Geometric Constructions

1. Front Matter

   Pages i-xi

   PDF

2. Euclidean Constructions

   • George E. Martin

   Pages 1-28

3. The Ruler and Compass

   • George E. Martin

   Pages 29-51

4. The Compass and the Mohr-Mascheroni Theorem

   • George E. Martin

   Pages 53-68

5. The Ruler

   • George E. Martin

   Pages 69-82

6. The Ruler and Dividers

   • George E. Martin

   Pages 83-96

7. The Poncelet-Steiner Theorem and Double Rulers

   • George E. Martin

   Pages 97-105

8. The Ruler and Rusty Compass

   • George E. Martin

   Pages 107-108

9. Sticks

   • George E. Martin

   Pages 109-121

10. The Marked Ruler

    • George E. Martin

    Pages 123-144

11. Paperfolding

    • George E. Martin

    Pages 145-159

12. Back Matter

    Pages 160-206

    PDF

Geometric constructions have been a popular part of mathematics throughout history. The ancient Greeks made the subject an art, which was enriched by the medieval Arabs but which required the algebra of the Renaissance for a thorough understanding. Through coordinate geometry, various geometric construction tools can be associated with various fields of real numbers. This book is about these associations. As specified by Plato, the game is played with a ruler and compass. The first chapter is informal and starts from scratch, introducing all the geometric constructions from high school that have been forgotten or were never seen. The second chapter formalizes Plato's game and examines problems from antiquity such as the impossibility of trisecting an arbitrary angle. After that, variations on Plato's theme are explored: using only a ruler, using only a compass, using toothpicks, using a ruler and dividers, using a marked rule, using a tomahawk, and ending with a chapter on geometric constructions by paperfolding. The author writes in a charming style and nicely intersperses history and philosophy within the mathematics. He hopes that readers will learn a little geometry and a little algebra while enjoying the effort. This is as much an algebra book as it is a geometry book. Since all the algebra and all the geometry that are needed is developed within the text, very little mathematical background is required to read this book. This text has been class tested for several semesters with a master's level class for secondary teachers.

• Mathematica
• Scratch
• algebra
• boundary element method
• class
• construction
• form
• geometry
• history of mathematics
• mathematics
• philosophy
• real number
• story
• theorem
• tool

• Department of Mathematics and Statistics, State University of New York at Albany, Albany, USA

  George E. Martin

Policies and ethics