Mathematical Omnibus: Thirty Lectures on Classic Mathematics

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American Mathematical Soc., 2007 - Mathematics - 463 pages
The book consists of thirty lectures on diverse topics, covering much of the mathematical landscape rather than focusing on one area. The reader will learn numerous results that often belong to neither the standard undergraduate nor graduate curriculum and will discover connections between classical and contemporary ideas in algebra, combinatorics, geometry, and topology. The reader's effort will be rewarded in seeing the harmony of each subject. The common thread in the selected subjects is their illustration of the unity and beauty of mathematics. Most lectures contain exercises, and solutions or answers are given to selected exercises. A special feature of the book is an abundance of drawings (more than four hundred), artwork by an accomplished artist, and about a hundred portraits of mathematicians. Almost every lecture contains surprises for even the seasoned researcher.
 

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Contents

Arithmetic and Combinatorics
5
Arithmetical Properties of Binomial Coefficients
28
On Collecting Like Terms on Euler Gauss and MacDonald
45
Equations
65
Equations of Degree Five
79
How Many Roots Does a Polynomial Have?
93
Chebyshev Polynomials
101
Geometry of Equations
109
Straight Lines
225
Twentyseven Lines
239
Web Geometry
253
The Crofton Formula
269
Polyhedra
285
Noninscribable Polyhedra
301
Impossible Tilings
319
Rigidity of Polyhedra
335

Envelopes and Singularities
125
Around Four Vertices
141
Segments of Equal Areas
159
On Plane Curves
171
Developable Surfaces
187
Paper Möbius Band
203
More on Paper Folding
213
Two Surprising Topological Constructions
359
Cone Eversion
373
The Poncelet Porism and Other Closure Theorems
403
Solutions to Selected Exercises
425
Bibliography
457
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