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Icons of Mathematics an EXPLORATION of TWENTY KEY IMAGES Claudi Alsina and Roger B AMS / MAA DOLCIANI MATHEMATICAL EXPOSITIONS VOL 45 Icons of Mathematics AN EXPLORATION OF TWENTY KEY IMAGES Claudi Alsina and Roger B. Nelsen i i “MABK018-FM” — 2011/5/16 — 19:53 — page i — #1 i i 10.1090/dol/045 Icons of Mathematics An Exploration of Twenty Key Images i i i i i i “MABK018-FM” — 2011/5/16 — 19:53 — page ii — #2 i i c 2011 by The Mathematical Association of America (Incorporated) Library of Congress Catalog Card Number 2011923441 Print ISBN 978-0-88385-352-8 Electronic ISBN 978-0-88385-986-5 Printed in the United States of America Current Printing (last digit): 10987654321 i i i i i i “MABK018-FM” — 2011/5/16 — 19:53 — page iii — #3 i i The Dolciani Mathematical Expositions NUMBER FORTY-FIVE Icons of Mathematics An Exploration of Twenty Key Images Claudi Alsina Universitat Politecnica` de Catalunya Roger B. Nelsen Lewis & Clark College Published and Distributed by The Mathematical Association of America i i i i i i “MABK018-FM” — 2011/5/16 — 19:53 — page iv — #4 i i DOLCIANI MATHEMATICAL EXPOSITIONS Committee on Books Frank Farris, Chair Dolciani Mathematical Expositions Editorial Board Underwood Dudley, Editor Jeremy S. Case Rosalie A. Dance Tevian Dray Thomas M. Halverson Patricia B. Humphrey Michael J. McAsey Michael J. Mossinghoff Jonathan Rogness Thomas Q. Sibley i i i i i i “MABK018-FM” — 2011/5/16 — 19:53 — page v — #5 i i The DOLCIANI MATHEMATICAL EXPOSITIONS series of the Mathematical As- sociation of America was established through a generous gift to the Association from Mary P. Dolciani, Professor of Mathematics at Hunter College of the City University of New York. In making the gift, Professor Dolciani, herself an exceptionally talented and successful expositor of mathematics, had the purpose of furthering the ideal of excellence in mathematical exposition. The Association, for its part, was delighted to accept the gracious gesture initiat- ing the revolving fund for this series from one who has served the Association with 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 stimulating mathematical content. Typically, they contain an ample supply of exercises, many with accompanying solutions. They are intended to be sufficiently elementary for the undergraduate and even the mathematically inclined high-school student to under- stand and enjoy, but also to be interesting and sometimes challenging to the more advanced mathematician. 1. Mathematical Gems, Ross Honsberger 2. Mathematical Gems II, Ross Honsberger 3. Mathematical Morsels, Ross Honsberger 4. Mathematical Plums, Ross Honsberger (ed.) 5. Great Moments in Mathematics (Before 1650), Howard Eves 6. Maxima and Minima without Calculus,IvanNiven 7. Great Moments in Mathematics (After 1650), Howard Eves 8. Map Coloring, Polyhedra, and the Four-Color Problem, David Barnette 9. Mathematical Gems III, Ross Honsberger 10. More Mathematical Morsels, Ross Honsberger 11. Old and New Unsolved Problems in Plane Geometry and Number Theory, Victor Klee and Stan Wagon 12. Problems for Mathematicians, Young and Old, 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. Linear Algebra Problem Book, Paul R. Halmos i i i i i i “MABK018-FM” — 2011/5/16 — 19:53 — page vi — #6 i i 17. From Erdos˝ to Kiev: Problems of Olympiad Caliber, Ross Honsberger 18. Which Way Did the Bicycle Go?:::and Other Intriguing Mathematical Mysteries, Joseph D. E. Konhauser, Dan Velleman, and Stan Wagon 19. In Polya’s´ Footsteps: Miscellaneous Problems and Essays, Ross Honsberger 20. Diophantus and Diophantine Equations, I. G. Bashmakova (Updated by Joseph Silverman and translated by Abe Shenitzer) 21. Logic as Algebra, Paul Halmos and Steven Givant 22. Euler: The Master of Us All, William Dunham 23. The Beginnings and Evolution of Algebra,I.G.BashmakovaandG.S. Smirnova (Translated by Abe Shenitzer) 24. Mathematical Chestnuts from Around the World, Ross Honsberger 25. Counting on Frameworks: Mathematics to Aid the Design of Rigid Structures, Jack E. Graver 26. Mathematical Diamonds, Ross Honsberger 27. Proofs that Really Count: The Art of Combinatorial Proof, Arthur T. Benjamin and Jennifer J. Quinn 28. Mathematical Delights, Ross Honsberger 29. Conics, Keith Kendig 30. Hesiod’s Anvil: falling and spinning through heaven and earth, Andrew J. Simoson 31. A Garden of Integrals, Frank E. Burk 32. A Guide to Complex Variables (MAA Guides #1), Steven G. Krantz 33. Sink or Float? Thought Problems in Math and Physics, Keith Kendig 34. Biscuits of Number Theory, Arthur T. Benjamin and Ezra Brown 35. Uncommon Mathematical Excursions: Polynomia and Related Realms,Dan Kalman 36. When Less is More: Visualizing Basic Inequalities, Claudi Alsina and Roger B. Nelsen 37. A Guide to Advanced Real Analysis (MAA Guides #2), Gerald B. Folland 38. A Guide to Real Variables (MAA Guides #3), Steven G. Krantz 39. Voltaire’s Riddle: Micromegas´ and the measure of all things, Andrew J. Simoson 40. A Guide to Topology, (MAA Guides #4), Steven G. Krantz 41. A Guide to Elementary Number Theory, (MAA Guides #5), Underwood Dudley 42. Charming Proofs: A Journey into Elegant Mathematics, Claudi Alsina and Roger B. Nelsen i i i i i i “MABK018-FM” — 2011/5/16 — 19:53 — page vii — #7 i i 43. Mathematics and Sports, edited by Joseph A. Gallian 44. A Guide to Advanced Linear Algebra, (MAA Guides #6), Steven H. Weintraub 45. Icons of Mathematics: An Exploration of Twenty Key Images, Claudi Alsina and Roger Nelsen MAA Service Center P.O. Box 91112 Washington, DC 20090-1112 1-800-331-1MAA FAX: 1-301-206-9789 i i i i i i “MABK018-FM” — 2011/5/16 — 19:53 — page viii — #8 i i Dedicated to the icons of our lives — our heroes, our mentors, and our loved ones. i i i i i i “MABK018-FM” — 2011/5/16 — 19:53 — page ix — #9 i i Preface Of all of our inventions for mass communication, pictures still speak the most universally understood language. Walt Disney An icon (from the Greek εικων´ , “image”) is defined as “a picture that is universally recognized to be representative of something.” The world is full of distinctive icons. Flags and shields represent countries, graphic designs represent commercial enterprises; paintings, photographs and even people themselves may evoke concepts, beliefs and epochs. Computer icons are es- sential tools for working with a great variety of electronic devises. What are the icons of mathematics? Numerals? Symbols? Equations? After many years working with visual proofs (also called “proofs without words”), we believe that certain geometric diagrams play a crucial role in visualizing mathematical proofs. In this book we present twenty of them, which we call icons of mathematics, and explore the mathematics that lies within and that can be created. All of our icons are two-dimensional; three- dimensional icons will appear in a subsequent work. Some of the icons have a long history both inside and outside of math- ematics (yin and yang, star polygons, the Venn diagram, etc.). But most of them are essential geometrical figures that enable us to explore an extraor- dinary range of mathematical results (the bride’s chair, the semicircle, the rectangular hyperbola, etc.). Icons of Mathematics is organized as follows. After the Preface we present a table with our twenty key icons. We then devote a chapter to each, illus- trating its presence in real life, its primary mathematical characteristics and how it plays a central role in visual proofs of a wide range of mathemati- cal facts. Among these are classical results from plane geometry, properties of the integers, means and inequalities, trigonometric identities, theorems from calculus, and puzzles from recreational mathematics. As the American ix i i i i i i “MABK018-FM” — 2011/5/16 — 19:53 — page x — #10 i i x Preface actor Robert Stack once said (speaking of icons of a different sort), “these are icons to be treasured.” Each chapter concludes with a selection of Challenges for the reader to explore further properties and applications of the icon. After the chapters we give solutions to all the Challenges in the book. We hope that many readers will find solutions that are superior to ours. Icons of Mathematics concludes with references and a complete index. As with our previous books with the MAA, we hope that both secondary school and college and university teachers may wish to use portions of it as a supplement in problem solving sessions, as enrichment material in a course on proofs and mathematical reasoning, or in a mathematics course for liberal arts students. Special thanks to Rosa Navarro for her superb work in the preparation of preliminary drafts of this manuscript. Thanks too to Underwood Dudley and the members of the editorial board of the Dolciani series for their careful reading of an earlier draft of the book and their many helpful suggestions. We would also like to thank Carol Baxter, Beverly Ruedi, and Rebecca Elmo of the MAA’s book publication staff for their expertise in preparing this book for publication. Finally, special thanks to Don Albers, the MAA’s editorial director for books, who as on previous occasions encouraged us to pursue this project and guided its final production.
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