Kepler's Conjecture: How Some of the Greatest Minds in History Helped Solve One of the Oldest Math Problems in the World Hardcover – 25 Mar 2003
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"...gives an entertaining and readable account of the history of the problem and the attempts to solve it..." (Nature, 11 September 2003)
"...an invigorating affirmation of math′s endless allure, and a neat lesson in why it pays to take nothing for granted..." (New Scientist, 13 September 2003)
"...the perfect balance of tone between mathematical explanation and historical exposition..." (M2 Best Books, 18 August 2003)
"...well–crafted piece of popular science writing..." (Mathematical Intelligencer, Vol 26: No. 1)
“…a nicely written approach…explained and presented in a conveniently readable manner.” (Zentralblatt MATH, March 2007)
From the Inside Flap
For 400 years, some of the best and brightest minds set out to prove Kepler′s conjecture–like Fermat′s celebrated last theorem, one of the oldest unproven mathematical conjectures–which raised one perplexing question: What is the best way to pack balls as densely as possible? Kepler′s Conjecture traces the fascinating history and progression of this geometric puzzle, illustrating how thoroughly this one simple question stymied the mathematical world.
Sometime toward the end of the 1590s, English nobleman and seafarer Sir Walter Raleigh set this great mathematical investigation in motion. While stocking his ship for yet another expedition, Raleigh asked his assistant, Thomas Harriot, to develop a formula that would allow him to know how many cannonballs were in a given stack simply by looking at the shape of the pile. Harriot solved the problem and took it one step further by attempting to discover how to maximize the number of cannonballs that would fit in the hold of a ship. And thus a problem was born.
After contemplating the question for a while, Harriot turned to one of the foremost mathematicians, physicists, and astronomers of the time, Johannes Kepler. Kepler did not reflect long, and came to the conclusion that the densest way to pack three–dimensional spheres was to stack them in the same manner that market vendors stack their apples, oranges, and melons.
Although this was fine for fruit vendors, until the conjecture could be proven, the mathematical world could not accept it. The first and only popular account of one of the greatest math problems of all time, Kepler′s Conjecture examines the attempts of many mathematical geniuses to prove this problem once and for all–from Danish astronomer Tycho Brahe to math greats Sir Isaac Newton and Carl Friedrich Gauss, from modern titans David Hilbert and Buckminster Fuller to Thomas Hales of the University of Michigan, who in 1998 submitted what seems to be the definitive proof.
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Top Customer Reviews
First, some of Szpiro's ridiculous statements (in quotation marks):
Page 118, [concerning Hilbert's first problem], "It is not either/or. One answer and its opposite are true".
Also, "Is the number of points in the smaller section 'countable', that is, somewhat less than a continuum?"
Page 119, "Both a statement and its opposite may be true simultaneously".
Page 33, "A dual problem is, in some sense, the opposite of the original problem".
Page 44, "He [Lagrange] is considered the foremost mathematician of the eighteenth century". No mention of Euler.
Page 122, "Some readers may have expected a convex tile, while Heesch's shape is concave".
Page 166, "Eight balls placed at the corners of a regular octahedron" [on p170 six balls suffice]. This is probably a very rare instance of a typo.
Page 192, "For example, with Carl Louis Ferdinand von Lindemann's proof in 1882 that pi is a transcendental number, it was established that this number has infinitely many digits". [What about the square root of 2? Or even 1/7?]
Page 77 is so full of garbage that my remarks must suffice. Szpiro thinks that Newton single-handedly invented calculus, in particular the differential calculus (no mention of Barrow, Fermat, and others who were differentiating before Newton was born); and most tellingly there is no mention of the magnificent 'Fundamental Theorem', which Newton (and Leibniz) did discover.Read more ›
Most Helpful Customer Reviews on Amazon.com (beta)
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.
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 ½ 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 Gödel is valid, however.)
the book at the bookstore, it seemed deceptively simple. It didn't seem to have much technical substance. I bought the book anyway, as it's really one of the only books on the subject. Thankfully, my initial reaction was wrong. The book contains alot more detail than appears. The book doesn't contain alot of math equations, in fact, it tends to gloss over even the simple theory a little too much. I found this a tad dissapointing. However, the author throws enough scraps of information to you, that if your really interested, you can pursue the matter in further detail.
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.