Peter Hilton Derek Holton Jean Pederson

MATHEMATICAL

REFLECTIONS

In a Room with Many Mirrors

Preface: Focusing Your Attention

1 Going Down the Drain

1.1 Constructions 1.2 Cobwebs 1.3 Consolidation 1.4 Fibonacci Strikes 1.5 Denouement 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 Pascal Sets 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 ℵ₀ 7.5 Schroder 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 Tent Map and the Logistic Map 8.4 Some More Sophisticated Material Final Break References Answers 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

Index

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Peter Hilton Derek Holton Jean Pederson

MATHEMATICAL

REFLECTIONS

In a Room with Many Mirrors

Contents