Mathematical Reflections: In a Room with Many Mirrors

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.