Preface pp. ixxBy Claudi Alsina and Roger Nelsen
Icons of Mathematics
Dolciani Mathematical Expositions
(No. 45)
Online ISBN: 9780883859865
Chapter DOI: http://dx.doi.org/10.5948/UPO9780883859865.001
Subjects: Geometry and topology 

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Of all of our inventions for mass communication, pictures still speak the most universally understood language.
Walt Disney
An icon (from the Greek εικών “image”) is defined as “a picture that is universally recognized to be representative of something.” The world is full of distinctive icons. Flags and shields represent countries, graphic designs represent commercial enterprises; paintings, photographs and even people themselves may evoke concepts, beliefs and epochs. Computer icons are essential tools for working with a great variety of electronic devises.
What are the icons of mathematics? Numerals? Symbols? Equations? After many years working with visual proofs (also called “proofs without words”), we believe that certain geometric diagrams play a crucial role in visualizing mathematical proofs. In this book we present twenty of them, which we call icons of mathematics, and explore the mathematics that lies within and that can be created. All of our icons are twodimensional; threedimensional icons will appear in a subsequent work.
Some of the icons have a long history both inside and outside of mathematics (yin and yang, star polygons, the Venn diagram, etc.). But most of them are essential geometrical figures that enable us to explore an extraordinary range of mathematical results (the bride's chair, the semicircle, the rectangular hyperbola, etc.).
Icons of Mathematics is organized as follows. After the Preface we present a table with our twenty key icons. We then devote a chapter to each, illustrating its presence in real life, its primary mathematical characteristics and how it plays a central role in visual proofs of a wide range of mathematical facts.
Frontmatter:  pp. iviii 
Preface:  pp. ixx 
Twenty Key Icons of Mathematics:  pp. xixii 
Contents:  pp. xiiixviii 
1  The Bride's Chair:  pp. 114 
2  Zhou Bi Suan Jing:  pp. 1520 
3  Garfield's Trapezoid:  pp. 2128 
4  The Semicircle:  pp. 2944 
5  Similar Figures:  pp. 4560 
6  Cevians:  pp. 6176 
7  The Right Triangle:  pp. 7790 
8  Napoleon's Triangles:  pp. 91102 
9  Arcs and Angles:  pp. 103116 
10  Polygons with Circles:  pp. 117130 
11  Two Circles:  pp. 131148 
12  Venn Diagrams:  pp. 149162 
13  Overlapping Figures:  pp. 163172 
14  Yin and Yang:  pp. 173182 
15  Polygonal Lines:  pp. 183200 
16  Star Polygons:  pp. 201220 
17  Selfsimilar Figures:  pp. 221232 
18  Tatami:  pp. 233242 
19  The Rectangular Hyperbola:  pp. 243252 
20  Tiling:  pp. 253260 
Solutions to the Challenges:  pp. 261308 
References:  pp. 309320 
Index:  pp. 321326 
About the Authors:  pp. 327328 