Icons of Mathematics: An Exploration of Twenty Key Images

Description

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. Content: ""Cover "" ""copyright page"" ""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""