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Introduction to Circle Packing: The Theory of Discrete Analytic Functions

Cambridge University Press
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Circle packing. Sounds like a neat topic, with lots of pretty pictures. After opening Kenneth Stephenson's book, I can still say it sounds like a neat topic, and indeed, there are lots of pretty pictures. But this book is not light-weight reading; Stephenson has embarked on an ambitious journey.

Part I provides an overview of circle packing, a brief 30 pages filled with pictures and slogans regarding the subject. It is here that I learn that circle packing involves both combinatorics and geometry, both rigidity and flexibility. I learn that packing is "dynamic business" with an "undercurrent of rigidity... reminiscent of rigidity in the classical setting of conformal/analytic mappings." Part II concerns maximal packings. Maximal packings showcase the rigidity of the subject, as I'm told, and along the way I'm reminded about many classical analysis results. Part III showcases the flexibility of the subject. Part IV takes up the theme of approximation, beginning with a conjecture of William Thurston's. The idea (now fact) is that discrete analytic functions — maps between circle packings preserving the structure of tangents and orientation — can be used to approximate classical conformal maps.

Stephenson is quite adept at reminding his readers of his philosophy throughout the text. These slogans helped keep me on track, and helped me keep sight of the main issues and accomplishments. Do not be fooled, however. This book is rigorous. The author does declare this in the preface, along with saying that, formally, there are no prerequisites needed for picking up this text. Maybe so; but Stephenson is more confident about this than I am. He claims that non-mathematicians can profit from reading this book, and that at least part of the book is suitable for a course geared to advanced undergraduates and graduate students. I think non-mathematicians will be nothing other than frustrated with this book, despite the wealth of philosophical comments. As for students, be advised that there are no exercises in the text. To be sure, there is a Practicum at the end of each of the four main parts of the book. These are some comments on things to try with the author's computer software called "Circlepack", available for free from his website.

And what is a circle packing? I'm told in the preface that it is not a sphere packing (darn! I thought I was going to learn something about sphere packings by reviewing this). After seeing lots of pictures of circle packings, I learn that a circle packing for a simplicial 2-complex K (also understood as an oriented surface) is a collection of circles, one for each vertex of the complex K, such that whenever two vertices determine an edge of K, the associated circles are tangent and such that whenever three vertices determine a postively oriented face of K, the associated circles determine a positively oriented triple.

Michele Intermont is an associate professor of mathematics at Kalamazoo College in Kalamazoo, MI.

Date Received: 
Wednesday, June 1, 2005
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Kenneth Stephenson
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Michele Intermont
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.
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Thursday, January 5, 2006