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.
Author(s): Mordechai Ben-Ari
Publisher: Springer
Year: 2022
Language: English
Pages: 231
City: Cham
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 ? 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
