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Cambridge University Press 978-0-521-82356-2 - Introduction to Circle Packing: The Theory of Discrete Analytic Functions Kenneth Stephenson Index More information

Index

algebraic curve, 287 brain flattening, 299, 301 parabolic, 52 algebraic number field, 287 branch point, 56, 142, 197, 344 spherical, 52 analytic function classical, 135, 311 universal covering, 122 classical, 133, 309 fractional, 208 composition, 173 discrete, 134, 139 multiplicity, 142 conductance, see random walk Andreev-Thurston Theorem, 332 order, 56, 141 cone angle, 275, 281, 304 angle sum, θR, 56 branch structure, 142 cone point, 275 boundary, 149, 161 infinite, 175 conformal, 137, 219, 281, 309 formulas, 57 polynomial, 197 automorphism, 42 angular derivative, 171 branch value, 142, 165 rectangle, 220, 315 angular resolution, 335 Brooks parameter, 328 ring domain, 315 Apollonian packing, 214 Brooks quadrilateral, 231, 327 torus, 120, 215 arclength element, ds, 41 Brooks spiral, 324, 330 conformal mapping area element, ds2, 41 Brownian motion, 234, 264, 266 classical, 249, 309 argument principle, 68, 141, 310 discrete, 137, 217, 249 argument, arg (z), 40 C-bound, 185, 259, 317 conformal structure, 50, 51, 276 asymptotic value, 188, 191 cadre, 106, 113, 171 classical, 275 asymptotically parabolic, 152, 257, Caratheodory´ Kernel Theorem, 253, discrete, 137, 219, 283 262 270, 315 conformal tiling, 290 atlas, 36, 50, 276 carrier, 58, 251 conformally equivalent, 50, 310 augmented boundary, 104, 106 Cauchy-Riemann equations, 312, convergence augmented carrier, 104 340 combinatorial, 150, 262 automorphism group, Aut( · ), 42, center function, 235 quasiconformal, 254, 279, 316 48, 140, 207, 231, 280, 289, chain (of faces), 38,59 covering theory, 202, 279 313 null, 60 classical, 120 circle packing, 51 discrete, 122 barycentric coordinates, 139, 316 affine, 215 Coxeter spiral, 324, 330 mapping, 316 rectangular, 162 cross ratio, 339 subdivision, 159, 276 alternate notions, 214 curvature, 35, 42, 48, 243 Beltrami equation, 314 circle packing embedding, 335 curvature flow, 305 Bely˘ı pair circular, see reasoning classical, 287 clone, 153 deBranges Theorem, 176 discrete, 210, 288 combinatorial defining equation, 287 Bely˘ı ’s Theorem, 287 closed disc, 54,62 degree, 38, 73, 227 Benjamini-Schramm Theorem, 175 flower, 38,56 bounded, 174, 181, 257, 279 Bieberbach Conj., see deBranges open disc, 54,73 Dehn twist, 156 Theorem rectangle, 162, 218, 223 Descartes Circle Theorem, 320 Blaschke product, 268 sphere, 54,72 dessin, 286, 286, 293, 294 classical, 163 surface, 116 development, 59, 81, 207, 209 discrete, 164, 174 compatibility, 103 dichotomy, 74, 128, 181, 225, boundary edge, 37 complex, 51,54 262 boundary value problem, 143, 148, compact , 54, 201 Dieudonne-Schwarz´ Lemma, 177 167 hyperbolic, 52 dilatation, see quasiconformal boundary vertex, 37 labeled, 55 disc algebra, 167

354

© Cambridge University Press www.cambridge.org Cambridge University Press 978-0-521-82356-2 - Introduction to Circle Packing: The Theory of Discrete Analytic Functions Kenneth Stephenson Index More information

Index 355

Distortion Lemma, Discrete, 161 ideal rectangle, 260, 302 Branched, 170 boundary, 46 ring domain, 225 domain construction, 160, 167, 196 triangle, 47, 213 torus, 229 doubling, 156, 162 vertex, 46, 154 modular function, 212, 215 Doyle spiral, 25, 94, 188, 273, 322, image surface, 165, 168, 200, 273, modular surface, 212 330 311 moduli space, 229, 230 Dubejko’s Theorem, 235 in situ packing, 218, 277, 281 Monodromy Theorem, 58, 142, 144 integrable systems, discrete, 339 monotonicity element, 104 interior edge, 37 angle, 63,95 entire function, discrete, 181 interior vertex, 37 angle sum, 65, 243 equilateral surface, 276, 283, 287 inversive distance packing, 332 area, 65 error function, erf(·), 189, 340 isometry, 42 combinatoric, 70 essentially unique, 52 isotopy, 99 global, 148 euclidean plane, C, 39,42 harmonic measure, 87, 112, 114 Euler characteristic, 39, 143 j-function, 212, 245 label, 70,74 exit probability, 242, 264 Jordan curve, 98, 310 labeled complex, 81 exponential, discrete, 188, 272 Jordan domain, 85, 220, 310 modulus, 225 extension of domain, 114 segment, 82, 83 extremal length, 220, 226 Konigsfunction,¨ 293 triangle, 63 Kakutani Theorem, 234 Farey triangulation, 214 Klein surface, 9, 245 Nehari’s condition, 180 Fatou Theorem, 175 Koebe function, 176 normal family, 151, 253, 270, 315 Fibonacci sequence, 321, 326, 335 Koebe’s 1/4-Theorem, 177 null homotopic, 39 finite topological type, 230 Koebe-Andreev-Thurston Theorem, fixed-point index, η, 98, 113, 171 72, 200, 267 oval of Descartes, 325 fundamental domain, 120, 127, 203, overlap packing, 284, 332 212 label, 55 fundamental group, 36, 39, 122 branched, 56, 141 packable surface, 229, 231 maximal, 70,80 packing algorithm, 29, 195, 243, gaussian curvature, 42 packing, 56 303 general position, 101 Laplace equation, 312 packing condition, 56, 133 genus, 36 Laplace operator, 42 packing label, 56 geodesic, 41 lattice group, 118 packing layout, 29, 58, 126, 306 geometric convergence, 79, 151, Length-Area Lemma, 250, 269 parabolic complex, 52,74 189 light interior mapping, 140, 310 parabolic surface, 231 ghost circle, 61, 117, 125, 127 Lindelof¨ Theorem, 175 pasting golden ratio ϕ, 321, 325, 329, 330 Liouville Theorem, 312 combinatorial, 154 Green’s function, 241 Discrete, 181 geometric, 157, 165, 205, 209, quasiconformal, 185, 314 282 harmonic function local modification, 38,60 Perron method, 66, 144, 148, 238 classical, 312 locally univalent, 94, 141, 311 phyllotaxis, 325 discrete, 234, 236 logarithmic branching, 188 Picard surface, 28, 211 harmonic measure, 112, 242, 264 Lowner’s¨ Lemma, Discrete, 171 Picard Theorem classical, 84 Discrete, 182 discrete, 86 Mobius¨ transformation, 44 classical, 312 He-Schramm Theorem, 266 maximal packing, 52, 70, 124, 333 polynomial, discrete, 182, 271 Hexagonal Packing Lemma, 251, branched, 164, 169, 174, 187 proper map, 68, 163, 182 259 Maximum Principle, 150 hexagonal refinement, 158, 277, meromorphic function, discrete, quasiconformal, 253, 314 279, 284, 298 181, 201, 208, 211 dilatation, 255, 279, 313, 317 holomorphic differential, 208 metric quasiregular, 314 holonomy, 207 dR, 55,81 homotopy, discrete, 38 ambient, 140, 213, 225 radial limit, 170 horocycle, 46, 213 intrinsic, 50, 202 random walk, 232 hyperbolic complex, 52,74 path, 55 conductance, 233, 237 hyperbolic contraction, 161, 167, minimal surface, discrete, 342 simple, 229, 232 312 Mod (modulus) tailored, 235, 266 hyperbolic density, 45 classical, 315 range construction, 160, 165, 199, hyperbolic cusp, 213 conformal, 220 205 hyperbolic plane, D, 39,45 graph, 222 ratio function f #, 139, 170, 179, hyperelliptic Riemann surface, 156 packing, 221, 225 222, 263

356 Index

rational function, discrete, 195, Schwarz-Pick Lemma, Discrete, triangulation, 37, 51, 327 345 161 inﬁnitesimal, 219 rational iteration, 293 classical, 312 tripartite complex, 210, 212, 286 reasoning, see circular Schwarzian derivative, 180 type, 75, 279 repack, 20,29 simplicial complex, 37, 122 type problem, 225, 229, 234, 293 Riemann Mapping Theorem, 249, simply connected, 36 267, 292, 312 sine, discrete, 194, 273 uniform neighbor model, 244 Discrete, 160 sphere packing, 31, 216 Uniformization Theorem, Discrete, Inverse, 173 spherical complex, 52,72 51, 160, 201 Riemann sphere, P, 39,43 square grid packing, 339 unit circle, T, 41 Riemann surface, 50, 201, 219, 275 squared rectangle, 224 univalence criterion, 180 discrete, 219 Standing Assumptions, 136, 202 univalent function Riemann-Hurwitz, 195, 204, 344 stereographic projection, 40 classical, 176 Ring Lemma, 73, 250, 318 Sto¨ılow’s Theorem, 141, 164, 182, discrete, 161 in the Large, 174, 183, 261, 270 196, 273, 288 universal cover, see covering theory Rodin-Sullivan Theorem, 250, 257, subdivision rule, 292 266 subordination, 173 van Kampen’s Theorem, 156, 165, subpacking, 66 200 schlicht, see univalent super-step method, 303 Schwarz Lemma, Discrete, 161 superharmonic function, 233 welding Branched, 169 superpacking, 144, 170 classical, 296 Schwarz reﬂection, 158, 192, 344 surface, 35 combinatorial, 156, 285, 297 Schwarz triangle, 344 Whitehead move, 230, 337 Schwarz-Christoffel method, 168, Teichmuller¨ space, 229, 280, 288 Williams Theorem, 298 224, 305 transition probability, 232, 241 winding number, w, 97, 310