The Geometry of Numbers (New Mathematical Library)

Rating:
Summary: Excellent guide to the mathematics of lattice points
Review: The lattice points of the plane, those where both coordinates are integers, is the topic of this book. In it, you learn how to hit them with lines and surround them with circles, triangles, polygons and other shapes. The title originated in the work of Hermann Minkowski, who used the inherent geometric properties of the plane to prove theorems concerning the number of lattice points inside shapes.
I found the book to be fascinating, in that shifting to the geometric approach makes so many of the hard problems much easier. One of the most difficult things to teach students who are just beginning to be exposed to proofs is that there are many ways to verify results. They just seem to get stuck on one approach and cannot see that there may be another, much simpler strategy. The examples in this book demonstrate cases where a problem appears hard, but in fact is fairly easy if geometric representations are used. This would make it very valuable as a tool to help build confidence and breadth of knowledge in students.
The book is split into two parts, where the first deals with lattice points on and in geometric figures and the second with Minkowski's fundamental theorem and its' consequences. Minkowski's theorem uses the geometric properties of points in the plane to solve Diophantine inequalities. This was new when first done, and provides a powerful and sometimes much simpler tool in the effort to solve such problems. The exposition is excellent, there are problem sets at the end of most sections and hints for solutions to almost all of them are in an appendix. These features make it ideal for a short course that combines geometry and basic number theory.

Published in Journal of Recreational Mathematics, reprinted with permission.