george g. szpiro - kepler's conjecture

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: permcoordinator@wiley.com. 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QA93 .S97 2002 510—dc21 2002014422 Printed in the United States of America 10 9 8 7 6 5 4 3 2 1 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 C O N T E N T S 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 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. vi P R E F A C E One further math note: throughout the text, numbers are truncated after three or four digits. In the mathematical literature this is usually written as, say, 0.883. . . . , to indicate that many more digits (possibly infinitely many) follow. In this book I do not always add the dots after the digits. I have found much valuable material at the Mathematics Library, the Har- man Science Library and the Edelstein Library for History and Philosophy of Science, all at the Hebrew University of Jerusalem. The library of the ETH in Zürich kindly supplied some papers that were not available any- where else, and even the library of the Israeli Atomic Energy Institute pro- vided a hard-to-find paper. I would like to thank all those institutions. The Internet proved, as always, to be a cornucopia of much useful informa- tion . . . and of much rubbish. For example, under the heading “On Johannes Kepler’s Early Life” I found the following gem: “There are no records of Johannes having any parents.” So much for that. Separating the e-wheat from the e-chaff will probably become the most important aspect of Internet search engines of the future. One of the most useful web sites I came across during the research for this book is the MacTutor History of Mathematics archive (www-groups.dcs.st-and.ac.uk/∼history), maintained by the School of Mathematics and Statistics of the University of Saint Andrews in Scotland. It stores a collection of biographies of about 1,500 mathematicians. Friends and colleagues read parts of the manuscript and made sugges- tions. I mention them in alphabetical order. Among the mathematicians and physicists who offered advice and explanations are Andras Bezdek, Benno Eckmann, Sam Ferguson, Tom Hales, Wu-Yi Hsiang, Robert Hunt, Greg Kuperberg, Wlodek Kuperberg, Jeff Lagarias, Christoph Lüthy, Robert MacPherson, Luigi Nassimbeni, Andrew Odlyzko, Karl Sigmund, Denis Weaire, and Günther Ziegler. I thank all of them for their efforts, most of all Tom and Sam, who were always ready with an e-mail clarifica- tion to any of my innumerable questions on the fine points of their proof. Thanks are also due to friends who took the time to read selected chapters: Elaine Bichler, Jonathan Dagmy, Ray and Jeanine Fields, Ies Friede, Jonathan Misheiker, Marshall Sarnat, Benny Shanon, and Barbara Zinn. Itay Almog did much more than just the artwork by correcting some errors and providing me with numerous suggestions for improvement. Special acknowledgment is reserved for my mother, who read the entire manu- script. (Needless to say, she found it fascinating.) I would also like to thank my agent, Ed Knappman, who encouraged me from the time when only a sample chapter and an outline existed, and Jeff Golick, the editor at John Wiley & Sons, who brought the manuscript into publishable form. P R E F A C E vii Finally, I want to express gratefulness and appreciation to my wife, For- tunée, and my children Sarit, Noam, and Noga. They always bore with me when I pointed out yet another instance of Kepler’s sphere arrangement. Their good humor is what makes it all worthwhile. This book was written in no little part to instill in them some love and admiration for science and mathematics. I hope I succeeded. My wife’s first name expresses it best and I want to end by saying, c’est moi qui est fortuné de vous avoir autour de moi! This book is dedicated to my parents, Simcha Binem Szpiro (from War- saw, Poland) and Marta Szpiro-Szikla (from Beregszasz, Hungary). viii P R E F A C E [...]... legs 2 20 KEPLER’S CONJECTURE First, in the construct-your-own-hexagon project, neighboring bees can share the work by building common walls, one of them working on either side In a build-your-own-circle scheme each bee would be on its own Next, Kepler claims that the straight edges of the hexagon give the hive more stability than do round walls, and make it less susceptible to crushing.3 Finally, and... to pack three-dimensional spheres was to stack them in the same manner that market vendors stack their apples, oranges, and melons In 1611 he published a little booklet that he presented as a New Year’s gift to his friend Wacker von Wackerfels It was called The Six-Cornered Snowflake, and in it he described a method of packing balls as tightly as possible This marks the birth of Kepler’s Conjecture We... volume—cylindrical bottles require less plastic than the cube-shaped ones But Eden containers do have an important advantage for the end user The 20-kilogram bottles can be rolled from the front door to the kitchen, while Neviot bottles must be carried Returning to the fruit stand, one method of displaying the wares is to just place them helter-skelter into a box With good reason, very few vendors choose... and foot and guarded day and night by two watchmen Since Kepler had to pay the guards’ salaries, he wrote a letter to the court asking whether one watchman would not suffice to guard this seventy-three-year-old woman, all the more so since she was shackled to a wall anyway Against all odds, Kepler managed to get a verdict of innocence for his mother Actually, Katharina herself was quite instrumental... numbers But Kepler did not only concern himself with big questions about the heavenly bodies, he also showed an interest in the small-scale workings of nature And here lies his importance for our investigation Kepler’s conjecture is contained in the little booklet The Six-Cornered Snowflake, which he wrote in 1611 as a New Year’s gift to his friend Wacker von Wackerfels Von Wackerfels, a traveler, intellectual,... the three-dimensional sphere is, of course, our cannonball Why stop at three dimensions? In fact, mathematicians—who don’t believe anything unless you give them a watertight proof—have no difficulty at all at defining something that nobody could ever see They simply define higher-dimensional spheres in the same manner as they defined lines, circles, and balls: the collection of points in n-dimensional... are, the faster they die, because of an increasing scarcity of resources 26 KEPLER’S CONJECTURE Having done away with bees, pomegranates, apples, and pears, Kepler was ready to attack his original project and get down to the nitty-gritty of snowflakes The first question was, Why are these objects flat rather than three-dimensional? After some deliberations Kepler came to the following conclusion: snowflakes... there is one problem: it’s not true The size of the interface between hot and cold air is large, and both two- or three-dimensional snowflakes could form there as long as their diameters are small The true reason for flatness will be discussed later Then Kepler turned his attention to the six-corneredness He suggested that the snowflake’s visible features may be caused by properties of the building... to be loaded onto airplanes This could be done most efficiently with boxes loaded with cube-shaped melons So why, the reader may ask, did nature evolve round melons (assuming, for illustration purposes, that melons are perfectly round 2 Japanese farmers have figured out how to grow cubic watermelons 6 KEPLER’S CONJECTURE objects)? And why are so many other fruits and vegetables approximately round?... minimizes the surface for a vegetable of a given volume? As the reader may guess, the answer is the sphere.3 If you compare two melons of the same weight, one cube-shaped and the other round, the round one has nearly 20 percent less surface than the cube-shaped one (see the appendix) By evolving round melons, nature strove to minimize the surface in order to reduce moisture loss By the way, this is another . 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. Eckmann, Sam Ferguson, Tom Hales, Wu-Yi Hsiang, Robert Hunt, Greg Kuperberg, Wlodek Kuperberg, Jeff Lagarias, Christoph Lüthy, Robert MacPherson, Luigi Nassimbeni, Andrew Odlyzko, Karl Sigmund, Denis. 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