Kepler's Conjecture: How Some of the Greatest Minds in History Helped Solve One of the Oldest Math Problems in the World

"No book in recent decades conveys more forcefully and beautifully the excitement of mathematical exploration than Dr. Szpiro's work."
- Clifford A. Pickover, author of The Mathematics of Oz

"A gripping and intelligent account of the solution of one of the great problems of mathematics-older than Fermat, and just as baffling. Kepler's Conjecture offers the nonspecialist genuine insights into the minds of research mathematicians when they are grappling with big, important questions. I enjoyed the book immensely."
- Ian Stewart, author of Flatterland and What Does a Martian Look Like?

Sir Walter Raleigh simply wanted to know the best and most efficient way to pack cannonballs in the hold of his ship. In 1611, German astronomer Johannes Kepler responded with the obvious answer: by piling them up the same way that grocers stack oranges or melons. For the next four centuries, Kepler's conjecture became the figurative loose cannon in the mathematical world as some of the greatest intellects in history set out to prove his theory. Kepler's Conjecture provides a mesmerizing account of this 400-year quest for an answer that would satisfy even the most skeptical mathematical minds.

The authors writing style is friendly; written definately for the laymen. Unfortunately, the author is also very sloppy in his writing style. He constantly throws out names, dates, and shifts back in forth in time spans over decades and even centuries. Sometimes you think to yourself "what century is he talking about now." Sometimes he'll throw out the name of a mathemtician, but won't even tell you when he did his research or won't give you a good frame of reference. This can be confusing if you are trying to build a mental timeline of the history of sphere packing research. If you read the book 20 times, you might be able to extrapolate through this bad writing style... but some questions are still left unaswered.

The majority of the discussion focuses on packing spheres in 1, 2 and 3 dimensions. He explains the history of this quite well over the course of several hundred pages. Occasionally, he'll
talk about higher dimensions and explain in laymens terms what you can do in higher dimensions that you can do in lower dimensions,.. and he tries to give you the gist of the idea of why mathemeticians even care about higher dimensions. I found this understandable, but dissapointing in that he didn't at least dedicate 1 chapter specifically to this topic. However, in fairness, the book is about Keplers conjecture, focusing on 3 dimensions. It's not really supposed to be about sphere packing in D > 3. As an a laymen enthusiast, I was dissapointed because I was hoping to learn more about higher dimensionality, as I really don't understand how that works.
But I think the excellent way in which the authorpresented the history has really motivated me to study this subject in more detail, so I'll seek alternative means to find out about higher dimensions.

The main emphasis of this book is the history of sphere packing. It all starts with the idea of how many cannonballs can be packed.. However, the author actually mentions that the problem actually dates to Greek times and was considered long before. I had no idea so many mathemeticians through history had worked on this problem, so the history was very rich and pleased me in the manner in which he dealt with it. Unfortunately, the author tends to gloss over the actual details of how various mathemeticians provided proofs. Many of these proofs were 20 to 100 pages, so I can see how it might be difficult to work this into a book.... but I definately think the author didn't put enough effort into this. I think that some math, could have been put into this book. Most of the history pertains to mathemeticians pursuits to find "upper bounds" for sphere packing. Lower bounds are not talked about much, but are mentioned briefly when the author say's the Riemann zeta function can somehow be used as a lower bound in higher dimensions...but he doesn't feel the need to explain this unfortunately.

Overall, I give the book 4-5 stars as a historical overview. I was very pleased in that aspect. I give the author 3 stars for his friendly writing style, but he aslo gets very sloppy by moving back and forward in time too much that it gets confusing. I'll give him 3 stars for "technical content" if we are to assume that the technical content is strictly for laymens. There really is no math content per se for anyone who has any higher college level math, but it might raise some thought provoking questions for those who'smaths skills aren't so good.

However, I strongly believe that any mathemeticians would be VERY happy to read this book. In fact, the author shows that several recent mathemeticians who worked on providing proofs for Keplers Conjecture were not even aware of the problem until well after they were professors... meaning that this book serves not so much a technical manual, but merely a source of history and motivation for those interested. Therefore, on average, I give this book maybe 3.5 - 4 stars.

Also, he doesn't discuss sphere packing as it applies to error correcting codes, but it should interest you nevertheless. If thats your interest, see Sloan and Conways bible on sphere Packing and Lattices; the ultimate tome on that subject.
-Hoffman


Very good but needed (more) proofreading.   April 14, 2005
Rod Ball (London United Kingdom)
6 out of 6 found this review helpful

This is an excellent book for the following reasons.
1. It provides a good historical review and an up-to-date description of a math topic with hardly any other books to meet the need. This book only covers the (2D&)3D versions of the kissing problem (proof that only 12 spheres can touch a central one) and the densest packing problem (which lacked a proof for 300 yrs. until recently). The only other books available are Zong and Conway & Sloane which are multi-dimensional specialized academic treatments - C&S being out of date and Zong only giving Hales' new proof superficial mention. Compared to the hoopla that greeted solutions to the 4-colur problem and Fermat's last theorem, solving Kepler's problem has gone almost unnoticed.
2. It is one of those all too rare "popular" books that isn't afraid to include a fair bit of mathematics, albeit mostly in appendices. This is mainly spherical trigonometry, but that's the nature of the problem and it gives a good idea of how Hales' elaborate proof procedure works, although for the voluminous ugly detail you need to consult Hales' website.
However, there are a lot of errors and misleading details such as;
p.25 Dodecahedron & Icosahedron labelled the wrong way round.
p.125 line 11 ratio of boxes wrong way round - can be corrected if the words "divided......by" changed to "divided.......into".
p.137 in the first paragraph the word "later" should be "after Blichfeldt" and the 2nd. occurrence of "Rankin" should read "Rogers & Lindsey".
p.222 Kelvin's tetrakaidekahedron isn't; it's Weaire and Phelan's structure.
p.232 footnote 7 "radius 3" should read "radius 4"
p.246 1st. equation - superfluous factor of on l.h.s. and the "3" in front of r.h.s. sqrt. should be deleted. 2nd equation - quotient slash "/" should be deleted between 3 and sqrt. on r.h.s.
p.255/6 line 14 expression for discriminant (determinant) has signs reversed throughout & should be multiplied by -1. Also the same capital D is inappropriately used for both the discriminant and the first auxilliary variable and lastly the 3 squared signs are twice missing from a.b.c partway down p.256.
p.277 equation - It is obvious that the total score is not 7.99961 as Szpiro claims but needs to be multiplied by 0.0553736 to convert it to 7.99961 points, where points are defined 100 pages earlier on p.171.
Also there is some confusion over FCC & HCP where on p.23 it is claimed the two packings "are exactly the same" (they're not, but they have the same density), whilst on p.230 it is implied that FCC has greater density because Gold, Silver & Platinum use it compared to the HCP of Cadmium , Cobalt & Zinc.
Perhaps it's time for a revised paperback edition !!
However, contra the reviewer below, Szpiro does not say that Cohen showed there must be cardinals between countability and the continuum: he actually says "some other notion of cardinality must exist between...." which is acceptable if a little unclear. (His criticism of Szpiro's comment on Goedel is valid, however.)



The venerable art of squeezing cannonballs   February 24, 2003
Royce E. Buehler (Cambridge, MA USA)
17 out of 19 found this review helpful

***1/2

It was Sir Walter Raleigh who first posed the question: How do you cram the largest number of cannonballs into the hold of a ship? The question found its way to the great astronomer Kepler, who replied that you can't do better than to imitate the grocer's stacks of melons. The melons take up 74.05% of their allotted space, and there's no more efficient way to pack spheres of equal radius.

So Kepler said. But he didn't provide a valid proof. And thereby hangs a tale.

Szpiro tells the tale, in a thorough overview of the many assaults over the centuries on a problem that turned out to be much harder than it looked. When it finally fell in 1998 at the hands of Hales and Ferguson, the solution required, among other things, computer examinations of thousands of simultaneous linear inequalities in over a hundred unknowns. Just as most of the solution is hidden away from mathematicians in gigabytes of computer output (though they are free to examine the programs), most of the mathematics is necessarily hidden away from the reader here. But Szpiro does a good job of presenting, in visualizable terms, the ideas of the ideas of the partial and final proofs. He lays out the story on three levels, with the more intense geometrical discussions set off in smaller type from the main narrative, so the casual reader can skip around them, and with the detailed (but accessible with no more than algebra and a little trig) derivation of formulas in appendices. So each reader can customize the book to his own comfort level.

I'd hoped to learn about the connections to deeper questions that have made the topic of sphere packing in higher dimensions so fascinating to mathematicians - the links to coding theory, to sporadic simple groups, and to Leech's lattice. Though he touches on them briefly, Szpiro sticks fairly closely to the two and three dimensional story. That's probably a good call. After all, it is only in those dimensions that the problem has actually been solved so far, and he was certainly in no danger of running out of material.

More worrisome is a certain carelessness that crops up too often. Sometimes the geometrical descriptions are unnecessarily ambiguous. A footnote says that Von Neumann "has been reported" to have been the model for Kubrick's Dr. Strangelove, seeming to imply that Kubrick must have said so. (As a Jew, Von Neumann would have been a poor model for Strangelove's Nazism!) A figure on p. 222 labeled "Kelvin's tetrakaidekahedron" is really a packing of two different kinds of polyhedra described several pages further on. We are told that Paul Cohen showed there must be cardinals "between countability and continuum" - whereas what he really proved was that one may assume such cardinals exist, or assume that they do not, without introducing contradictions. On the same page, we're informed that "Kurt Goedel showed that arithmetic is not free from contradictions", when in reality his great theorem showed only that arithmetic cannot be *proven* to be free from contradictions. Despite the distinct pleasures the book affords, flaws like these forced me to knock my rating down by half a star.


Suboptimal packing   September 10, 2004
Testudinal Terpsichore (Dartmouth, Nova Scotia Canada)
3 out of 6 found this review helpful

I've read and enjoyed many "mathematics for the non-mathematician" books. For example, Four Colours Suffice by Wilson, and Flatterland by Stewart, both of which I recommend. I regret to say that I cannot heartily commend Kepler's Conjecture by G. Szpiro.

It has good intentions. Kepler's Conjecture is a math problem with history galore, which Szpiro recounts. It is a problem easily understood by TC Pits - (The Celebrated Person In The Street).

The problem. We want to store as many balls of uniform size (e.g. soccer balls) as possible in a large storage container. Is the most efficient packing the "obvious" one? The obvious one, not being so obvious as everyone believes, is the one used to store cannonballs adjacent to cannons - nestle each ball into the indentation provided by three mutually adjacent balls, and repeat as necessary. The result - you've seen it many times - is known as the hexagonal closest packing, or HCP.

Mathematicians idealize this problem by using a very large room (i.e. Euclidean 3-space) and identical spheres.

The problem started with wanting to efficiently store cannonballs in the hold of a ship, or on the deck of a ship. For balls on a brass monkey (yes, this is where the saying originated) there is a simple formula. For the more general problem of Euclidean 3-space, the venerable CF Gauss proved that HCP is the most efficient lattice packing in the early 1800s.

But Kepler's conjecture was only recently solved by Thomas Hales, and the mathematical community hasn't exactly accepted his proof. What did Hales do that Gauss didn't? Well, Gauss proved a result for lattice packings; Hales did not assume the spheres were constrained to have their centres at lattice points.

Szpiro does tell the story, with generous dollops of colourful history. So what's not to like? He gets too many details wrong. His writing isn't polished, let alone scintillating. His mathematical excursions are seldom proofs, and often incorrect if they are meant to be proofs. He chastizes mathematicians who claim more than they prove, so I am holding him to the standard he asserts mathematicians should meet.

I don't expect from Szpiro the details that your math teachers expected in your homework or exams, and I don't expect a mathematical treatise. But I do expect his explanations to be correct except for the details, and I believe that, as often as not, they are not correct.

Having said some rather cutting things about Szpiro's book, I'll now darn him with vague praise. If you know a teenager fascinated by sphere-packing, sure, let him/her read this book. It won't hurt them, and if they aren't mathematically precocious, they'll likely enjoy it. If they are mathematically mature enough to see the flaws, they may enjoy the book despite the flaws.

I expected more than Szpiro has to offer, and I don't think I have unreasonable expectations. I've seen too many books in the genre that meet my expectations to offer praise where I don't feel it is deserved.

On the flip side, I've seen worse. I just can't commend them to you wholeheartedly.