Introduction to Circle Packing: The Theory of Discrete Analytic Functions

by Kenneth Stephenson

• ISBN13: 9780521823562
• ISBN10: 0521823560
• Format: Hardcover
• Copyright: 2005-04-18
• Publisher: Cambridge University Press

Summary

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 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.

Author Biography

Kenneth Stephenson is Professor of Mathematics at the University of Tennessee in Knoxville.

Table of Contents

Preface	xi
I An Overview of Circle Packing	1	(32)
1 A Circle Packing Menagerie	3	(12)
1.1. First Views	3	(3)
1.2. A Guided Tour	6	(9)
2 Circle Packings in the Wild	15	(16)
2.1. Basic Bookkeeping	15	(2)
2.2. The Storyline	17	(12)
Practicum I	29	(2)
Notes I	31	(2)
II Rigidity: Maximal Packings	33	(98)
3 Preliminaries: Topology, Combinatorics, and Geometry	35	(16)
3.1. Surfaces and Their Triangulations	35	(4)
3.2. The Classical Geometries	39	(9)
3.3. Circles, Automorphisms, Curvature	48	(1)
3.4. Riemann Surfaces	49	(2)
4 Statement of the Fundamental Result	51	(3)
5 Bookkeeping and Monodromy	54	(8)
5.1. Bookkeeping	54	(4)
5.2. The Monodromy Theorem	58	(4)
6 Proof for Combinatorial Closed Discs	62	(10)
6.1. A Mind Game	62	(1)
6.2. Monotonicities and Bounds	63	(3)
6.3. The Hyperbolic Proof	66	(6)
7 Proof for Combinatorial Spheres	72	(1)
8 Proof for Combinatorial Open Discs	73	(43)
8.1. Existence and Univalence	73	(8)
8.2. Completeness	81	(14)
8.3. Uniqueness	95	(17)
8.4. Proof of Lemma 8.7	112	(4)
9 Proof for Combinatorial Surfaces	116	(10)
9.1. A Discrete Torus	116	(2)
9.2. A Classical Torus	118	(2)
9.3. The Proof	120	(5)
9.4. One Final Example	125	(1)
Practicum II	126	(3)
Notes II	129	(2)
III Flexibility: Analytic Functions	131	(116)
10 The Intuitive Landscape	133	(6)
10.1. Think Geometry!	134	(1)
10.2. Fundamentals	135	(1)
10.3. Discrete Analytic Functions	136	(1)
10.4. Discrete Conformal Structures	137	(2)
11 Discrete Analytic Functions	139	(14)
11.1. Formal Definitions	139	(1)
11.2. Standard Mapping Properties	140	(1)
11.3. Branching	141	(2)
11.4. Boundary Value Problems	143	(6)
11.5. The Maximum Principle	149	(1)
11.6. Convergence	150	(3)
12 Construction Tools	153	(7)
12.1. Cutting Out	153	(1)
12.2. Slitting	153	(1)
12.3. Combinatorial Pasting	154	(2)
12.4. Doubling	156	(1)
12.5. Combinatorial Welding	156	(1)
12.6. Geometric Pasting	157	(1)
12.7. Schwarz Doubling	158	(1)
12.8. Refinement, etc.	158	(2)
13 Discrete Analytic Functions on the Disc	160	(21)
13.1. Schwarz and Distortion	160	(1)
13.2. Discrete Univalent Functions	161	(2)
13.3. Discrete Finite Blaschke Products	163	(4)
13.4. Discrete Disc Algebra Functions	167	(2)
13.5. Discrete Function Theory	169	(4)
13.6. Infinite Combinatorics	173	(3)
13.7. Experimental Challenges	176	(5)
14 Discrete Entire Functions	181	(14)
14.1. Liouville and (Barely) Beyond	181	(1)
14.2. Discrete Polynomials	182	(6)
14.3. Discrete Exponentials	188	(1)
14.4. The Discrete Error Function	189	(3)
14.5. The Discrete Sine Function	192	(2)
14.6. Further Examples?	194	(1)
15 Discrete Rational Functions	195	(6)
15.1. Basic Theory	195	(1)
15.2. A Fortuitous Example	196	(1)
15.3. Special Branched Situations	197	(2)
15.4. Range Constructions	199	(2)
16 Discrete Analytic Functions on Riemann Surfaces	201	(16)
16.1. Ground Rules	201	(1)
16.2. Discrete Branched Coverings	202	(7)
16.3. Discrete Belyi Functions	209	(2)
16.4. Further Examples	211	(2)
16.5. Alternate Notions of "Circle Packing"	213	(4)
17 Discrete Conformal Structure	217	(15)
17.1. A Key Example	217	(2)
17.2. Discrete Formulation	219	(1)
17.3. Extremal Length	220	(5)
17.4. The Type Problem	225	(4)
17.5. Packable Surfaces	229	(3)
18 Random Walks on Circle Packings	232	(11)
18.1. Random Walks and Electrical Networks	232	(1)
18.2. Technical Background	233	(1)
18.3. Random Walks and "Type"	234	(3)
18.4. Completion of a Proof	237	(3)
18.5. Geometric Walkers	240	(1)
18.6. New Intuition?	241	(2)
Practicum III	243	(2)
Notes III	245	(2)
IV Resolution: Approximation	247	(62)
19 Thurston's Conjecture	249	(8)
19.1. Geometric Ingredients	250	(1)
19.2. Proof of the Rodin-Sullivan Theorem	251	(3)
19.3. Example	254	(3)
20 Extending the Rodin-Sullivan Theorem	257	(11)
20.1. Outline	258	(1)
20.2. Proving the Lemmas	259	(4)
20.3. The Ratio Functions	263	(1)
20.4. Companion Notions	264	(2)
20.5. The He-Schramm Theorem	266	(2)
21 Approximation of Analytic Functions	268	(7)
21.1. Approximating Blaschke Products	268	(3)
21.2. Approximating Polynomials	271	(1)
21.3. Further Examples	272	(3)
22 Approximation of Conformal Structures	275	(11)
22.1. Polyhedral Surfaces	275	(1)
22.2. The Ten-Triangle-Toy	276	(1)
22.3. Convergence	277	(4)
22.4. More General in Situ Packings	281	(5)
23 Applications	286	(17)
23.1. "Dessins d'Enfants" of Grothendieck	286	(4)
23.2. Conformal Tilings	290	(5)
23.3. Conformal Welding	295	(4)
23.4. Brain Flattening	299	(3)
23.5. Summary	302	(1)
Practicum IV	303	(3)
Notes IV	306	(3)
Appendix A. Primer on Classical Complex Analysis	309	(9)
Appendix B. The Ring Lemma	318	(4)
Appendix C. Doyle Spirals	322	(5)
Appendix D. The Brooks Parameter	327	(4)
Appendix E. Inversive Distance Packings	331	(4)
Appendix F. Graph Embedding	335	(4)
Appendix G. Square Grid Packings	339	(4)
Appendix H. Schwarz and Buckyballs	343	(3)
Appendix I. Circle Pack	346	(1)
Bibliography	347	(7)
Index	354