeWorld,JanetAndersenandToddSwanson Writing Projects for Mathematics Courses: Crushed Clowns, Cars, and Coffee to Go, AnnalisaCrannell,GavinLaRose,ThomasRatliff,ElynRykken MAAServiceCenter P.O.Box91112 Washington,DC20090-1112 1-800-331-1MAA FAX:1-301-206-9789 Dedicatedto Professor BertholdSchweizer for alltheyears ofmathematical collaborationandfriendship Introduction “adullproofcanbesupplementedbyageometric analoguesosimpleandbeautifulthatthetruthof atheoremisalmostseenataglance” —MartinGardner “Behold!” —Bha¯skara Isitpossibletocreatemathematicaldrawingsthathelpstudentsunderstandmathematical ideas, proofs and arguments? We are convinced thatthe answer is yes and our objective inthis bookis toshow how some visualizationtechniques may be employedto produce picturesthathavebothmathematicalandpedagogicalinterest. Mathematical drawingsrelated toproofshave been producedsince antiquityinChina, Arabia,GreeceandIndiabutonlyinthelastthirtyyearshastherebeenagrowinginterestin so-called“proofswithoutwords.”HundredsofthesehavebeenpublishedinMathematics Magazineand The CollegeMathematicsJournal, as wellas inotherjournals,booksand ontheWorldWideWeb.Popularizingthisgenrewasthemotivationforthesecondauthor ofthisbookinpublishingthecollections[Nelsen,1993and2000]. Thefirstauthorbecameinterestedincreatingproofswithoutwordssomeyearsagoand more recently began a systematic study on how to teach others to design such pictures. Thisledhimtoorganize andpresentmany workshopsonthe topicdevotedtosecondary and university teachers. Consequently, we decided to join forces and prepare this book, extendingamathematicalcollaborationthatgoesbackmanyyears. Often times, a person encountering a “proof without words” may have the feeling that the pictures involved are the result of a serendipitous discovery or the consequence of exceptional ingenuity on the part of the picture’s creator. The next several chapters show that behind most of the pictures “proving” mathematical relations are some well- understoodmethodstofollow.Aswillbeseen,agivenmathematicalideaorrelationmay havemanydifferentimagesthatjustifyit,sointheend,dependingontheteachinglevelor theobjectivesforproducingthepictures,onecanchoosethebestalternative. Since ourmain objectiveinthispublicationistopresenta methodologyforproducing mathematicalvisualizations,wehavedividedthebookintothreeparts: PartI:Visualizingmathematicsbycreatingpictures; PartII:Visualizationintheclassroom; ix