Ruler and the Round

Classic Problems in Geometric Constructions

Nicholas D. Kazarinoff

• Publisher: Dover Publications
• Year: 2012
• eBook ISBN: 9780486143613
• Topic: Mathematics
• Subtopic: Geometry
• Index: Mathematics

Although easy to comprehend and fun to do, many geometric constructions defy completion with just a ruler and a compass. This book takes an intriguing look at the most famous of these "impossible" constructions.
In exploring ground rules, history, and angle trisection, the first part considers angle trisection and bird migration, constructed points, analytic geometry, algebraic classification of constructible numbers, fields of real numbers, cubic equations, and marked ruler, quadratix, and hyperbola (among other subjects). The second part treats nonconstructible regular polygons and the algebra associated with them; specifically, irreducibility and factorization, unique factorization of quadratic integers, finite dimensional vector spaces, algebraic fields, and nonconstructible regular polygons.
High school and college students as well as amateur mathematicians will appreciate this stimulating and provocative book, and its glimpses into the crucial role geometry plays in a wide range of mathematical applications.

PART ONE

Angle Trisection

CHAPTER ONE

Proof and Unsolved Problems

1.1. ANGLE TRISECTION AND BIRD MIGRATION

Our subject is the impossibility of several famous ruler and compass constructions—to construct an angle equal to one-third of a given angle, to construct a cube with twice the volume of a given cube, to subdivide a circle into any given number of equal parts. Interest in these problems remains high although they were all solved early in the nineteenth century. Unfortunately, this is not because the solutions are especially simple and beautiful. They are not simple, and their beauty is perhaps perceived only by mathematicians. Possibly interest in these problems remains high because the solutions contradict one’s intuition and frustrate one’s ego and because the solutions are much more difficult to explain than the problems are to state. Even so, that so many students and amateur mathematicians are captivated by these problems, especially the problem of angle trisection, is puzzling. Residents of the Great Lakes area exhibit much less interest in how migratory birds are able to return to their home territories, although at least as many Great Lakes area residents have observed migrating birds as have studied plane geometry. This is odd, since biologists do not yet have a satisfactory explanation of how many birds can find their way home to their birthplace across thousands of miles of land and ocean, whereas all mathematicians are agreed that the angle trisection problem is solved. Biologists do not receive dozens of letters each year with suggested explanations of bird migration, but mathematicians do receive dozens of letters each year containing, so it is claimed, constructions for trisecting any angle.

Junior high school and high school “angle trisectors” may be excused. It is less easy to excuse adult “angle trisectors.” To be candid, almost none of the latter can be convinced he has not made a great discovery; or, if he can be convinced that the particular construction he offers fails, he cannot be persuaded to cease his search, that the search is fruitless. I use the pronoun “he” deliberately. There are, I have concluded, almost no female angle trisectors! In fact, I know of none. Some of the more persistent angle trisectors are, on the other hand, among the most respected males in our society. They are physicians engaged in general practice! Is this because a doctor is used to “being his own boss” and having everyone accept his opinions without question? Confirmed angle trisectors either do not understand the difference between an unsolved and a solved problem in mathematics, or they do not admit the validity of indirect proofs. In any case, no one, least of all a mathematician, can convince a confirmed angle trisector that it is impo...

Table of contents

• Cover
• Title Page
• Copyright Page
• Dedication
• Contents
• Part One. Angle Trisection
• Part Two. Circle Division
• Appendix I
• Appendix II
• References
• Index