Mathematical Omnibus: Thirty Lectures on Classic Mathematics

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. Lecture 2. Lecture 3. Can a Number be Approximately Rational? 5 Arithmetical Properties of Binomial Coefficients 27 On Collecting Like Terms, on Euler, Gauss and MacDonald, and on Missed Opportunities 43 Part 2. Polynomials 63 Lecture 4. Lecture 5. Lecture 6. Lecture 7. Lecture 8. Equations of Degree Three and Four 65 Equations of Degree Five 77 How Many Roots Does a Polynomial Have? 91 Chebyshev Polynomials 99 Geometry of Equations 107 Geometry and Topology 119 Part 3. Envelopes and Singularities 121 Lecture 9. Cusps 123 Lecture 10. Lecture 11. Lecture 12. Around Four Vertices 139 Segments of Equal Areas 155 On Plane Curves 167 Part 4. Developable Surfaces 183 Lecture 13. Lecture 14. Lecture 15. Paper Sheet Geometry 185 Paper Mo¨bius Band 199 More on Paper Folding 207 Part 5. Straight Lines 217 Lecture 16. Lecture 17. Straight Lines on Curved Surfaces 219 Twenty Seven Lines 233 iii iv CONTENTS Lecture 18. Lecture 19. Web Geometry 247 The Crofton Formula 263 Part 6. Polyhedra 275 Lecture 20. Lecture 21. Lecture 22. Lecture 23. Lecture 24. Lecture 25. Curvature and Polyhedra 277 Non-inscribable Polyhedra 293 Can One Make a Tetrahedron out of a Cube? 299 Impossible Tilings 311 Rigidity of Polyhedra 327 Flexible Polyhedra 337 Part 7. [ 351 Lecture 26. Lecture 27. Alexander’s Horned Sphere 355 Cone Eversion 367 Part 8. On Ellipses and Ellipsoids 375 Lecture 28. Lecture 29. Lecture 30. Billiards in Ellipses and Geodesics on Ellipsoids 377 The Poncelet Porism and Other Closure Theorems 397 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, used in the 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 ... - tailieumienphi.vn