3.1416 And All That PHILIP J. DAVIS WILLIAM G. CHINN Birkhauser Boston. Basel. Stuttgart TO OUR FAMILIES All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without prior permission of the copyright owner. Second edition. First edition published in 1%9 by Simon & Schuster, New York. © Birkhauser Boston, 1985 ABCDEFGHIJ ISBN-13: 978-0-8176-3304-2 e-ISBN-13: 978-1-4615-8519-0 DOl: 10.1007/978-1-4615-8519-0 Library of Congress Cataloging in Publication Data Davis, Philip J., 1923- 3.1416 and all that. Bibliography: p. I. Mathematics-Popular works. I. Chinn, William G. II. Title. QA93.D3 1985 510 85-1314 CIP-Kurztitelaufnahme der Deutschen Bibliothek 3.1416 [Three onejour one six] and all that / Philip J. Davis and William G. Chinn.-2. ed. Boston; Basel; Stuttgart: Birkhiiuser, 1985. 1. AnI!. im VerI. Simon and Schuster, New York HE: Davis, PhilipJ. [Mitverf.]; Chinn, William G. [Mitverf.] CONTENTS Foreword vii Introduction ix 1. The Problem That Saved a Man's Life 1 2. The Code of the Primes 7 3. Pompeiu's Magic Seven 14 4. What Is an Abstraction? 20 5. Postulates-The Bylaws of Mathematics 27 6. The Logical Lie Detector 33 7. Number 40 8. The Philadelphia Story 63 9. Poinsot's Points and Lines 65 10. Chaos and Polygons 71 11. Numbers, Point and Counterpoint 79 12. The Mathematical Beauty Contest 88 13. The House That Geometry Built 94 14. Explorers of the Nth Dimension 101 15. The Band-Aid Principle 108 16. The Spider and the Fly 117 17. A Walk in the Neighborhood 123 18. Division in the Cellar 131 19. The Art of Squeezing 137 20. The Business of Inequalities 144 21. The Abacus and the Slipstick 152 22. Of Maps and Mathematics 159 23. "Mr. Milton, Mr. Bradley-Meet Andrey Andreyevich Markov" 164 24. 3.1416 and All That 172 Bibliography 177 Ancient and Honorable Society ofP i Watchers: 1984 Report 177 Bibliography 181 ACKNOWLEDGMENTS Most of the articles in this book originally appeared in Science World. Grateful acknowledgment is made to Science World for per mission to use them here. We thank the Scientific American for permission to reprint the article entitled "Number." The authorship of the individual articles is as follows: Philip J. Davis: 1,3,4,7,8,9, 12, 14, 15, 18,22,23, 24. William G. Chinn: 2, 5, 6, 10, 11, 13, 16, 17, 19,20,21. The first author would like to acknowledge his indebtedness to Professor Alexander Ostrowski and to Drs. Barry Bernstein and John Wrench for information incorporated in some of his articles. The second author would like to acknowledge his indebted ness to Professor G. Baley Price for permission to use materials from one of his articles. FOREWORD LYTTON STRACHEY tells the following story. In intervals of relaxation from his art, the painter Degas used to try his hand at writing sonnets. One day, while so engaged, he found that his in spiration had run dry. In desperation he ran to his friend Mallarme, who was a poet. "My poem won't come out," he said, "and yet I'm full of excellent ideas." "My dear Degas," Mallarme retorted, "poetry is not written with ideas, it is written with words." If we seek an application of Mallarme's words to mathematics we find that we shall want to turn his paradox around. We are led to say that mathematics does not consist of formulas, it consists of ideas. What is platitudinous about this statement is that mathe matics, of course, consists of ideas. Who but the most unregenerate formalist, asserting that mathematics is a meaningless game played with symbols, would deny it? What is paradoxical about the state ment is that symbols and formulas dominate the mathematical page, and so one is naturally led to equate mathematics with its formulas. + = Is not Pythagoras' Theorem a2 b2 c2? Does not the Binomial + = + + Theorem say that (a b) 2 a2 2ab b2? What more need be said-indeed, what more can be said? And yet, as every devotee who has tried to be creative in mathematics knows, formulas are not enough; for new formulas can be produced by the yard, while original thought remains as remote as hummingbirds in January. To appreciate mathematics at its deeper level we must pass from naked formulas to the ideas that lie behind them. In the present volume, we have selected for reprinting a num ber of pieces that appeared principally in Science World, a periodi cal with a wide circulation among students and teachers. In writing these articles it was our aim to deal with a number of diverse areas of current mathematical interest and, by concentrating on a limited aspect of each topic, to expose in a modest way the mathematical ideas that underlie it. It has not been possible, in the few pages allotted to each essay, to present the topics in the conventional text- vII viii Foreword book sense; our goal has been rather to provide a series of ap petizers or previews of coming attractions which might catch the reader's imagination and attract him to the thoroughgoing treat ments suggested in the bibliographies. Each article is essentially self-contained. What reason can we put forward for the study of mathematics by the educated man? Every generation has felt obliged to say a word about this. Some of the reasons given for this study are that mathematics makes one think logically, that mathematics is the Queen of the Sciences, that God is a geometer who runs His uni verse mathematically, that mathematics is useful in surveying fields, building pyramids, launching satellites. Other reasons are that life has become increasingly concerned with the manipulation of sym bols, and mathematics is the natural language of symbols; that "Euclid alone has looked upon beauty bare"; that mathematics can be fun. Each of these has its nugget of truth and must not be denied. Each undoubtedly can be made the basis of a course of instruction. But we should like to suggest a different reason. Mathematics has been cultivated for more than four thousand years. It was studied long before there were Democrats and Republicans or any of our present concerns. It has flourished in many lands, and the genius of many peoples has added to its stock of ideas. Mathe matics dreams of an order which does not exist. This is the source of its power; and in this dream it has exhibited a lasting quality that resists the crash of empire and the pettiness of small minds. Mathe matical thought is one of the great human achievements. The study of its ideas, its past and its present, can enable the individual to free himself from the tyranny of time and place and circumstance. Is not this what liberal education is about? Summer, 1968 INTRODUCTION 3.1416 AND ALL THAT is a beautiful book written by two masters of popularization. Philip J. Davis and William G. Chinn are math ematicians of many dimensions, who are interested in much more than mathematics. In the following twenty-four essays, you obtain pleasant introductions to powerful ideas of mathematics. In fact, you will have fun reading this book of mathematics. Essayists Davis and Chinn write with style, vigor, and a sense of the relationship of mathematics to other fields, including biology, history, music, philosophy, and even golf and woodworking. Professors Chinn and Davis are much aware of mathematics as a human activity. In their essays, they provide precious insights into how mathematicians think. They also show us that mathematics is alive and growing. If you like mathematics, this book will cause you to like it more. If you are wary of mathematics, then reading this book is likely to make you a mathematical convert. Donald J. Albers Second Vice-President The Mathematical Association of America ix 1 THE PROBLEM THAT SAVED A MAN'S LIFE TRUTH IS STRANGER THAN FICTION-even in the world of mathematics. It is hard to believe that the mathematical ideas of a small religious sect of ancient Greece were able to save-some twenty-five hundred years later-the life of a young man who lived in Germany. The story is related to some very important mathe matics and has a number of strange twists. * The sect was that of the Pythagoreans, whose founder, Pythag oras, lived ,in the town of Crotona in southern Italy around 500 B.C. The young man was Paul Wop· >kehl, who was professor of mathematics in Darmstadt, Germany, in the early 1900's. Between the two men lay a hundred generations, a thousand miles, and many dead civilizations. But between the two men stretched the binding cord of a mathematical idea-a thin cord, perhaps, but a strong and enduring one. Pythagoras was both a religious prophet and a mathematician. He founded a school of mathematics and a religion based upon the notion of the transmigration of souls. "All is number," said Pythag oras, and each generation of scientists discovers new reasons for • I have the story about the young man-Paul Wolfskehl-from the renowned mathematician Alexander Ostrowski. Professor Ostrowski himself heard the story many years ago and claims there is more to 'it than mere legend.-P.l.D. 1 2 3.1416 and All That thinking this might be true. "Do not eat beans, and do not touch white roosters," said Pythagoras, and succeeding generations won der whatever was in the man's head. "The square on the hypotenuse is equal to the sum of the squares on the sides," said Pythagoras, and each generation of geometry students has faithfully learned this relation-a relation that has retained its importance over the intervening years. Paul Wolfskehl was a modern man, somewhat of a romantic. He is as obscure to the world as Pythagoras is famous. Wolfskehl was a good mathematician, but not an extremely original one. The one elegant idea he discovered was recently printed in a textbook without so much as a credit line to its forgotten author. He is prin cipally remembered today as the donor of a prize of 100,000 marks -long since made worthless by inflation-for the solution of the famous unsolved "Fermat's Last Problem." But this prize is an im portant part of the story, and to begin it properly, we must return to the square on the hypotenuse. The Theorem of Pythagoras tells us that if a and b are the lengths of the legs of a right-angled triangle and if c is the length of its hypotenuse, then + a2 b2 = c2• The cases when a = 3, b = 4, c = 5 and when-a'= 5, b = 12, c = 13 are very familiar to us, and probably were familiar to the mathematicians of antiquity long before Pythagoras formulated his general theorem. The Greeks sought a formula for generating all the integer (whole number) solutions to Pythagoras' equation, and by the year 250 A.D. they had found it. At that time, the mathematician Dio phantus wrote a famous book on arithmetic (today we might prefer to call it the "theory of numbers"), and in his book he indicated the following: If m and n are any two integers, then the three in + tegers m2 - n2, 2mn, and m2 n2 are the two legs and the hypote nuse of a right triangle. For instance, if we select m = 2, n = 1, we obtain 3,4, and 5; if we select m = 3, n = 2, we obtain 5, 12, 13; and so on. Our story now skips about fourteen hundred years-while the The Problem That Saved a Man's Life 3 world of science slept soundly-to the year 1637, a time when mathematics was experiencing a rebirth. There was in France a king's counselor by the name of Pierre Fermat. Fermat's hobby was mathematics, but to call him an ama teur would be an understatement, for he was as skilled as any mathematician then alive. Fermat happened to own an edition of Diophantus, and it is clear that he must have read it from cover to cover many times, for he discovered and proved many wonderful things about numbers. On the very page where Diophantus talks + about the numbers m2 - n2, 2mn, and m2 n2, Fermat wrote in the margin: "It is impossible to separate a cube into two cubes, a fourth power into two fourth powers, or, generally, any power above the second into two powers of the same degree. I have dis covered a trulv marvelous demonstration which this margin is too narrow to contain." Let's see what Fermat meant by this. By Pythagoras' Theorem, a square, c2, is "separated" into two other squares, a2 and b2• If + we could do likewise for cubes, we would have c3 = a3 b3• If + we could do it for fourth powers, we would have c4 = a4 b4• But Fermat claims that this is impossible. Moreover, if n is any integer + greater than 2, Fermat claims, it is impossible to have CD = aD bD, where a, b, and c are positive integers. He left no indication of what his demonstration was. From 1637 to the present time, the most renowned mathe maticians of each century have attempted to supply a proof for Fermat's Last Theorem. Euler, Legendre, Gauss, Abel, Cauchy, Dirichlet, Lame, Kummer, Frobenius, and hosts of others have tackled it. In the United States, and closer to our own time, L. E. Dickson, who was professor at the University of Chicago, and H. S. Vandiver, a professor at the University of Texas, have gone after the problem with renewed vigor and cunning. But all have failed. However, many experts feel that the solution to the problem is now within the scope of methods that have been developed in the last few decades. Perhaps the most ingenious of all the contributions to the