Math Made Visual: Creating Images for Understanding Mathematics

208 Pages·2006·2.69 MB·English

by  Roger Nelsen

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THE MATHEMATICAL ASSOCIATION OF AMERICA
Claudi Alsina & Roger B. Nelsen ®
®
MAA
MATH MADE VISUAL
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MATH MADE VISUAL H Creating Images for Understanding Mathematics

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Creating Images for Understanding Mathematics A
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Claudi Alsina & Roger B. Nelsen
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Is it possibleto make mathematical drawings that help to understand mathematical ideas,
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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
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to produce pictures that have both mathematical and pedagogical interest. a
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Mathematical drawings related to proofs have been produced since antiquity in China,
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Arabia, Greece and India, but only in the last thirty years has there been a growing e
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interest in so-called “proofs without words.” Hundreds of these have been published in o
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Mathematics Magazineand The College Mathematics Journal, as well as in other journals, n
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books and on the World Wide Web. r
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Oftentimes, a person encountering a “proof without words” may have the feeling that in
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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
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i
well-understood methods. As the reader shall see, a given mathematical idea or relation c
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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.
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o Classroom Resource Materials
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Classroom Resource Materials ls
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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:[phone]
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;
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