Mathematical Surprises

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.

Supplementary material to the book can be found at https://github.com/motib/suprises.

Table of Contents

1 The Collapsing Compass

2 Trisection of an Angle

3 Squaring the Circle

4 The Five-Color Theorem

5 How to Guard a Museum

6 Induction

7 Solving Quadratic Equations

8 Ramsey Theory

9 Langford’s Problem

10 The Axioms of Origami

11 Lill’s Method and the Beloch Fold

12 Geometric Constructions Using Origami

13 A Compass Is Sufficient

14 A Straightedge and One Circle is Sufficient

16 Construction of a Regular Heptadecagon

A Theorems From Geometry and Trigonometry

A.1 Theorems About Triangles

A.2 Trigonometric Identities

A.3 The Angle Bisector Theorems

A.4 Ptolemy’s Theorem

A.5 Ceva’s Theorem

A.6 Menelaus’s Theorem

From the Back Cover

This 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.

About the Author

Mordechai (Moti) Ben-Ari is professor emeritus at the Department of Science Teaching of the Weizmann Institute of Science. He has written numerous textbooks including Mathematical Logic for Computer Science and Elements of Robotics (with Francesco Mondada), both published by Springer. He has received the ACM SIGCSE Award for Outstanding Contributions to Computer Science Education and the ACM Karl V. Karlstrom Outstanding Educator Award.