Sphere packing has been a hobby of mine for less than a year now, and I was drew to the subject via my interest in coding theory (error correction coding). At first glance, as I skimmed
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.