Mathematical Omnibus: Thirty Lectures on Classic Mathematics Dmitry Fuchs Serge Tabachnikov Department of Mathematics, University of California, Davis, CA 95616 . Department of Mathematics, Penn State University, University Park, PA 16802 . Contents Preface v Algebra and Arithmetics 1 Part 1. Arithmetic and Combinatorics 3 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 43 Part 2. Polynomials 63 Lecture 4. Equations of Degree Three and Four 65 Lecture 5. Equations of Degree Five 77 Lecture6. HowManyRootsDoesaPolynomialHave? 91 Lecture 7. Chebyshev Polynomials 99 Lecture 8. Geometry of Equations 107 Geometry and Topology 119 Part 3. Envelopes and Singularities 121 Lecture 9. Cusps 123 Lecture 10. Around Four Vertices 139 Lecture 11. Segments of Equal Areas 155 Lecture 12. On Plane Curves 167 Part 4. Developable Surfaces 183 Lecture 13. Paper Sheet Geometry 185 Lecture 14. Paper M¨obius Band 199 Lecture 15. More on Paper Folding 207 Part 5. Straight Lines 217 Lecture 16. Straight Lines on Curved Surfaces 219 Lecture 17. Twenty Seven Lines 233 iii iv CONTENTS Lecture 18. Web Geometry 247 Lecture 19. The Crofton Formula 263 Part 6. Polyhedra 275 Lecture 20. Curvature and Polyhedra 277 Lecture 21. Non-inscribable Polyhedra 293 Lecture 22. Can One Make a Tetrahedron out of a Cube? 299 Lecture 23. Impossible Tilings 311 Lecture 24. Rigidity of Polyhedra 327 Lecture 25. Flexible Polyhedra 337 Part 7. [ 351 Lecture 26. Alexander’s Horned Sphere 355 Lecture 27. Cone Eversion 367 Part 8. On Ellipses and Ellipsoids 375 Lecture 28. Billiards in Ellipses and Geodesics on Ellipsoids 377 Lecture 29. The Poncelet Porism and Other Closure Theorems 397 Lecture 30. Gravitational Attraction of Ellipsoids 409 Solutions to Selected Exercises 419 Bibliography 451 Index 455 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 Math- ematics?” 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 65 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 Figuren” 1 by Hans Rademacher and Otto Toeplitz. In his interview to the “Kvant” magazine, published 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 a considerable effort from the reader to follow the details of the arguments. We hope that, in reward, the reader, at least sometimes, will be filled with awe by the harmony of the subject (this feeling is what drives most of 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,usedinthe title, is “judged over a period of time to be of the highest quality and outstanding 1 “The enjoyment of mathematics”, in the English translation; the Russian title was a literal translation of the German original. v vi PREFACE of its kind”. 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 reach 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 computer as an experimental tool is given in Lecture 3 (see the discussion of “privileged exponents”). We did not try to make different lectures similar in their 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 sequence 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 200 exercises in the book, many provided with solutions or an- swers. 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). This book stems from a good many articles we wrote for the Russian magazine “Kvant” over the years 1970–1990 2 and from numerous lectures that we gave over the years to various audiences in the Soviet Union and the United States (where we live since 1990). These include advanced high school students – the participants of the Canada/USA Binational Mathematical Camp in 2001 and 2002, undergraduate 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 Berkeley. 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. To support the “coffee table book” claim, this volume is lavishly illustrated by an accomplished artist, Sergey Ivanov. Sergey was the artist-in-chief of the “Kvant” magazine in the 1980s, and then continued, in a similar position, in the 1990s, at its English-language cousin, “Quantum”. Being a physicist by education, Ivanov’s 2 Available, in Russian, online at http://kvant.mccme.ru/ PREFACE vii illustrations are not only aesthetically attractive but also reflect the mathematical content of the material. We started this preface with a quotation; let us finish with another one. Max Dehn, whose theorems are mentioned here more than once, thus characterized math- ematicians in his 1928 address [22]; we believe, his words apply to the subject of this book: At times the mathematician has the passion of a poet or a con- queror, the rigor of his arguments is that of a responsible states- man or, more simply, of a concerned father, and his tolerance and resignation are those of an old sage; he is revolutionary and conservative, skeptical and yet faithfully optimistic. Acknowledgments. This book is dedicated to V. I. Arnold on the occasion of his 70th anniversary; his style of mathematical research and exposition has greatly influenced the authors over the years. For two consecutive years, in 2005 and 2006, we participated in the “Research in Pairs” program at the Mathematics Institute at Oberwolfach. We are very grateful to this mathematicians’ paradise where the administration, the cooks and nature conspire to boost one’s creativity. Without our sojourns at MFO the completion of this project would still remain a distant future. The second author is also grateful to Max-Planck-Institut for Mathematics in Bonn for its invariable hospitality. Many thanks to John Duncan, Sergei Gelfand and G¨unter Ziegler who read the manuscript from beginning to end and whose detailed (and almost disjoint!) comments and criticism greatly improved the exposition. The second author gratefully acknowledges partial NSF support. Davis, CA and State College, PA December 2006 Algebra and Arithmetics [...]... 1.8.3 Why continued fractions are better than decimal fractions Decimal fractions for rational numbers are either finite or periodic infinite Decimal fractions √ for irrational numbers like e, π or 2 are chaotic Continued fractions for rational numbers are always finite Infinite periodic continued fractions correspond to “quadratic irrationalities”, that is, to roots of quadratic equations with rational coefficients... positive integers, and n ≥ 0 Proposition 1.3 Any rational number has a unique presentation as a finite continued fraction p Proof of existence For an irreducible fraction , we shall prove the existence q of a continued fraction presentation using induction on q For integers (q = 1), the existence is obvious Assume that a continued fraction presentation exists for all p p fractions with denominators less than... based on the geometric construction of Section 1.6 and on properties of so-called continued fractions which will be discussed in Section 1.8 But before considering continued fractions, we want to satisfy a natural curiosity of the reader who may want to see the number which exists according to Part (b) What is this most irrational irrational number, the number, most averse to rational approximation?... (we take positive roots of the quadratic equations) Thus, 2 √ α is the “golden ratio”; also 2 = β − 1 = [1; 2, 2, 2, ] 1.8.4 Why decimal fractions are better than continued fractions For decimal fractions, there are convenient algorithms for addition, subtraction, multiplication, and division (and even for extracting square roots) For continued fractions, there are almost no such algorithms Say,... (Roth) If α is a solution of an algebraic equation an xn + an−1 xn−1 + · · · + a1 x + a0 = 0 with integral coefficients, then for any ε > 0, there exist only finitely many fractions p such that q p 1 α− < 2+ε q q 1.13 Back to the trick In Section 1.3, we were given two 9-digit decimal fractions, of which one was obtained by a division of one 3-digit number by another, while the other one is a random sequence... parallelograms have equal areas (every two consecutive parallelograms have a common base and equal altitudes) Thus all of them have the same area as the parallelogram OA−2 BA−1 , and Proposition 1.2 (b) states that no one of them contains any point of Λ.2 (By the way, the polygonal lines A−2 A0 A2 A4 and A−1 A1 A3 may be constructed as “Newton polygons” Suppose that there is a nail at every point... pn−2 , an qn−1 + qn−2 ) = (qn α − pn , qn ) 2 Proposition 1.8 shows that convergents are the best rational approximations of real numbers In particular, the following holds Proposition 1.9 Let ε > 0 If for only finitely many convergents ε, then the whole set of fractions pn pn 2 < ,q α− qn n qn p p < ε is finite such that q 2 α − q q Proof The assumption implies that for some n, all the points An+1 , An+2... 33461 √ We mentioned the last two approximations of 2 in Section 1.2; in particular, we √ 99 stated that is the best approximation for 2 among the fractions with two-digit 70 denominators What is most surprising, there exists a beautiful formula for the indicators of quality of convergents 1, [1; 2] = LECTURE 1 CAN A NUMBER BE APPROXIMATELY RATIONAL? 19 1.11 Indicator of quality for convergents pn be... 90◦ and reflected in the x axis) The polygonal lines similar qn−1 to A−2 A0 A2 A4 and A−1 A1 A3 A5 are An An−2 A0 and An−1 An−3 A−1 The second one ends at a point A−1 on the x axis which means (as was noticed in qn Subsection 1.9.2) that is a finite continued fraction [an ; an−1 , an−2 , , a1 ], qn−1 as stated by Relation (3) Now, we divide Relation (1) by rn qn and compute λn : λn = 1... the best approximations (within the fragment of the continued fraction 19 199 given above) are (the error is ≈ 4 · 10−3 ) and (the error is ≈ 2.8 · 10−5 ) 7 71 For further information on the continued fraction for π and e, see [56], Appendix II 1.12 Proof of the Hurwitz-Borel Theorem Let α = [a0 ; a1 , a2 , ] be pn an irrational number We need to prove that for infinitely many convergents , qn √ 1 . Mathematical Omnibus: Thirty Lectures on Classic Mathematics Dmitry Fuchs Serge Tabachnikov Department of Mathematics, University. 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,