KEPLER’S CONJECTURE How Some of the Greatest Minds in History Helped Solve One of the Oldest Math Problems in the World George G. Szpiro John Wiley & Sons, Inc. KEPLER’S CONJECTURE How Some of the Greatest Minds in History Helped Solve One of the Oldest Math Problems in the World George G. Szpiro John Wiley & Sons, Inc. This book is printed on acid-free paper.●∞ Copyright © 2003 by George G. Szpiro. All rights reserved Published by John Wiley & Sons, Inc., Hoboken, New Jersey Published simultaneously in Canada Illustrations on pp. 4, 5, 8, 9, 23, 25, 31, 32, 34, 45, 47, 50, 56, 60, 61, 62, 66, 68, 69, 73, 74, 75, 81, 85, 86, 109, 121, 122, 127, 130, 133, 135, 138, 143, 146, 147, 153, 160, 164, 165, 168, 171, 172, 173, 187, 188, 218, 220, 222, 225, 226, 228, 230, 235, 236, 238, 239, 244, 245, 246, 247, 249, 250, 251, 253, 258, 259, 261, 264, 266, 268, 269, 274, copyright © 2003 by Itay Almog. All rights reserved Photos pp. 12, 37, 54, 77, 100, 115 © Nidersächsische Staats- und Universitätsbib- liothek, Göttingen; p. 52 © Department of Mathematics, University of Oslo; p. 92 © AT&T Labs; p. 224 © Denis Weaire No part of this publication may be reproduced, stored in a retrieval system, or trans- mitted in any form or by any means, electronic, mechanical, photocopying, record- ing, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4470, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, email: [email protected]. Limit of Liability/Disclaimer of Warranty: While the publisher and the author have used their best efforts in preparing this book, they make no representations or war- ranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the pub- lisher nor the author shall be liable for any loss of profit or any other commercial dam- ages, including but not limited to special, incidental, consequential, or other damages. For general information about our other products and services, please contact our Customer Care Department within the United States at (800) 762-2974, outside the United States at (317) 572-3993 or fax (317) 572-4002. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books. For more information about Wiley products, visit our web site at www.wiley.com. Library of Congress Cataloging-in-Publication Data: Szpiro, George, date. Kepler’s conjecture : how some of the greatest minds in history helped solve one of the oldest math problems in the world / by George Szpiro. p. cm. Includes bibliographical references and index. ISBN 0-471-08601-0 (cloth : acid-free paper) 1. Mathematics—Popular works. I. Title. QA93 .S97 2002 510—dc21 2002014422 Printed in the United States of America 10987654321 Contents Preface v 1 Cannonballs and Melons 1 2 The Puzzle of the Dozen Spheres 10 3 Fire Hydrants and Soccer Players 33 4 Thue’s Two Attempts and Fejes-Tóth’s Achievement 49 5 Twelve’s Company, Thirteen’s a Crowd 72 6 Nets and Knots 82 7 Twisted Boxes 99 8 No Dancing at This Congress 112 9 The Race for the Upper Bound 124 10 Right Angles for Round Spaces 140 11 Wobbly Balls and Hybrid Stars 156 12 Simplex, Cplex, and Symbolic Mathematics 181 13 But Is It Really a Proof ? 201 14 Beehives Again 215 15 This Is Not an Epilogue 229 Mathematical Appendixes Chapter 1 234 Chapter 2 238 Chapter 3 239 Chapter 4 243 Chapter 5 247 Chapter 6 249 Chapter 7 254 Chapter 9 258 iii iv CONTENTS Chapter 11 263 Chapter 13 264 Chapter 15 279 Bibliography 281 Index 287 Preface This book describes a problem that has vexed mathematicians for nearly four hundred years. In 1611, the German astronomer Johannes Kepler con- jectured that the way to pack spheres as densely as possible is to pile them up in the same manner that greengrocers stack oranges or tomatoes. Until recently, a rigorous proof of that conjecture was missing. It was not for lack of trying. The best and the brightest attempted to solve the problem for four centuries. Only in 1998 did Tom Hales, a young mathematician from the University of Michigan, achieve success. And he had to resort to computers. The time and effort that scores of mathemati- cians expended on the problem is truly surprising. Mathematicians rou- tinely deal with four and higher dimensional spaces. Sometimes this is difficult; it often taxes the imagination. But at least in three-dimensional space we know our way around. Or so it seems. Well, this isn’t so, and the intellectual struggles that are related in this book attest to the immense dif- ficulties. After Simon Singh published his bestseller on Fermat’s problem, he wrote in New Scientist that “a worthy successor for Fermat’s Last Theo- rem must match its charm and allure. Kepler’s sphere-packing conjecture is just such a problem—it looks simple at first sight, but reveals its subtle hor- rors to those who try to solve it.” I first met Kepler’s conjecture in 1968, as a first-year mathematics stu- dent at the Swiss Federal Institute of Technology (ETH). A professor of geometry mentioned in an unrelated context that “one believes that the densest packing of spheres is achieved when each sphere is touched by twelve others in a certain manner.” He mentioned that Kepler had been the first person to state this conjecture and went on to say that together with Fermat’s famous theorem this was one of the oldest unproven mathemati- cal conjectures. I then forgot all about it for a few decades. Thirty years and a few career changes later, I attended a conference in Haifa, Israel. It dealt with the subject of symmetry in academic and artistic v vi PREFACE disciplines. I was working as a correspondent for a Swiss daily, the Neue Zürcher Zeitung (NZZ). The seven-day conference turned out to be one of the best weeks of my journalistic career. Among the people I met in Haifa was Tom Hales, the young professor from the University of Michigan, who had just a few weeks previously completed his proof of Kepler’s conjecture. His talk was one of the highlights of the conference. I subsequently wrote an article on the conference for the NZZ, featuring Tom’s proof as its cen- terpiece. Then I returned to being a political journalist. The following spring, while working up a sweat on my treadmill one afternoon, an idea suddenly hit. Maybe there are people, not necessarily mathematicians, who would be interested in reading about Kepler’s con- jecture. I got off the treadmill and started writing. I continued to write for two and a half years. During that time, the second Palestinian uprising broke out and the peace process was coming apart. It was a very sad and frustrating period. What kept my spirits up in these trying times was that during the night, after the newspaper’s deadline, I was able to work on the book. But then, just as I was putting the finishing touches to the last chap- ters, an Islamic Jihad suicide bomber took the life of one my closest friends. A few days later, disaster hit New York, Washington, and Pennsylvania. If only human endeavor could be channeled into furthering knowledge instead of seeking to visit destruction on one’s fellow men. Would it not be nice if newspapers could fill their pages solely with stories about arts, sports, and scientific achievements, and spice up the latter, at worst, with news on priority disputes and academic battles? This book is meant for the general reader interested in science, scientists, and the history of science, while trying to avoid short-changing mathe- maticians. No knowledge of mathematics is needed except for what one usually learns in high school. On the other hand, I have tried to give as much mathematical detail as possible so that people who would like to know more about what mathematicians do will also find the book of inter- est. (Readers interested in knowing more about the people who helped solve Kepler’s conjecture and the circumstances of their work will also be able to find additional material at www.GeorgeSzpiro.com.) Those readers more interested in the basic story may want to skip the more esoteric mathematical points; for that reason, some of the denser mathemati- cal passages are set in a different font. Even more esoteric material is banished to appendixes. I should point out that the mathematics is by no means rigor- ous. My aim was to give the general idea of what constitutes a mathematical proof, not to get lost in the details. Emphasis is placed on vividness and some- times only an example is given rather than a stringent argument.