Contents
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
Chapter 11 263
Chapter 13 264
Chapter 15 279
Bibliography 281
Index 287
Preface
This book describes a problem that has vexed mathematicians for nearlyfour hundred years. In 1611, the German astronomer Johannes Kepler conjectured
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 mathematicians
expended on the problem is truly surprising. Mathematicians routinely
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 difficulties.
After Simon Singh published his bestseller on Fermat’s problem,
he wrote in New Scientist that “a worthy successor for Fermat’s Last Theorem
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 horrors
to those who try to solve it.”
I first met Kepler’s conjecture in 1968, as a first-year mathematics student
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 mathematical
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 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 centerpiece.
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 conjecture.
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 chapters,
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 mathematicians.
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 interest.
(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 mathematical
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 rigorous.
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 sometimes
only an example is given rather than a stringent argument.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 Harman
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 anywhere
else, and even the library of the Israeli Atomic Energy Institute provided
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 information
. . . 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 suggestions.
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 clarification
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 manuscript.
(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.Finally, I want to express gratefulness and appreciation to my wife, Fortunée,
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!
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