Ruler and the Round: Classic Problems in Geometric Constructions

RULER

AND THE ROUND

Classic Problems in Geometric Constructions

NICHOLAS D. KAZARINOFF

Copyright

Copyright © 1970 by Prindle, Weber & Schmidt, Incorporated All rights reserved.

Bibliographical Note

This Dover edition, first published in 2003, is an unabridged republication of the work published as Ruler and the Round or Angle Trisection and Circle Division by Prindle, Weber & Schmidt, Incorporated, Boston, in 1970.

Library of Congress Cataloging-in-Publication Data

Kazarinoff, Nicholas D.

Ruler and the round : classic problems in geometric constructions/Nicholas D. Kazarinoff.

      p.cm.

Originally published: Boston : Prindle, Weber & Schmidt, 1970, in series: The Prindle, Weber & Schmidt complementary series in mathematics ; v. 15.

Includes bibliographical references and index.

eISBN 13: 9-780-48614361-3

  1. Geometry–Problems, Famous. I. Title.

QA466 .K38 2003

516.2′04–dc21

2002031510

Manufactured in the United States by Courier Corporation

42515002

www.doverpublications.com

in memory of Vasya

a golden man

Vassily Nikolaevich Vyedernikov

1925–1966

Preface

Geometric constructions are easy to comprehend and fun to do. Yet, some are impossible to complete with ruler and compass alone. The most famous of these impossible constructions are discussed in this book. Ground rules, history, and angle trisection are treated in Part I; nonconstructible regular polygons and the algebra associated with them are treated in Part II.

I wrote this book because college and high school audiences have expressed strong interest in its subject, because the mathematics in it is mathematics worth mastering, and because I found writing it challenging and enjoyable.

The book is intended for readers in three groups: high school students who have studied plane geometry for at least one term (particularly, Part I is accessible to those with this minimal background), for high school seniors and college students (to whom the whole book should be accessible), and for amateur mathematicians of any age.

I thank Ivan Niven for valuable comments and advice, and I thank Prindle, Weber & Schmidt Inc., the one publishing company willing to make this work available to its intended audience.

NICHOLAS D. KAZARINOFF

Ann Arbor, Michigan

January 1970

Contents

PART ONE. ANGLE TRISECTION

CHAPTER ONE. PROOF AND UNSOLVED PROBLEMS

1.1 Angle Trisection and Bird Migration

1.2 Proof

1.3 Solved and Unsolved Problems

1.4 Things to Come

CHAPTER TWO. GROUND RULES AND THEIR ALGEBRAIC INTERPRETATION

2.1 Constructed Points

2.2 Analytic Geometry

CHAPTER THREE. SOME HISTORY

CHAPTER FOUR. FIELDS

4.1 Fields of Real Numbers

4.2 Quadratic Fields

4.3 Iterated Quadratic Extensions of R

4.4 Algebraic Classification of Constructible Numbers

CHAPTER FIVE. ANGLES, CUBES, AND CUBICS

5.1 Cubic Equations

5.2 Angles of 20°

5.3 Doubling a Unit Cube

5.4 Some Trisectable and Nontrisectable Angles

5.5 Trisection with n Points Given

CHAPTER SIX. OTHER MEANS

6.1 Marked Ruler, Quadratrix, and Hyperbola

6.2 Approximate Trisections

PART II. CIRCLE DIVISION

CHAPTER SEVEN. IRREDUCIBILITY AND FACTORIZATION

7.1 Why Irreducibility?

7.2 Unique Factorization

7.3 Eisenstein’s Test

CHAPTER EIGHT. UNIQUE FACTORIZATION OF QUADRATIC INTEGERS

CHAPTER NINE. FINITE DIMENSIONAL VECTOR SPACES

9.1 Definitions and Examples

9.2 Linear Dependence and Linear Independence

9.3 Bases and Dimension

9.4 Bases for Iterated Quadratic Extensions of R

CHAPTER TEN. ALGEBRAIC FIELDS

10.1 Algebraic Fields as Vector Spaces

10.2 The Last Link

CHAPTER ELEVEN. NONCONSTRUCTIBLE REGULAR POLYGONS

11.1 Construction of a Regular Pentagon

11.2 Constructibility of Regular Pentagons, a Second View

11.3 Irreducible Polynomials and Regular (2n + 1)-gons

11.4 Nonconstructible Regular Polygons

11.5 Regular ph-gons

11.6 Squaring a Circle

Appendix I

Appendix II

References

Index

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 impossible to trisect a 60° angle with unmarked straightedge and compasses alone and that a proof of this is a solution to the problem of angle trisection.

1.2. PROOF

Our subject is mysterious to some people and challenging to most people. There is confusion in the minds of some people as to whether or not the classical construction problems are solved and what it means to assert that they have been solved. Therefore, it is now wise and proper to give a brief discussion of what a proof in mathematics is and what an unsolved problem is.

The standard by which practically all the world’s mathematicians judge a proof is this: A proof is that which has convinced and now convinces the intelligent reader. Of course, one asks, who are the intelligent readers? The best answer I can give to that question is that within a given culture the intelligent readers of mathematical proofs are those people who are generally accepted to be mathematicians. Moreover, proof is relative: What is good mathematics in this culture in this age may not be considered good mathematics in this or another culture in a future age, just as today we consider much mathematics of past cultures and ages to be incomplete or incorrect. Next, attention should be focused on the point that a proof is an argument that has convinced and now concinces. The use of past and present tenses is deliberate. I maintain that an argument is not a proof until it has been articulated, heard or read, and, finally, found to be convincing, so convincing that there exist live men who are presently convinced of it. A mathematical proof is a temporal, communicable phenomenon in the minds of living men. Mathematical proofs are not arguments written on tablets of gold in Heaven (or on Earth); they are certain collections of thoughts that many people, intelligent readers, hold in common.*

Saying this much is already to invite much philosophical dispute. To say more is to become more involved and technical than is proper here. What about the proofs in this little book? They are restatements of proofs accepted by a wide variety of mathematicians, those people who conjecture and prove theorems, and they have convinced several intelligent readers. I hope they convince you too.

1.3. SOLVED AND UNSOLVED PROBLEMS

One must not confuse the impossibility of a geometric construction with an unsolved problem—or with the unsolvability of a problem! Consider the following example: to construct the longest straight line segment. The construction is impossible, and the problem is thereby solved. It is not an unsolved problem or an insoluble one. (For, suppose one were able to construct a line segment longer than any other. An axiom of plane geometry is that a straight line segment can be extended indefinitely beyond each of its end points. Thus the constructed longest segment would be extendable, and each of its extensions would be longer than itself so that it could not be the longest segment.) Another example is: to construct a square whose side length is a whole number of units and whose area is two square units. Clearly, there exists no such square, and a proof of this statement constitutes a solution to the problem. On the other hand, if we change our rules slightly and admit as candidates for solutions to our problem squares of any side length, then we can solve the problem affirmatively. (Given a straight line segment AB of unit length, we construct a second segment AC perpendicular to AB at A and also of unit length. Then BC is a side of a square of area 2. The length of BC is , which is not a whole number.) Analogously, if we change the classical rules and permit the use of another instrument in addition to an