New horizons in geometry

528 Pages·2012·0.64 MB·English

by  Apostol

NEW HORIZONS
IN GEOMETRY
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(cid:2)c 2012by
The MathematicalAssociationof America (Incorporated)
Illustrations (cid:2)c Mamikon A. Mnatsakanian
Library of Congress CatalogCard Number 2012949754
Print Edition ISBN 978-0-88385-354-2
Electronic Edition ISBN 978-1-61444-210-3
Printedin South Korea
by Charles Allen Imaging Experts, Pasadena, CA
Current Printing (last digit):
10 9 8 7 6 5 4 3 2 1
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The DolcianiMathematicalExpositions
NUMBER FORTY-SEVEN
NEW HORIZONS
IN GEOMETRY
Tom M. Apostol
California Institute of Technology
and
Mamikon A. Mnatsakanian
California Institute of Technology
Published and Distributed by
The Mathematical Association of America
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The DOLCIANI MATHEMATICAL EXPOSITIONS series of the Mathematical
Association of America was established through a generous gift to the Association
fromMaryP.Dolciani,ProfessorofMathematicsatHunterCollegeoftheCityUni-
versityofNewYork. Inmakingthegift,Professor Dolciani,herselfanexceptionally
talentedandsuccessful expositorofmathematics,hadthe purpose offurtheringthe
ideal of excellence in mathematical exposition.
The Association, for its part, was delighted to accept the gracious gesture initi-
atingtherevolvingfundforthisseriesfromonewhohasservedtheAssociationwith
distinction, both as a member of the Committee on Publications and as a member
of the Board of Governors. It was with genuine pleasure that the Board chose to
name the series in her honor.
The books in the series are selected for their lucid expository style and stimu-
lating mathematicalcontent. Typically,they contain an ample supply of exercises,
manywithaccompanyingsolutions. Theyareintendedtobesufficientlyelementary
for the undergraduate and even the mathematicallyinclinedhigh-schoolstudent to
understand and enjoy, but also to be interesting and sometimes challenging to the
more advanced mathematician.
Committee on Books
Frank Farris, Chair
Dolciani Mathematical Expositions Editorial Board
Underwood Dudley,Editor
Jeremy S. Case
RosalieA. Dance
Christopher Dale Goff
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1. MathematicalGems,Ross Honsberger
2. MathematicalGems II,Ross Honsberger
3. MathematicalMorsels,Ross Honsberger
4. MathematicalPlums,Ross Honsberger (ed.)
5. GreatMoments in Mathematics (Before1650),Howard Eves
6. Maxima andMinima withoutCalculus,Ivan Niven
7. GreatMoments in Mathematics (After1650),Howard Eves
8. Map Coloring,Polyhedra,and the Four-ColorProblem,David Barnette
9. MathematicalGems III,Ross Honsberger
10. More MathematicalMorsels,Ross Honsberger
11. Old and New Unsolved Problems in Plane Geometry and Number Theory,
VictorKlee and Stan Wagon
12. Problemsfor Mathematicians,YoungandOld,Paul R. Halmos
13. Excursions in Calculus: An Interplay of the Continuous and the Discrete, Robert
M.Young
14. The Wohascum County Problem Book, George T. Gilbert, Mark Krusemeyer, and
Loren C.Larson
15. Lion Hunting and Other Mathematical Pursuits: A Collection of Mathematics,
Verse, and Stories by Ralph P. Boas, Jr., edited by Gerald L. Alexanderson and
Dale H.Mugler
16. LinearAlgebraProblem Book,Paul R.Halmos
17. From Erdo˝s to Kiev: Problemsof Olympiad Caliber,Ross Honsberger
18. WhichWay Did the BicycleGo? ...andOther IntriguingMathematicalMysteries,
Joseph D. E. Konhauser, Dan Velleman,and Stan Wagon
19. In P´olya’sFootsteps: MiscellaneousProblemsand Essays,Ross Honsberger
20. Diophantus and Diophantine Equations, I. G. Bashmakova (Updated by Joseph
Silverman and translated by Abe Shenitzer)
21. Logicas Algebra,Paul Halmos and Steven Givant
22. Euler: The Master of Us All,William Dunham
23. The Beginnings and Evolution of Algebra, I. G. Bashmakova and G. S. Smirnova
(Translated by Abe Shenitzer)
24. MathematicalChestnutsfrom Aroundthe World,Ross Honsberger
25. CountingonFrameworks: MathematicstoAidtheDesignofRigidStructures,Jack
E.Graver
26. MathematicalDiamonds,Ross Honsberger
27. ProofsthatReallyCount: TheArtofCombinatorialProof,ArthurT.Benjaminand
Jennifer J.Quinn
28. MathematicalDelights,Ross Honsberger
29. Conics,Keith Kendig
30. Hesiod’sAnvil: fallingand spinningthroughheavenandearth,Andrew J.Simoson
31. A Gardenof Integrals,Frank E.Burk
32. A Guideto Complex Variables (MAA Guides #1), Steven G.Krantz
33. Sinkor Float? ThoughtProblemsin Math and Physics,Keith Kendig
34. Biscuitsof Number Theory,Arthur T.Benjamin and Ezra Brown
35. UncommonMathematicalExcursions: PolynomiaandRelatedRealms,DanKalman
36. When Less is More: Visualizing Basic Inequalities, Claudi Alsina and Roger B.
Nelsen
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37. A Guideto AdvancedReal Analysis (MAAGuides #2),Gerald B. Folland
38. A Guideto Real Variables (MAA Guides #3), Steven G.Krantz
39. Voltaire’s Riddle: Microm´egas and the measure of all things, Andrew J.
Simoson
40. A Guideto Topology,(MAAGuides #4),Steven G. Krantz
41. A Guideto ElementaryNumber Theory,(MAA Guides #5), Underwood Dudley
42. Charming Proofs: A Journey into Elegant Mathematics, Claudi Alsina and Roger
B.Nelsen
43. Mathematicsand Sports,edited by Joseph A.Gallian
44. A Guideto AdvancedLinearAlgebra,(MAAGuides #6),Steven H.Weintraub
45. Icons of Mathematics: An Exploration of Twenty Key Images, Claudi Alsina and
Roger B. Nelsen
46. A Guideto PlaneAlgebraicCurves,(MAA Guides #7), KeithKendig
47. New Horizonsin Geometry,TomM. Apostol and Mamikon A.Mnatsakanian
48. A Guideto Groups,Rings,and Fields,(MAAGuides #8),Fernando Gouvˆea
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CONTENTS*
Preface ...................................................................... ix
Introduction ................................................................. xi
Foreword .................................................................... xiii
Chapter 1. Mamikon’s Sweeping -Tangent Theorem ......................... 1
Chapter 2. Cycloids and Trochoids .......................................... 31
Chapter 3. Cyclogons and Trochogons ....................................... 65
Chapter 4. Circumgons and Circumsolids .................................... 101
Chapter 5. The Method of Punctured Containers ............................ 135
Chapter 6. Unwrapping Curves from Cylinders and Cones ................... 169
Chapter 7. New Descriptions of Conics via Twisted Cylinders,
Focal Disks, and Directors ............................................... 213
Chapter 8. Ellipse to Hyperbola: “With This String I Thee Wed” ........... 243
Chapter 9. Trammels ........................................................ 267
Chapter 10. Isoperimetric and Isoparametric Problems ...................... 295
Chapter 11. Arclength and Tanvolutes .......................................331
Chapter 12. Centroids ....................................................... 375
Chapter 13. New Balancing Principles with Applications .................... 401
Chapter 14. Sums of Squares ................................................ 443
Chapter 15. Appendix ....................................................... 473
Bibliography ................................................................ 501
Index ....................................................................... 505
About the Authors .......................................................... 513
*Detailed contents for each chapter appear at the beginning ofthe chapter
vii
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PREFACE
This book is a compendium of joint work produced by the authors during the pe-
riod1998–2012,mostofit published intheAmerican Mathematical Monthly, Math
Horizons, Mathematics Magazine, and The Mathematical Gazette. The published
papershavebeenedited,augmented,andrearrangedinto15chapters. Eachchapter
ispreceded byasampleofproblemsthatcanbesolvedbythemethodsdevelopedin
thatchapter. Eachopeningpagecontainsabriefabstract ofthe chapter’scontents.
Chapter 1, entitled “Mamikon’sSweeping-Tangent Theorem,” was the starting
pointofthiscollaboration. Itdescribesaninnovativeandvisualapproachforsolving
manystandardcalculusproblemsbyageometricmethodthatmakeslittleornouse
of formulas. The method was conceived in 1959 by my co-author (who prefers to
be called Mamikon),when he was an undergraduate student at Yerevan University
in Armenia. When young Mamikon showed his method to Soviet mathematicians
they dismissed it out of hand and said “It can’t be right. You can’t solve calculus
problems that easily.”
Mamikonwent onto get a Ph.D.in physics, was appointeda professor ofastro-
physicsattheUniversityofYerevan,andbecameaninternationalexpertinradiative
transfer theory,allthewhilecontinuingtodevelophispowerfulgeometricmethods.
Mamikoneventuallypublishedapaper outliningthemin1981,butitseems tohave
escaped notice, probably because it appeared in Russian in an Armenian journal
with limitedcirculation. (Reference [59]in the Bibliography.)
Mamikoncame to Californiain 1990to learn more about earthquake prepared-
ness for Armenia. When the Soviet government collapsed he was stranded in the
UnitedStates withoutavisa. Withthe helpofafewmathematicianshehadmetin
Sacramento andat UC Davisandwho recognized hisremarkabletalents,Mamikon
was granted status as an “alien of extraordinary ability.” While working at UC
Davis and for the California Department of Education, he further developed his
methods into a universal teaching tool using hands-on and computer activities, as
wellas diagrams. He has taught these methods at UC Davisand in Northern Cali-
fornia classrooms, ranging from Montessori elementary schools to inner-city public
high schools, and he has demonstrated them at teacher conferences. Students and
teachers alike have responded enthusiastically, because the methods are vivid and
dynamic and don’t require the algebraic formalismof trigonometry or calculus.
A few years later,Mamikonvisited Caltech andconvinced me that hismethods
havethepotentialtomakeasignificantimpactonmathematicseducation,especially
if they are combined with visualizationtools of modern technology. Since then, we
ix
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