Mathematical Omnibus Thirty Lectures on Classic Mathematics Dmitry Fuchs Serge Tabachnikov Mathematical Omnibus Thirty Lectures on Classic Mathematics http://dx.doi.org/10.1090/mbk/046 Mathematical Omnibus Thirty Lectures on Classic Mathematics Dmitry Fuchs Serge Tabachnikov 2000 Mathematics Subject Classification. Primary 00A05. For additional information and updates on this book, visit www.ams.org/bookpages/mbk-46 Library of Congress Cataloging-in-Publication Data Fuchs, Dmitry Mathematical omnibus : thirty lectures on classic mathematics / Dmitry Fuchs, Serge Tabach- nikov. p. cm. Includes bibliographical references and index. ISBN 978-0-8218-4316-1 (alk. paper) 1. Mathematics. I. Tabachnikov, Serge. II. Title. QA37.3.F83 2007 510—dc22 2007060824 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. 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Visit the AMS home page at http://www.ams.org/ 10 9 8 7 6 5 4 3 2 16 15 14 13 12 11 To Vladimir Arnold with admiration Contents Preface ix Art and Photo Credits xiii Algebra and Arithmetics Chapter 1. Arithmetic and Combinatorics Lecture 1. Can a Number be Approximately Rational? 5 Lecture 2. Arithmetical Properties of Binomial Coefficients 27 Lecture 3. On Collecting Like Terms, on Euler, Gauss, and MacDonald, and on Missed Opportunities 45 Chapter 2. Equations Lecture 4. Equations of Degree Three and Four 67 Lecture 5. Equations of Degree Five 79 Lecture 6. How Many Roots Does a Polynomial Have? 93 Lecture 7. Chebyshev Polynomials 101 Lecture 8. Geometry of Equations 109 Geometry and Topology Chapter 3. Envelopes and Singularities Lecture 9. Cusps 125 Lecture 10. Around Four Vertices 141 Lecture 11. Segments of Equal Areas 159 Lecture 12. On Plane Curves 171 Chapter 4. Developable Surfaces Lecture 13. Paper Sheet Geometry 189 Lecture 14. Paper M¨obiusBand 203 Lecture 15. More on Paper Folding 213 Chapter 5. Straight Lines Lecture 16. Straight Lines on Curved Surfaces 225 vii viii CONTENTS Lecture 17. Twenty-seven Lines 239 Lecture 18. Web Geometry 253 Lecture 19. The Crofton Formula 269 Chapter 6. Polyhedra Lecture 20. Curvature and Polyhedra 285 Lecture 21. Non-inscribable Polyhedra 301 Lecture 22. Can One Make a Tetrahedron out of a Cube? 307 Lecture 23. Impossible Tilings 319 Lecture 24. Rigidity of Polyhedra 335 Lecture 25. Flexible Polyhedra 345 Chapter 7. Two Surprising Topological Constructions Lecture 26. Alexander’s Horned Sphere 361 Lecture 27. Cone Eversion 373 Chapter 8. On Ellipses and Ellipsoids Lecture 28. Billiards in Ellipses and Geodesics on Ellipsoids 383 Lecture 29. The Poncelet Porism and Other Closure Theorems 403 Lecture 30. Gravitational Attraction of Ellipsoids 415 Solutions to Selected Exercises 425 Bibliography 457 Index 461 Preface For more than two thousand years some familiarity with mathe- matics has been regarded as an indispensable part of the intellec- tual equipment of every cultured person. Today the traditional place of mathematics in education is in grave danger. These opening sentences to the preface of the classical book What Is Mathematics? were written by Richard Courant in 1941. It is somewhat soothing to learn that the problems that we tend to associate with the current situation were equally acute sixty-five years ago (and, most probably, way earlier as well). This is not to say that there are no clouds on the horizon, and by this book we hope to make a modest contribution to the continuation of the mathematical culture. The first mathematical book that one of our mathematical heroes, Vladimir Arnold, read at the age of twelve was Von Zahlen und Figuren1 by Hans Rade- macher and Otto Toeplitz. In his interview given to the “Kvant” magazine, pub- lished in 1990, Arnold recalls that he worked on the book slowly, a few pages a day. We cannot help hoping that our book will play a similar role in the mathematical development of some prominent mathematician of the future. We hope that this book will be of interest to anyone who likes mathematics, from high school students to accomplished researchers. We do not promise an easy ride: the majority of results are proved, and it will take considerable effort from the reader to follow the details of the arguments. We hope that as a reward the reader, at least sometimes, will be filled with awe by the harmony of the subject (this feeling is what drives most mathematicians in their work!). To quote from A Mathematician’s Apology by G. H. Hardy, The mathematician’s patterns, like the painter’s or the poet’s, must be beautiful; the ideas, like the colors or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics. For us too, beauty is the first test in the choice of topics for our own research, as well as the subject for popular articles and lectures, and consequently, in the choice of material for this book. We did not restrict ourselves to any particular area (say, number theory or geometry); our emphasis is on the diversity and the unity of mathematics. If, after reading our book, the reader becomes interested in a more systematic exposition of any particular subject, (s)he can easily find good sources in the literature. About the subtitle: the dictionary definition of the word classic is “judged over a period of time to be of the highest quality and outstanding of its kind”. 1The Enjoyment of Mathematics, in the English translation; the Russian title was a literal translation of the German original. ix xPREFACE We tried to select mathematics satisfying this rigorous criterion. The reader will find here theorems of Isaac Newton and Leonhard Euler, Augustin Louis Cauchy and Carl Gustav Jacob Jacobi, Michel Chasles and Pafnuty Chebyshev, Max Dehn and James Alexander, and many other great mathematicians of the past. Quite often we include recent results of prominent contemporary mathematicians, such as Robert Connelly, John Conway and Vladimir Arnold. There are about four hundred figures in this book. We fully agree with the dictum that a picture is worth a thousand words. The figures are mathematically precise—so a cubic curve is drawn by a computer as a locus of points satisfying an equation of degree three. In particular, the figures illustrate the importance of accurate drawing as an experimental tool in geometrical research. Two examples are given in Lecture 29: the Money-Coutts Theorem, discovered by accurate drawing as late as in the 1970s, and a very recent theorem by Richard Schwartz on the Poncelet grid which he discovered by computer experimentation. Another example of using the computer as an experimental tool is given in Lecture 3 (see the discussion of “privileged exponents”). We did not try to make the lectures similar in length and level of difficulty: some are quite long and involved whereas others are considerably shorter and lighter. One lecture, “Cusps”, stands out: it contains no proofs but only numerous examples, richly illustrated by figures; many of these examples are rigorously treated in other lectures. The lectures are independent of each other, but the reader will notice some themes that reappear throughout the book. We do not assume much by way of preliminary knowledge: a standard calculus course will do in most cases, and quite often even calculus is not required (and this relatively low threshold does not leave out mathematically inclined high school students). We also believe that any reader, no matter how sophisticated, will find surprises in almost every lecture. There are about two hundred exercises in the book, many provided with so- lutions or answers. They further develop the topics discussed in the lectures; in many cases, they involve more advanced mathematics (then, instead of a solution, we give references to the literature). More difficult exercises are marked by a single or a double asterisk. This book stems from a good many articles we wrote for the Russian magazine “Kvant” over the years 1970–19902 and from numerous lectures that we gave over the years to various audiences in the Soviet Union and the United States (where we have lived since 1990). These include advanced high school students—the partici- pants of the Canada/USA Binational Mathematical Camp in 2001 and 2002, under- graduate students attending the Mathematics Advanced Study Semesters (MASS) program at Penn State over the years 2000–2006, high school students—along with their teachers and parents—attending the Bay Area Mathematical Circle at Berke- ley. The book may be used for an undergraduate Honors Mathematics Seminar (there is more than enough material for a full academic year), various topics courses, Mathematical Clubs at high school or college, or simply as a “coffee table book” to browse through, at one’s leisure.