**Dolciani Mathematical Expositions**

Volume: 56;
2011;
327 pp;
Softcover

**Print ISBN: 978-1-4704-5616-0
Product Code: DOL/56**

List Price: $60.00

AMS Member Price: $45.00

MAA Member Price: $45.00

**Electronic ISBN: 978-1-4704-5617-7
Product Code: DOL/56.E**

List Price: $60.00

AMS Member Price: $45.00

MAA Member Price: $45.00

# Icons of Mathematics: An Exploration of Twenty Key Images

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*Claudi Alsina; Roger B. Nelsen*

MAA Press: An Imprint of the American Mathematical Society

The authors present twenty icons of mathematics, that is, geometrical shapes such as the right triangle, the Venn diagram, and the yang and yin symbol and explore mathematical results associated with them. As with their previous books (Charming Proofs, When Less is More, Math Made Visual) proofs are visual whenever possible. The results require no more than high-school mathematics to appreciate and many of them will be new even to experienced readers. Besides theorems and proofs, the book contains many illustrations and it gives connections of the icons to the world outside of mathematics. There are also problems at the end of each chapter, with solutions provided in an appendix. The book could be used by students in courses in problem solving, mathematical reasoning, or mathematics for the liberal arts. It could also be read with pleasure by professional mathematicians, as it was by the members of the Dolciani editorial board, who unanimously recommend its publication.

#### Reviews & Endorsements

Images, whether real or in the imagination, are a foundational component of mathematics. In this book the authors begin with 20 of the most fundamental real images and develop a series of consequences with proofs based on those images. A short section of challenge problems are given at the end of each chapter with solutions included in an appendix. Some of the 20 iconic images used are: Two circles, Venn diagrams, Polygons with circles, Right triangles, The semicircle, and The bride's chair. A set of works in geometry, the book could be used as a text in a college course in Euclidean geometry; it is an excellent study item to prepare high school teachers of geometry. People currently teaching high school geometry will find it a valuable resource for more challenging problems to present to the students. Others with just an interest in geometry will find it worthy of an in-depth look.

-- Charles Ashbacher, Journal of Recreational Mathematics

Treating mainly elementary geometry, this book can be enjoyed by amateurs and professionals alike. All that is needed is some secondary background in Euclidean geometry and trigonometry, seasoned with imagination. The twenty "icons," or geometrical diagrams, some of historical interest, act as an organizing principle. Each sets the stage for a chain of related results, most established in an informal manner by standard Euclidean arguments, algebraic and trigonometric manipulations, and "proofs without words" using partitions and figure-shifting. Frequent digressions provide historical background, short biographies, notes about mathematical artefacts and information about how geometry intervenes in everyday life. Apart from standard results on circles and triangles, the authors discuss a variety of topics, including Dido's isoperimetric problem, regular solids, reptiles, cevians, the butterfly theorem, Reuleaux polygons, polygonal numbers, triangulation of polygons, the cycloid, star polygons, self-similarity, and spirals and tilings. This book is particularly recommended for secondary mathematics students...

-- E.J. Barbeau, Mathematical Reviews

Certain images in mathematics prompt an immediate reaction, similar to the way a smell can trigger a memory. In this provocative collection, the images chosen have what Alsina (Polytechnic Univ. of Catalonia, Spain) and Nelsen (Lewis and Clark College) believe to be iconic value, that is, they are universally recognized. The authors identify and name each image, and explain the image's history, everyday appearance, and mathematical roles. Along with the classical results, they consider generalizations that are not well known but very engaging. For example, cevians make the list for their role in identifying the many special points of triangles. The authors also discuss Stewart's theorem and a generalization to circles. Each of the volume's 20 chapters ends with a "Challenges" section. This unusual work is a welcome addition to any library; all readers will find something to inform and even delight them.

-- J. McCleary, CHOICE

# Table of Contents

## Icons of Mathematics: An Exploration of Twenty Key Images

- Front Cover 11
- title page 44
- Preface 1010
- Twenty Key Icons of Mathematics 1212
- Contents 1414
- 1 The Bride’s Chair 2020
- 2 Zhou Bi Suan Jing 3434
- 3 Garfield’s Trapezoid 4040
- 4 The Semicircle 4848
- 4.1 Thales’ triangle theorem 4949
- 4.2 The right triangle altitude theorem and the geometric mean 5050
- 4.3 Queen Dido’s semicircle 5151
- 4.4 The semicircles of Archimedes 5353
- 4.5 Pappus and the harmonic mean 5656
- 4.6 More trigonometric identities 5757
- 4.7 Areas and perimeters of regular polygons 5858
- 4.8 Euclid’s construction of the five Platonic solids 5959
- 4.9 Challenges 6060

- 5 Similar Figures 6464
- 6 Cevians 8080
- 7 The Right Triangle 9696
- 8 Napoleon’s Triangles 110110
- 9 Arcs and Angles 122122
- 10 Polygons with Circles 136136
- 11 Two Circles 150150
- 11.1 The eyeball theorem 151151
- 11.2 Generating the conics with circles 152152
- 11.3 Common chords 154154
- 11.4 Vesica piscis 156156
- 11.5 The vesica piscis and the golden ratio 157157
- 11.6 Lunes 158158
- 11.7 The crescent puzzle 160160
- 11.8 Mrs. Miniver’s problem 160160
- 11.9 Concentric circles 162162
- 11.10 Challenges 163163

- 12 Venn Diagrams 168168
- 13 Overlapping Figures 182182
- 14 Yin and Yang 192192
- 15 Polygonal Lines 202202
- 16 Star Polygons 220220
- 17 Self-similar Figures 240240
- 18 Tatami 252252
- 19 The Rectangular Hyperbola 262262
- 20 Tiling 272272
- Solutions to the Challenges 280280
- Chapter 1 280280
- Chapter 2 283283
- Chapter 3 284284
- Chapter 4 286286
- Chapter 5 289289
- Chapter 6 291291
- Chapter 7 295295
- Chapter 8 299299
- Chapter 9 302302
- Chapter 10 304304
- Chapter 11 306306
- Chapter 12 309309
- Chapter 13 312312
- Chapter 14 314314
- Chapter 15 316316
- Chapter 16 317317
- Chapter 17 321321
- Chapter 18 322322
- Chapter 19 324324
- Chapter 20 325325

- References 328328
- Index 340340
- About the Authors 346346
- Back Cover 347347