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Since it was first published three decades ago, Excursions Into Mathematics has been one of the most popular mathematical books written for a general audience. Taking the reader for short excursions into several specific disciplines of mathematics, it makes mathematical concepts accessible to a wide audience. The Millennium Edition is updated with current research and new solutions to outstanding problems that have been discovered since the last edition was printed, such as the solution to the well-known four-color problem. Excursions Into Mathematics: The Millennium Edition is an exciting revision of the original, much-loved classic. Everyone with an interest in mathematics should read this book. Preface Contents Foreword Note to the Instructor Chapter 1. Euler’s Formula for Polyhedra, and Related Topics by Donald W. Crowe Section 1. Introduction 1A. Prelude and Fantasy 1B. Euler’s Formula 1. Exercises 2. Theorem (Euler’s Theorem) 3. Exercises Section 2. Regular Polyhedra Example 1 Example 2 Section 3. Deltahedra Section 4. Polyhedra without Diagonals Section 5. The n-Dimensional Cube and the Tower of Hanoi 5A. The n-Cube 5B. The Tower of Hanoi 5C. Chinese Rings 5D. An Application Section 6. The Four-Color Problem and the Five-Color Theorem Section 7. The Conquest of Saturn, the Scramble for Africa, and Other Problems 7A. The Conquest of Saturn 7B. The Scramble for Africa 7C. Some Problems Equivalent to the Four-Color Problem References Chapter 2. The Search for Perfect Numbers by Michael N. Bleicher Section 1. Introduction Section 2. Prime Numbers and Factorization Section 3. Euclidean Perfect Numbers Section 4. Primes and Their Distribution Section 5. Factorization Techniques Section 6. Non-Euclidean Perfect Numbers Section 7. Extensions and Generalizations References Chapter 3. What is Area? by Anatole Beck Section 1. Introduction Section 2. Rectangles and Grid Figures Section3. Triangles Section 4. Polygons Section 5. Polygonal Regions Section 6. Area in General Section 7. Pathology References Chapter 4. Some Exotic Geometries by Donald W. Crowe Section 1. Historical Background Section 2. Spherical Geometry Section 3. Absolute Geometry Section 4. More History—Saccheri, Bolyai, and Lobachevsky Section 5. Hyperbolic Geometry Section 6. New Beginnings Section 7. Analytic Geometry—A Reminder Section 8. Finite Arithmetics Section 9. Finite Geometries Section 10. Application Section 11. Circles and Quadratic Equations Section 12. Finite Affine Planes Section 13. Finite Projective Planes Section 14. Ovals in a Finite Plane 14A. Ovals in Planes of Odd Order 14B. Ovals in Planes of Even Order (Optional) Section 15. A Finite Version of Poincare’s Universe Definitions Postulates Common Notions Propositions References Chapter 5. Games by Anatole Beck Section 1. Introduction Section 2. Some Tree Games Section 3. The Game of Hex Section 4. The Game of Nim Section 5. Games of Chance Section 6. Matrix Games Section 7. Applications of Matrix Gaines Section 8. Positive-sum Games Section 9. Sharing Section 10. Cooperative Games References Chapter 6. What's in a Name? by Michael N. Bleicher Section 1. Introduction Section 2. Historical Background Section 3. Place Notation for the Base b Section 4. Some Properties of Natural Numbers Related to Notation Section 5. Fractions: First Comments Section 6. Farey Fractions Section 7. Egyptian Fractions Section 8. The Euclidean Algorithm Section 9. Continued Fractions Section 10. Decimal Fractions Section 11. Concluding Remarks References Glossary of Symbols Index Appendix 2000 by Anatole Beck, Michael N. Bleicher, Donald W. Crowe Pages Referenced 49, 55 and 74 93, 97, 103, 107, 108, 110, 123 136, 138, 140, 327 and 386, 332 338 396, 403, 414, 424, 429, 434 438, 451, 452, 467 "Since it was first published three decades ago, Excursions into mathematics has been one of the most popular mathematical books written for a general audience. Taking the reader for short "excursions" into several specific disciplines of mathematics, it makes mathematical concepts accessible to a wide audience. The Millennium Edition is updated with current research and new solutions to outstanding problems that have been discovered since the last edition was printed, such as the solution to the well-known "four-color problem." Excursions into mathematics: The Millennium edition is an exciting revision of the original, much-loved classic. Everyone with an interest in mathematics should read this book." --Publisher description Updated with current research and new solutions to outstanding problems that have been discovered since the last edition, such as the solution to the well-known "four-color problem." Excursions Into Mathematics: This is an exciting revision of the original, much-loved classic. Everyone with an interest in mathematics should read this book.