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Book Categories |
Foreword | xix | |
Note to the Instructor | xxi | |
Chapter 1 | Euler's Formula for Polyhedra, and Related Topics | |
Section 1 | Introduction | 3 |
Section 2 | Regular Polyhedra | 12 |
Section 3 | Deltahedra | 21 |
Section 4 | Polyhedra without Diagonals | 31 |
Section 5 | The n-dimensional Cube and the Tower of Hanoi | 40 |
Section 6 | The Four-color Problem and the Five-color Theorem | 55 |
Section 7 | The Conquest of Saturn, the Scramble for Africa, and Other Problems | 64 |
Chapter 2 | The Search for Perfect Numbers | |
Section 1 | Introduction | 81 |
Section 2 | Prime Numbers and Factorization | 84 |
Section 3 | Euclidean Perfect Numbers | 93 |
Section 4 | Primes and Their Distribution | 102 |
Section 5 | Factorization Techniques | 110 |
Section 6 | Non-Euclidean Perfect Numbers | 124 |
Section 7 | Extensions and Generalizations | 137 |
Chapter 3 | What is Area? | |
Section 1 | Introduction | 147 |
Section 2 | Rectangles and Grid Figures | 153 |
Section 3 | Triangles | 161 |
Section 4 | Polygons | 166 |
Section 5 | Polygonal Regions | 177 |
Section 6 | Area in General | 185 |
Section 7 | Pathology | 201 |
Chapter 4 | Some Exotic Geometries | |
Section 1 | Historical Background | 213 |
Section 2 | Spherical Geometry | 217 |
Section 3 | Absolute Geometry | 221 |
Section 4 | More History--Saccheri, Bolyai, and Lobachevsky | 226 |
Section 5 | Hyperbolic Geometry | 230 |
Section 6 | New Beginnings | 242 |
Section 7 | Analytic Geometry--A Reminder | 248 |
Section 8 | Finite Arithmetics | 255 |
Section 9 | Finite Geometries | 262 |
Section 10 | Application | 273 |
Section 11 | Circles and Quadratic Equations | 279 |
Section 12 | Finite Affine Planes | 289 |
Section 13 | Finite Projective Planes | 292 |
Section 14 | Ovals in a Finite Plane | 296 |
Section 15 | A Finite Version of Poincare's Universe | 303 |
Appendix | Excerpts from Euclid | 308 |
Chapter 5 | Games | |
Section 1 | Introduction | 317 |
Section 2 | Some Tree Games | 318 |
Section 3 | The Game of Hex | 327 |
Section 4 | The Game of Nim | 340 |
Section 5 | Games of Chance | 359 |
Section 6 | Matrix Games | 365 |
Section 7 | Applications of Matrix Games | 378 |
Section 8 | Positive-sum Games | 381 |
Section 9 | Sharing | 383 |
Section 10 | Cooperative Games | 385 |
Chapter 6 | What's in a Name? | |
Section 1 | Introduction | 391 |
Section 2 | Historical Background | 392 |
Section 3 | Place Notation for the Base b | 396 |
Section 4 | Some Properties of Natural Numbers Related to Notation | 402 |
Section 5 | Fractions: First Comments | 414 |
Section 6 | Farey Fractions | 416 |
Section 7 | Egyptian Fractions | 421 |
Section 8 | The Euclidean Algorithm | 435 |
Section 9 | Continued Fractions | 440 |
Section 10 | Decimal Fractions | 444 |
Section 11 | Concluding Remarks | 470 |
Glossary of Symbols | 477 | |
Index | 479 | |
Appendix 2000 | 491 |
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