Mathematical vistas : from a room with many windows /
The goal of Mathematical Vistas is to stimulate the interest of bright people in mathematics. The book consists of nine related mathematical essays which will intrigue and inform the curious reader. In order to offer a broad spectrum of exciting developments in mathematics, topics are treated at dif...
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| Other Authors: | , |
| Format: | eBook |
| Language: | English |
| Published: |
New York :
Springer,
[2002]
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| Series: | Undergraduate texts in mathematics.
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| Subjects: | |
| Online Access: | Connect to the full text of this electronic book |
Table of Contents:
- Machine generated contents note: 1 Paradoxes in Mathematics 1
- 1.1 Introduction: Don't Believe Everything You See and Hear 1
- 1.2 Are Things Equal to the Same Thing Equal to One
- Another? (Paradox 1) 4
- 1.3 Is One Student Better Than Another? (Paradox 2)6
- 1.4 Do Averages Measure Prowess? (Paradox 3)8
- 1.5 May Procedures Be Justified Exclusively by Statistical
- Tests? (Paradox 4)11
- 1.6 A Basic Misunderstanding -and a Salutary Paradox
- About Sailors and Monkeys (Paradox 5)14
- References20
- 2 Not the Last of Fermat 23
- 2.1 Introduction: Fermat's Last Theorem (FLT)23
- 2.2 Something Completely Different24
- 2.3 Diophantus26
- 2.4 Enter Pierre de Fermat-27
- 2.5 Flashback to Pythagoras28
- 2.6 Scribbles in Margins32
- 2.7 n = 433
- 2.8 Euler Enters the Fray36
- 2.9 I Had to Solve It40
- References46
- 3 Fibonacci and Lucas Numbers: Their Connections and
- Divisibility Properties 49
- 3.1 Introduction: A Number Trick and Its Explanation 49
- 3.2 A First Set of Results on the Fibonacci and Lucas Indices 54
- 3.3 On Odd Lucasian Numbers56
- 3.4 A Theorem on Least Common Multiples62
- 3.5 The Relation Between the Fibonacci and Lucas Indices .63
- 3.6 On Polynomial Identities Relating Fibonacci and
- Lucas Numbers64
- References69
- 4 Paper-Folding, Polyhedra-Building, and Number Theory 71
- 4.1 Introduction: Forging the Link Between Geometric
- Practice and Mathematical Theory71
- 4.2 What Can Be Done Without Euclidean Tools73
- 4.3 Constructing All Quasi-Regular Polygons93
- 4.4 How to Build Some Polyhedra (Hands-On Activities)95
- 4.5 The General Quasi-Order Theorem114
- References124
- 5 Are Four Colors Really Enough? 127
- 5.1 Introduction: A Schoolboy Invention127
- 5.2 The Four-Color Problem127
- 5.3 Graphs130
- 5.4 Touring with Euler136
- 5.5 Why Graphs?138
- 5.6 Another Concept142
- 5.7 Planarity144
- 5.8 The End148
- 5.9 Coloring Edges149
- 5.10 A Beginning?153
- References157
- 6 From Binomial to Trinomial Coefficients and Beyond 159
- 6.1 Introduction and Warm-Up159
- 6.2 Analogues of the Generalized Star of Da, id Theorems .177
- 6.3 Extending the Pascal Tetrahedron and the
- Pascal m-simplex188
- 6.4 Some Variants and Generalizations190
- 6.5 The Geometry of the 3-Dimensional Analogue of the
- Pascal Hexagon193
- References 198
- 7 Catalan Numbers 199
- 7.1 Introduction: Three Ideas About the Same Mathematics199
- 7.2 A Fourth Interpretation208
- 7.3 Catalan Numbers215
- 7.4 Extending the Binomial Coefficients218
- 7.5 Calculating Generalized Catalan Numbers220
- 7.6 Counting p-Good Paths223
- 7.7 A Fantasy- and the Awakening227
- References 233
- 8 Symmetry 235
- 8.1 Introduction: A Really Big Idea235
- 8.2 Symmetry in Geometry239
- 8.3 Homologues 254
- 8.4 The P61ya Enumeration Theorem257
- 8.5 Even and Odd Permutations263
- References269
- 9 Parties 271
- 9.1 Introduction: Cliques andAnticliques 271
- 9.2 Ramsey and Erd6s 275
- 9.3 Further Progress277
- 9.4 N (r, r) 281
- 9.5 Even More Ramsey283
- 9.6 Birthdays and Coincidences285
- 9.7 Come to the Dance287
- 9.8 Philip Hall290
- 9.9 Back to Graphs292
- 9.10 Epilogue295
- References297.