Mathematical Surprises

Description

This is open access book provides plenty of pleasant mathematical surprises. There are many fascinating results that do not appear in textbooks although they are accessible with a good knowledge of secondary-school mathematics. This book presents a selection of these topics including the mathematical formalization of origami, construction with straightedge and compass (and other instruments), the five- and six-color theorems, a taste of Ramsey theory and little-known theorems proved by induction. Among the most surprising theorems are the Mohr-Mascheroni theorem that a compass alone can perform all the classical constructions with straightedge and compass, and Steiner's theorem that a straightedge alone is sufficient provided that a single circle is given. The highlight of the book is a detailed presentation of Gauss's purely algebraic proof that a regular heptadecagon (a regular polygon with seventeen sides) can be constructed with straightedge and compass. Although the mathematics used in the book is elementary (Euclidean and analytic geometry, algebra, trigonometry), students in secondary schools and colleges, teachers, and other interested readers will relish the opportunity to confront the challenge of understanding these surprising theorems. Foreword Preface What Is a Surprise? An Overview of the Contents Style Acknowledgments Contents Chapter 1 The Collapsing Compass 1.1 Construction with a Straightedge and Compass 1.2 Fixed Compasses and Collapsing Compasses 1.3 Euclid’s Construction for Copying a Line Segment 1.4 A Flawed Construction for Copying a Line Segment 1.5 Don’t Trust a Diagram What Is the Surprise? Sources Chapter 2 Trisection of an Angle 2.1 Approximate Trisections 2.1.1 First Approximate Trisection 2.1.2 Second Approximate Trisection 2.2 Trisection Using a Neusis 2.3 Doubling the Cube with a Neusis 2.4 Trisection Using a Quadratrix 2.5 Constructible Numbers 2.6 Constructible Numbers As Roots of Polynomials 2.7 Impossibility of the Classical Constructions What Is the Surprise? Sources Chapter 3 Squaring the Circle 3.1 Kocha´nski’s Construction 3.2 Ramanujan’s First Construction 3.3 Ramanujan’s Second Construction 3.4 Squaring a Circle Using a Quadratrix What Is the Surprise? Sources Chapter 4 The Five-Color Theorem 4.1 Planar Maps and Graphs 4.2 Euler’s Formula 4.3 Non-planar Graphs 4.4 The Degrees of the Vertices 4.5 The Six-Color Theorem 4.6 The Five-Color Theorem 4.7 Kempe’s Incorrect Proof of the Four-Color Theorem What Is the Surprise? Sources Chapter 5 How to Guard a Museum 5.1 Coloring Triangulated Polygons 5.2 From Coloring of Polygons to Guarding a Museum 5.3 Any Polygon Can Be Triangulated What Is the Surprise? Sources Chapter 6 Induction 6.1 The Axiom of Mathematical Induction 6.2 Fibonacci Numbers 6.3 Fermat Numbers 6.4 McCarthy’s 91-function 6.5 The Josephus Problem What Is the Surprise? Sources Chapter 7 Solving Quadratic Equations 7.1 Traditional Methods for Solving Quadratic Equations 7.2 The Relation Between the Roots and the Coefficients 7.3 Examples of Loh’s Method 7.4 Derivation of the Traditional Formula 7.5 Al-Khwarizmi’s Geometric Solution of Quadratic Equations 7.6 Cardano’s Construction for Solving Cubic Equations 7.7 They Weren’t Intimidated by Imaginary Numbers 7.8 Lill’s Method and Carlyle’s Circle 7.9 Numerical Computation of the Roots What Is the Surprise? Sources Chapter 8 Ramsey Theory 8.1 Schur triples 8.2 Pythagorean Triples 8.3 Van der Waerden’s problem 8.4 Ramsey’s Theorem 8.5 The Probabilistic Method 8.6 SAT Solving 8.6.1 Propositional Logic and the SAT Problem 8.6.2 Schur triples 8.6.3 Pythagorean Triples 8.6.4 An Overview of the DPLL Algorithm 8.7 Pythagorean Triples in Babylonian Mathematics What Is the Surprise? Sources Chapter 9 Langford’s Problem 9.1 Langford’s Problem as a Covering Problem 9.2 For Which Values of 𝑵 Is Langford’s Problem Solvable? 9.3 Solution for L(4) What Is the Surprise? Sources Chapter 10 The Axioms of Origami 10.1 Axiom 1 10.2 Axiom 2 10.3 Axiom 3 10.4 Axiom 4 10.5 Axiom 5 10.6 Axiom 6 10.6.1 Derivation of the Equation of a Fold 10.6.2 Derivation of the Equations of the Reflections 10.6.3 Tangents to a Parabola 10.7 Axiom 7 What Is the Surprise? Sources Chapter 11 Lill’s Method and the Beloch Fold 11.1 A Magic Trick 11.2 Specification of Lill’s Method 11.2.1 Lill’s Method as an Algorithm 11.2.2 Negative Coefficients 11.2.3 Zero Coefficients 11.2.4 Non-integer Roots 11.2.5 The Cube Root of Two 11.3 Proof of Lill’s Method 11.4 The Beloch Fold What Is the Surprise? Sources Chapter 12 Geometric Constructions Using Origami 12.1 Abe’s Trisection of an Angle 12.2 Martin’s Trisection of an Angle 12.3 Messer’s Doubling of a Cube 12.4 Beloch’s Doubling of a Cube 12.5 Construction of a Regular Nonagon What Is the Surprise? Sources Chapter 13 A Compass Is Sufficient 13.1 What Is a Construction With Only a Compass? 13.2 Reflection of a Point 13.3 Construction of a Circle With a Given Radius 13.4 Addition and Subtraction of Line Segments 13.5 Construction of a Line Segment as a Ratio of Segments 13.6 Construction of the Intersection of Two Lines 13.7 Construction of the Intersection of a Line and a Circle What Is the Surprise? Sources Chapter 14 A Straightedge and One Circle is Sufficient 14.1 What Is a Construction With Only a Straightedge? 14.2 Construction of a Line Parallel to a Given Line 14.3 Construction of a Perpendicular to a Given Line 14.4 Copying a Line Segment in a Given Direction 14.5 Construction of a Line Segment as a Ratio of Segments 14.6 Construction of a Square Root 14.7 Construction of the Intersection of a Line and a Circle 14.8 Construction of the Intersection of Two Circles What Is the Surprise? Sources Chapter 15 Are Triangles with Equal Areas and Perimeters Congruent? 15.1 From a Triangle to an Elliptic Curve 15.2 Solving the Equation for the Elliptic Curve 15.3 Derivation of a Triangle From the Elliptic Curve What Is the Surprise? Sources Chapter 16 Construction of a Regular Heptadecagon 16.1 Construction of Regular Polygons 16.2 The Fundamental Theorem of Algebra 16.3 Roots of Unity 16.4 Gauss’s Proof That a Heptadecagon Is Constructible 16.5 Derivation of Gauss’s Formula 16.6 Construction of a Heptadecagon 16.7 Construction of a Regular Pentagon What Is the Surprise? Sources 16.7.1 Trigonometry 16.7.2 Geometry Appendix A Theorems From Geometry and Trigonometry A.1 Theorems About Triangles A.1.1 Computing the Area of a Triangle A.2 Trigonometric Identities A.2.1 The Sine and Cosine of the Sum and Difference of Two Angles A.2.2 The Cosine of a Triple Angle A.2.3 The Sine and Cosine of a Half-Angle A.2.4 The Law of Cosines A.2.5 The Tangent of the Sum of Two Angles A.2.6 The Tangent of a Half-Angle A.2.7 The Product of Three Tangents A.2.8 The Limit of sin A.3 The Angle Bisector Theorems A.4 Ptolemy’s Theorem A.4.1 A Trapezoid Circumscribed by a Circle A.4.2 Proof of Ptolemy’s Theorem A.5 Ceva’s Theorem A.6 Menelaus’s Theorem Sources References Index