
Book DetailsDolciani Mathematical ExpositionsVolume: 45; 2011; 327 pp
Reprinted edition available: DOL/56
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 highschool 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.

Table of Contents

Chapters

Chapter 1. The Bride’s Chair

Chapter 2. Zhou Bi Suan Jing

Chapter 3. Garfield’s Trapezoid

Chapter 4. The Semicircle

Chapter 5. Similar Figures

Chapter 6. Cevians

Chapter 7. The Right Triangle

Chapter 8. Napoleon’s Triangles

Chapter 9. Arcs and Angles

Chapter 10. Polygons with Circles

Chapter 11. Two Circles

Chapter 12. Venn Diagrams

Chapter 13. Overlapping Figures

Chapter 14. Yin and Yang

Chapter 15. Polygonal Lines

Chapter 16. Star Polygons

Chapter 17. Selfsimilar Figures

Chapter 18. Tatami

Chapter 19. The Rectangular Hyperbola

Chapter 20. Tiling


Additional Material

Reviews

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 indepth 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 figureshifting. 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, selfsimilarity, 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

 Book Details
 Table of Contents
 Additional Material
 Reviews
Reprinted edition available: DOL/56
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 highschool 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.

Chapters

Chapter 1. The Bride’s Chair

Chapter 2. Zhou Bi Suan Jing

Chapter 3. Garfield’s Trapezoid

Chapter 4. The Semicircle

Chapter 5. Similar Figures

Chapter 6. Cevians

Chapter 7. The Right Triangle

Chapter 8. Napoleon’s Triangles

Chapter 9. Arcs and Angles

Chapter 10. Polygons with Circles

Chapter 11. Two Circles

Chapter 12. Venn Diagrams

Chapter 13. Overlapping Figures

Chapter 14. Yin and Yang

Chapter 15. Polygonal Lines

Chapter 16. Star Polygons

Chapter 17. Selfsimilar Figures

Chapter 18. Tatami

Chapter 19. The Rectangular Hyperbola

Chapter 20. Tiling

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 indepth 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 figureshifting. 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, selfsimilarity, 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