by George E. Martin

Springer-Verlag 1998

PREFACE

Books are to be called for and supplied on the assumption that the process of reading is not half-sleep, but in the highest sense an exercise, a gymnastic struggle; that the reader is to do something for himself.

Walt Whitman

The old games are the best games. One of the oldest is geometric constructions. As specified by Plato, the game is played with a ruler and a compass, where the ruler can be used only to draw the line through two given points and the compass can be used only to draw the circle with a given center and through a given point. Skilled players of the game sometimes give themselves a handicap, such as restricting the compass to a fixed opening. A more severe restriction is to use only the ruler, after drawing exactly one circle (Chapter 6). On the other hand, a master player of Plato's game need not use the ruler at all (Chapter 3). Some prefer to play the game of geometric constructions with other tools, even toothpicks (Chapter 8). The most famous of the other construction tools is the marked ruler, which is simply a ruler with two marks on its edge (Chapter 9). We can do more constructions with only the marked ruler than with the ruler and compass. For example, we will prove that angle trisection is generally impossible with only the ruler and compass (Chapter 2), and we will see how to trisect any given angle with a marked ruler. The first chapter starts from scratch and reminds us of all the euclidean constructions from high school that we have forgotten or never seen. The last chapter covers geometric constructions by paperfolding.

Although many of our construction problems are inherited from antiquity, we take advantage of modern algebra and the resultant coordinate geometry to analyze and classify these problems. We necessarily encounter algebra in exploring the constructions. Various geometric construction tools are associated with various algebraic fields of numbers. This book is about these associations. Some readers will find this theoretical association a fascinating end in itself. Some will be stimulated to seek out elegant means of accomplishing those constructions that the theory proves exist and will know to avoid those proposed constructions that the theory proves do not exist. It is important to know what cannot be done in order to avoid wasting time in attempting impossible constructions. The reader of this book will not be among those few persons who turn up every year to proclaim they have "solved a construction problem that has stumped mathematicians for over two thousand years." The principal purposes for reading this book are to learn a little geometry and a little algebra and to enjoy the exercise.

Very little mathematical background is required of the reader. Abstract algebra, in general, and Galois theory, in particular, are not prerequisite. Once the ideas introduced in the second chapter become familiar, the rest of the book follows smoothly. Even though the format is that of a textbook, there are so many hints and answers to be found in the lengthy section called The Back of the Book that the individual studying alone should have no problem testing comprehension against some of the exercises. A lozenge indicates that a given exercise has an entry in The Back of the Book.

By skipping over the optional Chapter 8 to get to the essential Chapter 9, an instructor can expect to cover the material in one semester. A new instructor should be warned that, although students will at first balk at the schemes that are introduced in the first chapter, the students will very quickly learn to use them and that the instructor's problem will be turning the schemes off when they are no longer appropriate.

If the figures in the text have a home-made look, it is partly because they have been made by an author learning to use The Geometer's Sketchpad, Dynamic Geometry for the Macintosh, published by Key Curriculum Press. The dynamic power of this software helped in making the figures and suggests a challenging follow-up seminar that attacks the question, What points can theoretically be constructed with this software? The task would be to consider the mathematical aspects of formulating a new chapter with the geometric construction tool motivated by The Geometer's Sketchpad.

The material has been class tested for many semesters with a master's level class for secondary teachers. The students in these classes have helped shape this book. The text jelled in the summer of 1984 with the then new Macintosh. Notes from that time show that we had class elections to determine the official definitions for the semester. The preliminary version of the text then carried the dedication "For Gillygaloos Everywhere." A residue of these classes can be seen in the somewhat unconventional Chapter 7, where there is a possibility of hands-on learning about mathematical structure.

I would like to thank the editors at Springer-Verlag for accepting "Geometric Constructions" for this distinguished series. There are three wonderful women at Springer-Verlag New York who have steered the text from manuscript to bound book. They are Ina Lindemann, Anne Fossella, and Victoria Evarretta. I also wish to thank Mademoiselle Claude Jacir, Documentaliste au Musee, l'Ordre de la Legion d'honneur, for providing information on Pierre Joseph Glotin. Finally, I am very much indebted to my friend and colleague Hugh Gordon, who made many helpful suggestions while teaching from preliminary versions of this book.

George E. Martin

martin@math.albany.edu

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