Mathematical Vistas

by Hilton, Peter John; Holton, Derek Allan; Pedersen, Jean

• ISBN13: 9780387950648
• ISBN10: 0387950648
• Format: Hardcover
• Copyright: 2002-03-01
• Publisher: Springer Nature

Summary

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 different levels of depth and thoroughness. Some chapters can be understood completely with little background, others can be thought of as appetizers for further study. A number of breaks are included in each chapter. These are problems designed to test the reader¿s understanding of the material thus far in the chapter. This book is a sequel to the authors¿ popular book Mathematical Reflections (ISBN 0-387-94770-1) and can be read independently.

Table of Contents

Preface: Focusing Your Attention	vii
Paradoxes in Mathematics	1	(22)
Introduction: Don't Believe Everything You See and Hear	1	(3)
Are Things Equal to the Same Thing Equal to One Another? (Paradox 1)	4	(2)
Is One Student Better Than Another? (Paradox 2)	6	(2)
Do Averages Measure Prowess? (Paradox 3)	8	(3)
May Procedures Be Justified Exclusively by Statistical Tests? (Paradox 4)	11	(3)
A Basic Misunderstanding -- and a Salutary Paradox About Sailors and Monkeys (Paradox 5)	14	(9)
References	20	(3)
Not the Last of Fermat	23	(26)
Introduction: Fermat's Last Theorem (FLT)	23	(1)
Something Completely Different	24	(2)
Diophantus	26	(1)
Enter Pierre de Fermat	27	(1)
Flashback to Pythagoras	28	(4)
Scribbles in Margins	32	(1)
n = 4	33	(3)
Euler Enters the Fray	36	(4)
I Had to Solve It	40	(9)
References	46	(3)
Fibonacci and Lucas Numbers: Their Connections and Divisibility Properties	49	(22)
Introduction: A Number Trick and Its Explanation	49	(5)
A First Set of Results on the Fibonacci and Lucas Indices	54	(2)
On Odd Lucasian Numbers	56	(6)
A Theorem on Least Common Multiples	62	(1)
The Relation Between the Fibonacci and Lucas Indices	63	(1)
On Polynomial Identities Relating Fibonacci and Lucas Numbers	64	(7)
References	69	(2)
Paper-Folding, Polyhedra-Building, and Number Theory	71	(56)
Introduction: Forging the Link Between Geometric Practice and Mathematical Theory	71	(2)
What Can Be Done Without Euclidean Tools	73	(20)
Constructing All Quasi-Regular Polygons	93	(2)
How to Build Some Polyhedra (Hands-On Activities)	95	(19)
The General Quasi-Order Theorem	114	(13)
References	124	(3)
Are Four Colors Really Enough?	127	(32)
Introduction: A Schoolboy Invention	127	(1)
The Four-Color Problem	127	(3)
Graphs	130	(6)
Touring with Euler	136	(2)
Why Graphs?	138	(4)
Another Concept	142	(2)
Planarity	144	(4)
The End	148	(1)
Coloring Edges	149	(4)
A Beginning?	153	(6)
References	157	(2)
From Binomial to Trinomial Coefficients and Beyond	159	(40)
Introduction and Warm-Up	159	(18)
Analogues of the Generalized Star of David Theorems	177	(11)
Extending the Pascal Tetrahedron and the Pascal m-simplex	188	(2)
Some Variants and Generalizations	190	(3)
The Geometry of the 3-Dimensional Analogue of the Pascal Hexagon	193	(6)
References	198	(1)
Catalan Numbers	199	(36)
Introduction: Three Ideas About the Same Mathematics	199	(9)
A Fourth Interpretation	208	(7)
Catalan Numbers	215	(3)
Extending the Binomial Coefficients	218	(2)
Calculating Generalized Catalan Numbers	220	(3)
Counting p-Good Paths	223	(4)
A Fantasy-and the Awakening	227	(8)
References	233	(2)
Symmetry	235	(36)
Introduction: A Really Big Idea	235	(4)
Symmetry in Geometry	239	(15)
Homologues	254	(3)
The Polya Enumeration Theorem	257	(6)
Even and Odd Permutations	263	(8)
References	269	(2)
Parties	271	(28)
Introduction: Cliques and Anticliques	271	(4)
Ramsey and Erdos	275	(2)
Further Progress	277	(4)
N(r, r)	281	(2)
Even More Ramsey	283	(2)
Birthdays and Coincidences	285	(2)
Come to the Dance	287	(3)
Philip Hall	290	(2)
Back to Graphs	292	(3)
Epilogue	295	(4)
References	297	(2)
Selected Answers to Breaks	299	(26)
Index	325