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The Geometry of Numbers

Mathematical Association of America
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Minkowski discovered that geometry can be a powerful tool for studying many questions in number theory, such as how well irrational numbers can be approximated by rationals, or which integers are sums of two squares. The interplay between geometric methods and apparently non-geometric questions is intriguing, and many of the proofs are very elegant. Much of the geometry of numbers can be explained at a relatively elementary level, but there are few books on the subject aimed at beginning students. This book is likely the most accessible treatment of this material ever written. It should play a valuable role in exposing bright high school students, or college math majors, to the geometry of numbers.

Unfortunately, this book shows substantial editing problems. For example, on page 19 it refers to another book for a proof that if m and n have g.c.d. 1, then there exist p and q such that mp - nq = 1. It might be reasonable to omit the proof, but the theorem is in fact proved on page 7! Similarly, on page 12 the book explains what a lemma is, while a "preliminary lemma" already occurs on page 7. There are also a number of typos, such as repeated confusion of "l.c.m." and "g.c.d." in the proof of the fundamental theorem of arithmetic. An experienced reader will immediately realize that when the book says "Since the least common multiple l.c.m.(c,a) = 1, we must have integers s and t such that cs + at = 1," it really means the g.c.d. However, the New Mathematical Library series is meant to be accessible to inexperienced readers. Problems of this sort occur less frequently later in the book, but they still occur frequently enough to bother me. I hope that a second edition will someday correct them.

Beyond these superficial problems, this book is a charming and generally clear account of a selection of basic results of the geometry of numbers. Some of the choices of topics are excellent. For example, Section 1.6 discusses how wide a strip between parallel lines can be without containing any lattice points, and shows exactly how the answer depends on the slope of the lines. More sophisticated books would not devote much space to this elementary topic, but it is valuable for helping beginners develop intuition about lattice points. Another useful feature of this book is that the authors do not restrict their attention to the geometry of numbers proper. There are several interesting digressions, such as a discussion of counting lattice points in disks.

Although I was impressed with some of the authors' choices, many others were not to my taste. For example, the question of which primes are sums of two squares is discussed several times, but the book does not even mention the geometry of numbers proof that every prime congruent to 1 modulo 4 is a sum of two squares. This is a puzzling omission, since that is one of the most famous applications of the geometry of numbers, and Section 8.6 does include an overview of the analogous proof for sums of four squares, which would be much easier to grasp with the simpler proof for two squares as motivation. Furthermore, at several points in this book, curiosities are placed on an equal footing with deep results, and a few theorems are taken out of context. For example, Section 2.3 proves strange-looking summation identities involving the greatest integer function, without mentioning that they are actually useful (for example, for proving quadratic reciprocity).

Overall, I was often disappointed when well-known, illuminating examples or comments were not included. Of course, the book is not meant as an encyclopedia or a treatise for experts, so some omissions are only to be expected. However, there were times when I felt the readers were not being taken seriously enough, and if the book had said more, they might have found it interesting or enlightening.

A non-technical example of what I refer to occurs on page 153:

Interest in an optimal [sphere packing] density for two dimensions is well founded. Its solution is important to many practical design issues such as determining how best to place wires in a cross-section of cable.

First, this supposed application is trivial. It is hard to imagine anyone interested in practical design issues needing a mathematician to explain how to pack disks in the plane, or a skeptical student accepting this as justification for studying two-dimensional sphere packing. More importantly, it misrepresents how wires are actually packed into cables. Many types of wires that are often bundled in cables, such as telephone or ethernet wires, occur in "twisted pairs" of strands. Twisted pairs cannot be packed as efficiently as individual wires. At Bell Labs, the question arose of how to predict theoretically how many twisted pairs could be packed into a given width of cable. This is more subtle than two-dimensional sphere packing, and much more interesting; the answer is less than 63% of the sphere packing density. For details, see "The packing problem for twisted pairs" by E. N. Gilbert (Bell System Tech. J. 58 (1979), 2143-2162). I do not mean to suggest that the book under review should necessarily mention twisted pairs, or that a detailed discussion would be appropriate or useful in a short book aimed at beginners. However, this sort of superficial coverage occurs a number of times in the book, in mathematical as well as non-mathematical contexts.

Despite the issues mentioned above, this book presents some beautiful mathematics to readers who may not be prepared to study other treatments.

Henry Cohn is a researcher in the theory group at Microsoft Research, and holds a five-year fellowship from the American Institute of Mathematics. His primary mathematical interests are number theory, combinatorics, and the theory of computation.

Date Received: 
Tuesday, June 6, 2000
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C. D. Olds, Anneli Lax, and Giuliana Davidoff
Anneli Lax New Mathematical Library 41
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Henry Cohn
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Modify Date: 
Tuesday, October 30, 2007