Mathematical Omnibus: Thirty Lectures on Classic Mathematics

The book consists of thirty lectures on diverse topics, covering much of the mathematical landscape rather than focusing on one area. The reader will learn numerous results that often belong to neither the standard undergraduate nor graduate curriculum and will discover connections between classical and contemporary ideas in algebra, combinatorics, geometry, and topology. The reader's effort will be rewarded in seeing the harmony of each subject. The common thread in the selected subjects is their illustration of the unity and beauty of mathematics. Most lectures contain exercises, and solutions or answers are given to selected exercises. A special feature of the book is an abundance of drawings (more than four hundred), artwork by an award-winning artist, and about a hundred portraits of mathematicians. Almost every lecture contains surprises for even the seasoned researcher. Lecture 1. Can A Number Be Approximately Rational? Lecture 2. Arithmetical Properties Of Binomial Coefficients Lecture 3. On Collecting Like Terms, On Euler, Gauss, And Macdonald, And On Missed Opportunities Lecture 4. Equations Of Degree Three And Four Lecture 5. Equations Of Degree Five Lecture 6. How Many Roots Does A Polynomial Have? Lecture 7. Chebyshev Polynomials Lecture 8. Geometry Of Equations Lecture 9. Cusps Lecture 10. Around Four Vertices Lecture 11. Segments Of Equal Areas Lecture 12. On Plane Curves Lecture 13. Paper Sheet Geometry Lecture 14. Paper Möbius Band Lecture 15. More On Paper Folding Lecture 16. Straight Lines On Curved Surfaces Lecture 17. Twenty-seven Lines Lecture 18. Web Geometry Lecture 19. The Crofton Formula Lecture 20. Curvature And Polyhedra Lecture 21. Non-inscribable Polyhedra Lecture 22. Can One Make A Tetrahedron Out Of A Cube? Lecture 23. Impossible Tilings Lecture 24. Rigidity Of Polyhedra Lecture 25. Flexible Polyhedra Lecture 26. Alexander's Horned Sphere Lecture 27. Cone Eversion Lecture 28. Billiards In Ellipses And Geodesics On Ellipsoids Lecture 29. The Poncelet Porism And Other Closure Theorems Lecture 30. Gravitational Attraction Of Ellipsoids Lecture 31. Solutions To Selected Exercises. Dmitry Fuchs, Serge Tabachnikov. Includes Bibliographical References (p. 457-459) And Index. The book consists of thirty lectures on diverse topics, covering much of the mathematical landscape rather than focusing on one area. The reader will learn numerous results that often belong to neither the standard undergraduate nor graduate curriculum and will discover connections between classical and contemporary ideas in algebra, combinatorics, geometry, and topology. The reader's effort will be rewarded in seeing the harmony of each subject. The common thread in the selected subjects is their illustration of the unity and beauty of mathematics. Most lectures contain exercises, and solutions or answers are given to selected exercises. A special feature of the book is an abundance of drawings (more than four hundred), artwork by an accomplished artist, and about a hundred portraits of mathematicians. Almost every lecture contains surprises for even the seasoned researcher. -- Back cover