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Softcover ISBN:  9781470456160 
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MAA Member Price:  $93.75 $71.25 
AMS Member Price:  $93.75 $71.25 
Softcover ISBN:  9781470456160 
Product Code:  DOL/56 
List Price:  $65.00 
MAA Member Price:  $48.75 
AMS Member Price:  $48.75 
eBook ISBN:  9781470456177 
Product Code:  DOL/56.E 
List Price:  $60.00 
MAA Member Price:  $45.00 
AMS Member Price:  $45.00 
Softcover ISBN:  9781470456160 
eBook ISBN:  9781470456177 
Product Code:  DOL/56.B 
List Price:  $125.00 $95.00 
MAA Member Price:  $93.75 $71.25 
AMS Member Price:  $93.75 $71.25 

Book DetailsDolciani Mathematical ExpositionsVolume: 56; 2011; 327 pp
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

Front Cover

title page

Preface

Twenty Key Icons of Mathematics

Contents

1 The Bride’s Chair

1.1 The Pythagorean theorem—Euclid’s proof and more

1.2 The Vecten configuration

1.3 The law of cosines

1.4 Grebe’s theorem and van Lamoen’s extension

1.5 Pythagoras and Vecten in recreational mathematics

1.6 Challenges

2 Zhou Bi Suan Jing

2.1 The Pythagorean theorem—a proof from ancient China

2.2 Two classical inequalities

2.3 Two trigonometric formulas

2.4 Challenges

3 Garfield’s Trapezoid

3.1 The Pythagorean theorem—the Presidential proof

3.2 Inequalities and Garfield’s trapezoid

3.3 Trigonometric formulas and identities

3.4 Challenges

4 The Semicircle

4.1 Thales’ triangle theorem

4.2 The right triangle altitude theorem and the geometric mean

4.3 Queen Dido’s semicircle

4.4 The semicircles of Archimedes

4.5 Pappus and the harmonic mean

4.6 More trigonometric identities

4.7 Areas and perimeters of regular polygons

4.8 Euclid’s construction of the five Platonic solids

4.9 Challenges

5 Similar Figures

5.1 Thales’ proportionality theorem

5.2 Menelaus’s theorem

5.3 Reptiles

5.4 Homothetic functions

5.5 Challenges

6 Cevians

6.1 The theorems of Ceva and Stewart

6.2 Medians and the centroid

6.3 Altitudes and the orthocenter

6.4 Anglebisectors and the incenter

6.5 Circumcircle and circumcenter

6.6 Nonconcurrent cevians

6.7 Ceva’s theorem for circles

6.8 Challenges

7 The Right Triangle

7.1 Right triangles and inequalities

7.2 The incircle, circumcircle, and excircles

7.3 Right triangle cevians

7.4 A characterization of Pythagorean triples

7.5 Some trigonometric identities and inequalities

7.6 Challenges

8 Napoleon’s Triangles

8.1 Napoleon’s theorem

8.2 Fermat’s triangle problem

8.3 Area relationships among Napoleon’s triangles

8.4 Escher’s theorem

8.5 Challenges

9 Arcs and Angles

9.1 Angles and angle measurement

9.2 Angles intersecting circles

9.3 The power of a point

9.4 Euler’s triangle theorem

9.5 The Taylor circle

9.6 The Monge circle of an ellipse

9.7 Challenges

10 Polygons with Circles

10.1 Cyclic quadrilaterals

10.2 Sangaku and Carnot’s theorem

10.3 Tangential and bicentric quadrilaterals

10.4 Fuss’s theorem

10.5 The butterfly theorem

10.6 Challenges

11 Two Circles

11.1 The eyeball theorem

11.2 Generating the conics with circles

11.3 Common chords

11.4 Vesica piscis

11.5 The vesica piscis and the golden ratio

11.6 Lunes

11.7 The crescent puzzle

11.8 Mrs. Miniver’s problem

11.9 Concentric circles

11.10 Challenges

12 Venn Diagrams

12.1 Threecircle theorems

12.2 Triangles and intersecting circles

12.3 Reuleaux polygons

12.4 Challenges

13 Overlapping Figures

13.1 The carpets theorem

13.2 The irrationality of sqrt 2 and sqrt 3

13.3 Another characterization of Pythagorean triples

13.4 Inequalities between means

13.5 Chebyshev’s inequality

13.6 Sums of cubes

13.7 Challenges

14 Yin and Yang

14.1 The great monad

14.2 Combinatorial yin and yang

14.3 Integration via the symmetry of yin and yang

14.4 Recreational yin and yang

14.5 Challenges

15 Polygonal Lines

15.1 Lines and line segments

15.2 Polygonal numbers

15.3 Polygonal lines in calculus

15.4 Convex polygons

15.5 Polygonal cycloids

15.6 Polygonal cardioids

15.7 Challenges

16 Star Polygons

16.1 The geometry of star polygons

16.2 The pentagram

16.3 The Star of David

16.4 The star of Lakshmi and the octagram

16.5 Star polygons in recreational mathematics

16.6 Challenges

17 Selfsimilar Figures

17.1 Geometric series

17.2 Growing figures iteratively

17.3 Folding paper in half twelve times

17.4 The spira mirabilis

17.5 The Menger sponge and the Sierpinski carpet

17.6 Challenges

18 Tatami

18.1 The Pythagorean theorem—Bhaskara’s proof

18.2 Tatami mats and Fibonacci numbers

18.3 Tatami mats and representations of squares

18.4 Tatami inequalities

18.5 Generalized tatami mats

18.6 Challenges

19 The Rectangular Hyperbola

19.1 One curve, many definitions

19.2 The rectangular hyperbola and its tangent lines

19.3 Inequalities for natural logarithms

19.4 The hyperbolic sine and cosine

19.5 The series of reciprocals of triangular numbers

19.6 Challenges

20 Tiling

20.1 Lattice multiplication

20.2 Tiling as a proof technique

20.3 Tiling a rectangle with rectangles

20.4 The Pythagorean theorem—infinitely many proofs

20.5 Challenges

Solutions to the Challenges

Chapter 1

Chapter 2

Chapter 3

Chapter 4

Chapter 5

Chapter 6

Chapter 7

Chapter 8

Chapter 9

Chapter 10

Chapter 11

Chapter 12

Chapter 13

Chapter 14

Chapter 15

Chapter 16

Chapter 17

Chapter 18

Chapter 19

Chapter 20

References

Index

About the Authors

Back Cover


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


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 Book Details
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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.

Front Cover

title page

Preface

Twenty Key Icons of Mathematics

Contents

1 The Bride’s Chair

1.1 The Pythagorean theorem—Euclid’s proof and more

1.2 The Vecten configuration

1.3 The law of cosines

1.4 Grebe’s theorem and van Lamoen’s extension

1.5 Pythagoras and Vecten in recreational mathematics

1.6 Challenges

2 Zhou Bi Suan Jing

2.1 The Pythagorean theorem—a proof from ancient China

2.2 Two classical inequalities

2.3 Two trigonometric formulas

2.4 Challenges

3 Garfield’s Trapezoid

3.1 The Pythagorean theorem—the Presidential proof

3.2 Inequalities and Garfield’s trapezoid

3.3 Trigonometric formulas and identities

3.4 Challenges

4 The Semicircle

4.1 Thales’ triangle theorem

4.2 The right triangle altitude theorem and the geometric mean

4.3 Queen Dido’s semicircle

4.4 The semicircles of Archimedes

4.5 Pappus and the harmonic mean

4.6 More trigonometric identities

4.7 Areas and perimeters of regular polygons

4.8 Euclid’s construction of the five Platonic solids

4.9 Challenges

5 Similar Figures

5.1 Thales’ proportionality theorem

5.2 Menelaus’s theorem

5.3 Reptiles

5.4 Homothetic functions

5.5 Challenges

6 Cevians

6.1 The theorems of Ceva and Stewart

6.2 Medians and the centroid

6.3 Altitudes and the orthocenter

6.4 Anglebisectors and the incenter

6.5 Circumcircle and circumcenter

6.6 Nonconcurrent cevians

6.7 Ceva’s theorem for circles

6.8 Challenges

7 The Right Triangle

7.1 Right triangles and inequalities

7.2 The incircle, circumcircle, and excircles

7.3 Right triangle cevians

7.4 A characterization of Pythagorean triples

7.5 Some trigonometric identities and inequalities

7.6 Challenges

8 Napoleon’s Triangles

8.1 Napoleon’s theorem

8.2 Fermat’s triangle problem

8.3 Area relationships among Napoleon’s triangles

8.4 Escher’s theorem

8.5 Challenges

9 Arcs and Angles

9.1 Angles and angle measurement

9.2 Angles intersecting circles

9.3 The power of a point

9.4 Euler’s triangle theorem

9.5 The Taylor circle

9.6 The Monge circle of an ellipse

9.7 Challenges

10 Polygons with Circles

10.1 Cyclic quadrilaterals

10.2 Sangaku and Carnot’s theorem

10.3 Tangential and bicentric quadrilaterals

10.4 Fuss’s theorem

10.5 The butterfly theorem

10.6 Challenges

11 Two Circles

11.1 The eyeball theorem

11.2 Generating the conics with circles

11.3 Common chords

11.4 Vesica piscis

11.5 The vesica piscis and the golden ratio

11.6 Lunes

11.7 The crescent puzzle

11.8 Mrs. Miniver’s problem

11.9 Concentric circles

11.10 Challenges

12 Venn Diagrams

12.1 Threecircle theorems

12.2 Triangles and intersecting circles

12.3 Reuleaux polygons

12.4 Challenges

13 Overlapping Figures

13.1 The carpets theorem

13.2 The irrationality of sqrt 2 and sqrt 3

13.3 Another characterization of Pythagorean triples

13.4 Inequalities between means

13.5 Chebyshev’s inequality

13.6 Sums of cubes

13.7 Challenges

14 Yin and Yang

14.1 The great monad

14.2 Combinatorial yin and yang

14.3 Integration via the symmetry of yin and yang

14.4 Recreational yin and yang

14.5 Challenges

15 Polygonal Lines

15.1 Lines and line segments

15.2 Polygonal numbers

15.3 Polygonal lines in calculus

15.4 Convex polygons

15.5 Polygonal cycloids

15.6 Polygonal cardioids

15.7 Challenges

16 Star Polygons

16.1 The geometry of star polygons

16.2 The pentagram

16.3 The Star of David

16.4 The star of Lakshmi and the octagram

16.5 Star polygons in recreational mathematics

16.6 Challenges

17 Selfsimilar Figures

17.1 Geometric series

17.2 Growing figures iteratively

17.3 Folding paper in half twelve times

17.4 The spira mirabilis

17.5 The Menger sponge and the Sierpinski carpet

17.6 Challenges

18 Tatami

18.1 The Pythagorean theorem—Bhaskara’s proof

18.2 Tatami mats and Fibonacci numbers

18.3 Tatami mats and representations of squares

18.4 Tatami inequalities

18.5 Generalized tatami mats

18.6 Challenges

19 The Rectangular Hyperbola

19.1 One curve, many definitions

19.2 The rectangular hyperbola and its tangent lines

19.3 Inequalities for natural logarithms

19.4 The hyperbolic sine and cosine

19.5 The series of reciprocals of triangular numbers

19.6 Challenges

20 Tiling

20.1 Lattice multiplication

20.2 Tiling as a proof technique

20.3 Tiling a rectangle with rectangles

20.4 The Pythagorean theorem—infinitely many proofs

20.5 Challenges

Solutions to the Challenges

Chapter 1

Chapter 2

Chapter 3

Chapter 4

Chapter 5

Chapter 6

Chapter 7

Chapter 8

Chapter 9

Chapter 10

Chapter 11

Chapter 12

Chapter 13

Chapter 14

Chapter 15

Chapter 16

Chapter 17

Chapter 18

Chapter 19

Chapter 20

References

Index

About the Authors

Back Cover

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