Introduction to Circle Packing: The Theory of Discrete Analytic Functions

The topic of 'circle packing' was born of the computer age but takes its inspiration and themes from core areas of classical mathematics. A circle packing is a configuration of circles having a specified pattern of tangencies, as introduced by William Thurston in 1985. This book, first published in 2005, lays out their study, from first definitions to latest theory, computations, and applications. The topic can be enjoyed for the visual appeal of the packing images - over 200 in the book - and the elegance of circle geometry, for the clean line of theory, for the deep connections to classical topics, or for the emerging applications. Circle packing has an experimental and visual character which is unique in pure mathematics, and the book exploits that to carry the reader from the very beginnings to links with complex analysis and Riemann surfaces. There are intriguing, often very accessible, open problems throughout the book and seven Appendices on subtopics of independent interest. This book lays the foundation for a topic with wide appeal and a bright future.

Table of Contents:
Part I. An Overview of Circle Packing: 1. A circle packing menagerie; 2. Circle packings in the wild; Part II. Rigidity: Maximal Packings: 3. Preliminaries: topology, combinatorics, and geometry; 4. Statement of the fundamental result; 5. Bookkeeping and monodromy; 6. Proof for combinatorial closed discs; 7. Proof for combinatorial spheres; 8. Proof for combinatorial open discs; 9. Proof for combinatorial surfaces; Part III. Flexibility: Analytic Functions: 10. The intuitive landscape; 11. Discrete analytic functions; 12. Construction tools; 13. Discrete analytic functions on the disc; 14. Discrete entire functions; 15. Discrete rational functions; 16. Discrete analytic functions on Riemann surfaces; 17. Discrete conformal structure; 18. Random walks on circle packings; Part IV: 19. Thurston's Conjecture; 20. Extending the Rodin/Sullivan theorem; 21. Approximation of analytic functions; 22. Approximation of conformal structures; 23. Applications; Appendix A. Primer on classical complex analysis; Appendix B. The ring lemma; Appendix C. Doyle spirals; Appendix D. The brooks parameter; Appendix E. Schwarz and buckyballs; Appendix F. Inversive distance packings; Appendix G. Graph embedding; Appendix H. Square grid packings; Appendix I. Experimenting with circle packings.

About the Author :
Kenneth Stephenson is Professor of Mathematics at the University of Tennessee in Knoxville, where he has established an active research program in complex function theory. He has had visiting positions at the University of Hawaii and Florida State University, and sabbatical appointments at the Open University and the University of Cambridge. Over the last fifteen years he has centered his research on circle packing. In this book he formulates circle packing as a discrete incarnation of classical analytic function theory.

Review :
'This book lays out the study of circle packing, from first definitions to the latest theory, computations, and applications. … The topic can be enjoyed for the visual appeal of the packing images - over 200 in the book - and the elegance of circle geometry, for the clean line of theory, for the deep connections to classical topics, or for the emerging applications. Circle packing has an experimental and visual character that is unique in pure mathematics, and the book exploits that character to carry the reader from the very beginnings to links with complex analysis and Riemann surfaces. … The author uses both discrete functions and discrete conformal structures in several settings of active research interest, ranging from number theory to conformal tilings to (of all things!) human 'brain mapping'. These are all settings involving classically defined structures for which no numerical approximation methods were available until circle packing arrived on the scene. There are intriguing, often very accessible, open problems throughout the book and nine Appendices on subtopics of independent interest: Primer on classical complex analysis, The ring lemma, Doyle spirals, The Brooks parameter, Inversive distance packings, Graph embedding, Square grid packings, Schwarz and buckyballs, Circle packings.' Zentralblatt MATH 'this beautifully produced book is an inviting introduction to an emerging area of mathematics that hs both an immediate visual appeal, with plenty of opportunities for computer-driven experimentation, and a rapidly developing clean line of theory … Stephenson is one of the leading pioneers in this exciting development and his stimulating book, written in an enthusiastic, almost conversational, style, will surely attract new workers into this new field. For, as he aptly remarks in the Preface, 'Circle packing has opened a discrete world that both parallels and approximates the classical world of conformal geometry - a 'quantum' classical analysis that is classical in the limit.' The Mathematical Gazette '… a splendid work of academic art. … The overall effect is that of a stunning menagerie of images complementing beautifully scripted text. … Ken Stephenson has produced in this textbook an effective and enjoyable tour of both the basic theory of circle packing and its use in deriving an intricate theory of discrete analytic functions. … I expect Introduction to Circle Packing: the Theory of Discrete Analytic Functions to be the source for student and researcher for many years to come.' Bulletin of the American Mathematical Society