Introduction to Circle Packing
The Theory of Discrete Analytic Functions

This book, first published in 2005, introduces a new mathematical topic known as 'circle packing', taking the reader from first definitions to late-breaking results.

Kenneth Stephenson (Author)

9780521823562, Cambridge University Press

Hardback, published 18 April 2005

370 pages, 190 b/w illus. 10 colour illus.
26 x 18.5 x 2.5 cm, 0.929 kg

'… 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

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.

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.

Subject Areas: Complex analysis, complex variables [PBKD]