Softcover ISBN: | 978-1-4704-5616-0 |
Product Code: | DOL/56 |
List Price: | $65.00 |
MAA Member Price: | $48.75 |
AMS Member Price: | $48.75 |
eBook ISBN: | 978-1-4704-5617-7 |
Product Code: | DOL/56.E |
List Price: | $60.00 |
MAA Member Price: | $45.00 |
AMS Member Price: | $45.00 |
Softcover ISBN: | 978-1-4704-5616-0 |
eBook: ISBN: | 978-1-4704-5617-7 |
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 |
Softcover ISBN: | 978-1-4704-5616-0 |
Product Code: | DOL/56 |
List Price: | $65.00 |
MAA Member Price: | $48.75 |
AMS Member Price: | $48.75 |
eBook ISBN: | 978-1-4704-5617-7 |
Product Code: | DOL/56.E |
List Price: | $60.00 |
MAA Member Price: | $45.00 |
AMS Member Price: | $45.00 |
Softcover ISBN: | 978-1-4704-5616-0 |
eBook ISBN: | 978-1-4704-5617-7 |
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 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.
-
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 Angle-bisectors and the incenter
-
6.5 Circumcircle and circumcenter
-
6.6 Non-concurrent 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 Three-circle 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 Self-similar 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 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
-
-
RequestsReview Copy – for publishers of book reviewsAccessibility – to request an alternate format of an AMS title
- Book Details
- Table of Contents
- Additional Material
- Reviews
- Requests
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.
-
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 Angle-bisectors and the incenter
-
6.5 Circumcircle and circumcenter
-
6.6 Non-concurrent 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 Three-circle 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 Self-similar 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 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