Introduction to circle packing : the theory of discrete analytic functions / Kenneth Stephenson

Bibliogr. p. 347-353. Index

A circle packing is a configuration of circles having a special pattern of tangencies. In 1985, W. Thurston linked this topic to analytic functions and conjectured how discrete analytic functions built with circle packings should approximate the Riemann uniformization mapping of a simply connected bounded open set in the plane.
This conjecture and the (positive) answer given by B. Rodin and D. Sullivan in 1987 were the starting point for a great amount of research in the past 20 years.
This book is an overview of this topic. It lays out the study of circle packings, from first definitions to the latest theory, computations and applications. The experimental and visual character of circle packings is exploited to carry the reader from the very beginnings to links with complex analysis and Riemann surfaces. The questions of existence, uniqueness, convergence are addressed, widely using manipulations and displays.
Let us briefly outline the way this book is structured. Part I is devoted to an informal and largely visual tour of the topic. Part II contains a complete and essentially self-contained proof of the fundamental result of existence and uniqueness of a circle packing with prescribed combinatorics. Removing topological conditions in the latter result gives a wealth of flexibility which is studied in Part III. Part IV deals with approximation of classical analytic functions by their discrete counterparts.
This text is both mathematically rigorous and accessible to the novice mathematician, enabling him to penetrate deeply into the subject. The reading is pleasant, the style is lively and the enthusiasm of the author is quite communicative (MSN)