A relaxed and informal presentation conveying the joy of mathematical discovery and insight. Frequent questions lead readers to see mathematics as an accessible world of thought, where understanding can turn opaque formulae into beautiful and meaningful ideas. The text presents eight topics that illustrate the unity of mathematical thought as well as the diversity of mathematical ideas. Drawn from both "pure" and "applied" mathematics, they include: spirals in nature and in mathematics; the modern topic of fractals and the ancient topic of Fibonacci numbers; Pascals Triangle and paper folding; modular arithmetic and the arithmetic of the infinite. The final chapter presents some ideas about how mathematics should be done, and hence, how it should be taught. Presenting many recent discoveries that lead to interesting open questions, the book can serve as the main text in courses dealing with contemporary mathematical topics or as enrichment for other courses. It can also be read with pleasure by anyone interested in the intellectually intriguing aspects of mathematics.

Inhalt

1 Going Down the Drain.- 1.1 Constructions.- 1.2 Cobwebs.- 1.3 Consolidation.- 1.4 Fibonacci Strikes.- 1.5 Dénouement.- Final Break.- References.- Answers for Final Break.- 2 A Far Nicer Arithmetic.- 2.1 General Background: What You Already Know.- 2.2 Some Special Moduli: Getting Ready for the Fun.- 2.3 Arithmetic mod p: Some Beautiful Mathematics.- 2.4 Arithmetic mod Non-primes: The Same But Different.- 2.5 Primes, Codes, and Security.- 2.6 Casting Out 9's and 11's: Tricks of the Trade.- Final Break.- Answers for Final Break.- 3 Fibonacci and Lucas Numbers.- 3.1 A Number Trick.- 3.2 The Explanation Begins.- 3.3 Divisibility Properties.- 3.4 The Number Trick Finally Explained.- 3.5 More About Divisibility.- 3.6 A Little Geometry!.- Final Break.- References.- Answers for Final Break.- 4 Paper-Folding and Number Theory.- 4.1 Introduction: What You Can Do With-and Without-Euclidean Tools.- I Simple Paper-Folding.- 4.2 Going Beyond Euclid: Folding 2-Period Regular Polygons.- 4.3 Folding Numbers.- 4.4 Some Mathematical Tidbits.- II General Paper-Folding.- 4.5 General Folding Procedures.- 4.6 The Quasi-Order Theorem.- 4.7 Appendix: A Little Solid Geometry.- Final Break.- References.- 5 Quilts and Other Real-World Decorative Geometry.- 5.1 Quilts.- 5.2 Variations.- 5.3 Round and Round.- 5.4 Up the Wall.- Final Break.- References.- Answers for Final Break.- 6 Pascal, Euler, Triangles, Windmills.- 6.1 Introduction: A Chance to Experiment.- I Pascals Set the Scene.- 6.2 The Binomial Theorem.- 6.3 The Pascal Triangle and Windmill.- 6.4 The Pascal Flower and the Generalized Star of David.- II Euler Takes the Stage.- 6.5 Eulerian Numbers and Weighted Sums.- 6.6 Even Deeper Mysteries.- References.- 7 Hair and Beyond.- 7.1 A Problem with Pigeons, and Related Ideas.- 7.2 The Biggest Number.- 7.3 The Big Infinity.- 7.4 Other Sets of Cardinality ?0.- 7.5 Schröder and Bernstein.- 7.6 Cardinal Arithmetic.- 7.7 Even More Infinities?.- Final Break.- References.- Answers for Final Break.- 8 An Introduction to the Mathematics of Fractal Geometry.- 8.1 Introduction to the Introduction: What's Different About Our Approach.- 8.2 Intuitive Notion of Self-Similarity.- 8.3 The lént Map and the Logistic Map.- 8.4 Some More Sophisticated Material.- Final Break.- References.- Answers for Final Break.- An Introduction to the Mathematics of Fractal Geometry.- 8.1 Introduction to the Introduction: What's Different About Our Approach.- 8.2 Intuitive Notion of Self-Similarity.- 8.3 The tent Map and and the Logistic Map.- 8.4 Some more Sophisticated Material.- Final Break.- References.- Answer for Final Break.- 9 Some of Our Own Reflections.- 9.1 General Principles.- 9.2 Specific Principles.- 9.3 Appendix: Principles of Mathematical Pedagogy.- References.