INTRODUCTION TO CIRCLE PACKING: A REVIEW

This is a review of Introduction to Circle Packing: The Theory of Discrete Analytic Functions, by Kenneth Stephenson, Cambridge University Press, Cambridge UK, 2005, pp. i-xii, 1–356, £42, ISBN-13 978-0-521-82356-2. 1. The Context: A Personal Reminiscence Two important stories in the recent history of mathematics are those of the geometrization of topology and the discretization of geometry. Having come of age during the unfolding of these stories as both observer and practitioner, this reviewer does not hold the detachment of the historian and, perhaps, can be forgiven the personal accounting that follows, along with its idiosyncratic telling. The first story begins at a time when the mathematical world is entrapped by abstraction. Bourbaki reigns and generalization is the cry of the day. Coxeter is a curious doddering uncle, at best tolerated, at worst vilified as a practitioner of the unsophisticated mathematics of the nineteenth century. 1.1. The geometrization of topology. It is 1978 and I have just begun my graduate studies in mathematics. There is some excitement in the air over ideas of Bill Thurston that purport to offer a way to resolve the Poincaré conjecture by using nineteenth century mathematics—specifically, the noneuclidean geometry of Lobachevski and Bolyai—to classify all 3-manifolds. These ideas finally appear in a set of notes from Princeton a couple of years later, and the notes are both fascinating and infuriating—theorems are left unstated and often unproved, chapters are missing never to be seen, the particular dominates—but the notes are bulging with beautiful and exciting ideas, often with but sketches of intricate arguments to support the landscape that Thurston sees as he surveys the topology of 3-manifolds. Thurston’s vision is a throwback to the previous century, having much in common with the highly geometric, highly particular landscape that inspired Felix Klein and Max Dehn. These geometers walked around and within Riemann surfaces, one of the hot topics of the day, knew them intimately, and understood them in their particularity, not from the rarified heights that captured the mathematical world in general, and topology in particular, in the period from the 1930’s until the 1970’s. The influence of Thurston’s Princeton notes on the development of topology over the next 30 years would be pervasive, not only in its mathematical content, but AMS SUBJECT CLASSIFICATION: 52C26