Mathematical Vistas: From a Room with Many Windows

Couverture
Springer Science & Business Media, 8 janv. 2002 - 337 pages
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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Table des matières

Paradoxes in Mathematics
3
THING EQUAL TO ONE ANOTHER? PARADOX 1
6
13 IS ONE STUDENT BETTER THAN ANOTHER? PARADOX 2
8
14 DO AVERAGES MEASURE PROWESS? PARADOX 3
10
MAY PROCEDURES BE JUSTIFIED EXCLUSIVELY BY STATISTICAL TESTS? PARADOX 4
13
PARADOX ABOUT SAILORS AND MONKEYS PARADOX 5
16
Not the Last of Fermat
25
22 SOMETHING COMPLETELY DIFFERENT
26
54 TOURING WITH EULER
138
55 WHY GRAPHS?
140
56 ANOTHER CONCEPT
144
57 PLANARITY
146
58 THE END
150
59 COLORING EDGES
151
510 A BEGINNING?
155
REFERENCES
159

23 DIOPHANTUS
28
24 ENTER PIERRE DE FERMAT
29
25 FLASHBACK TO PYTHAGORAS
30
26 SCRIBBLES IN MARGINS
34
27 n 4
35
28 EULER ENTERS THE FRAY
38
29 I HAD TO SOLVE IT
42
Fibonacci and Lucas Numbers Their Connections and Divisibility Properties
51
THE FIBONACCI AND LUCAS INDICES
56
33 ON ODD LUCASIAN NUMBERS
58
34 A THEOREM ON LEAST COMMON MULTIPLES
64
35 THE RELATION BETWEEN THE FIBONACCI AND LUCAS INDICES
65
36 ON POLYNOMIAL IDENTITIES RELATING FIBONACCI AND LUCAS NUMBERS
66
REFERENCES
71
PaperFolding Polyhedra Building and Number Theory
73
42 WHAT CAN BE DONE WITHOUT EUCLIDEAN TOOLS
75
43 CONSTRUCTING ALL QUASIREGULAR POLYGONS
95
44 HOW TO BUILD SOME POLYHEDRA HANDSON ACTIVITIES
97
45 THE GENERAL QUASIORDER THEOREM
116
Are Four Colors Really Enough?
129
53 GRAPHS
132
From Binomial to Trinomial Coefficients and Beyond
161
62 ANALOGUES OF THE GENERALIZED STAR OF DAVID THEOREMS
179
REFERENCES
185
63 EXTENDING THE PASCAL TETRAHEDRON AND THE PASCAL mSIMPLEX
190
64 SOME VARIANTS AND GENERALIZATIONS
192
65 THE GEOMETRY OF THE 3DIMENSIONAL ANALOGUE OF THE PASCAL HEXAGON
195
REFERENCES
200
Catalan Numbers
201
72 A FOURTH INTERPRETATION
210
73 CATALAN NUMBERS
217
74 EXTENDING THE BINOMIAL COEFFICIENTS
220
75 CALCULATING GENERALIZED CATALAN NUMBERS
222
76 COUNTING pGOOD PATHS
225
76 Counting pGood Paths 225
227
REFERENCES
235
Symmetry
237
96 Birthdays and Coincidences
287
Selected Answers to Breaks
301
Index
327
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