Mathematical Vistas: From a Room with Many Windows

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Springer Science & Business Media, Jan 8, 2002 - Mathematics - 337 pages
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Focusing YourAttention We have called this book Mathematical Vistas because we have already published a companion book MathematicalRefiections in the same series;1 indeed, the two books are dedicated to the same principal purpose - to stimulate the interest ofbrightpeople in mathematics.Itis not our intention in writing this book to make the earlier book aprerequisite, but it is, of course, natural that this book should contain several references to its predecessor. This is especially - but not uniquely- true of Chapters 3, 4, and 6, which may be regarded as advanced versions of the corresponding chapters in Mathematical Reflections. Like its predecessor, the present work consists of nine chapters, each devoted to a lively mathematical topic, and each capable, in principle, of being read independently of the other chapters.' Thus this is not a text which- as is the intention of most standard treatments of mathematical topics - builds systematically on certain common themes as one proceeds 1Mathematical Reflections - In a Room with Many Mirrors, Springer Undergraduate Texts in Math ematics, 1996; Second Printing 1998. We will refer to this simply as MR. 2There was an exception in MR; Chapter 9 was concerned with our thoughts on the doing and teaching of mathematics at the undergraduate level.
 

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Contents

Paradoxes in Mathematics
1
THING EQUAL TO ONE ANOTHER? PARADOX 1
4
13 IS ONE STUDENT BETTER THAN ANOTHER? PARADOX 2
6
14 DO AVERAGES MEASURE PROWESS? PARADOX 3
8
MAY PROCEDURES BE JUSTIFIED EXCLUSIVELY BY STATISTICAL TESTS? PARADOX 4
11
PARADOX ABOUT SAILORS AND MONKEYS PARADOX 5
14
REFERENCES
20
Not the Last of Fermat
23
Are Four Colors Really Enough?
127
53 GRAPHS
130
54 TOURING WITH EULER
136
55 WHY GRAPHS?
138
56 ANOTHER CONCEPT
142
57 PLANARITY
144
58 THE END
148
59 COLORING EDGES
149

22 SOMETHING COMPLETELY DIFFERENT
24
23 DIOPHANTUS
26
24 ENTER PIERRE DE FERMAT
27
25 FLASHBACK TO PYTHAGORAS
28
26 SCRIBBLES IN MARGINS
32
27 n 4
33
28 EULER ENTERS THE FRAY
36
29 I HAD TO SOLVE IT
40
REFERENCES
46
Fibonacci and Lucas Numbers Their Connections and Divisibility Properties
49
THE FIBONACCI AND LUCAS INDICES
54
33 ON ODD LUCASIAN NUMBERS
56
34 A THEOREM ON LEAST COMMON MULTIPLES
62
35 THE RELATION BETWEEN THE FIBONACCI AND LUCAS INDICES
63
36 ON POLYNOMIAL IDENTITIES RELATING FIBONACCI AND LUCAS NUMBERS
64
REFERENCES
69
PaperFolding Polyhedra Building and Number Theory
71
42 WHAT CAN BE DONE WITHOUT EUCLIDEAN TOOLS
73
43 CONSTRUCTING ALL QUASIREGULAR POLYGONS
93
44 HOW TO BUILD SOME POLYHEDRA HANDSON ACTIVITIES
95
45 THE GENERAL QUASIORDER THEOREM
114
REFERENCES
124
510 A BEGINNING?
153
REFERENCES
157
From Binomial to Trinomial Coefficients and Beyond
159
62 ANALOGUES OF THE GENERALIZED STAR OF DAVID THEOREMS
177
REFERENCES
184
63 EXTENDING THE PASCAL TETRAHEDRON AND THE PASCAL mSIMPLEX
188
64 SOME VARIANTS AND GENERALIZATIONS
190
65 THE GEOMETRY OF THE 3DIMENSIONAL ANALOGUE OF THE PASCAL HEXAGON
193
REFERENCES
198
Catalan Numbers
199
72 A FOURTH INTERPRETATION
208
73 CATALAN NUMBERS
215
74 EXTENDING THE BINOMIAL COEFFICIENTS
218
75 CALCULATING GENERALIZED CATALAN NUMBERS
220
76 COUNTING pGOOD PATHS
223
76 Counting pGood Paths 225
225
REFERENCES
233
Symmetry
235
96 Birthdays and Coincidences
285
Selected Answers to Breaks
299
Index
325
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