DOCSLIB.ORG

Apollonian Circle Packings: Dynamics and Number Theory

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Apollonian Circle Packings: Dynamics and Number Theory APOLLONIAN CIRCLE PACKINGS: DYNAMICS AND NUMBER THEORY HEE OH Abstract. We give an overview of various counting problems for Apol- lonian circle packings, which turn out to be related to problems in dy- namics and number theory for thin groups. This survey article is an expanded version of my lecture notes prepared for the 13th Takagi lec- tures given at RIMS, Kyoto in the fall of 2013. Contents 1. Counting problems for Apollonian circle packings 1 2. Hidden symmetries and Orbital counting problem 7 3. Counting, Mixing, and the Bowen-Margulis-Sullivan measure 9 4. Integral Apollonian circle packings 15 5. Expanders and Sieve 19 References 25 1. Counting problems for Apollonian circle packings An Apollonian circle packing is one of the most of beautiful circle packings whose construction can be described in a very simple manner based on an old theorem of Apollonius of Perga: Theorem 1.1 (Apollonius of Perga, 262-190 BC). Given 3 mutually tangent circles in the plane, there exist exactly two circles tangent to all three. Figure 1. Pictorial proof of the Apollonius theorem 1 2 HEE OH Figure 2. Possible configurations of four mutually tangent circles Proof. We give a modern proof, using the linear fractional transformations ^ of PSL2(C) on the extended complex plane C = C [ f1g, known as M¨obius transformations: a b az + b (z) = ; c d cz + d where a; b; c; d 2 C with ad − bc = 1 and z 2 C [ f1g. As is well known, a M¨obiustransformation maps circles in C^ to circles in C^, preserving angles between them. (In the whole article, a line in C is treated as a circle in C^). In particular, it maps tangent circles to tangent circles. For given three mutually tangent circles C1;C2;C3 in the plane, denote by p the tangent point between C1 and C2, and let g 2 PSL2(C) be an element which maps p to 1. Then g maps C1 and C2 to two circles tangent at 1, that is, two parallel lines, and g(C3) is a circle tangent to these parallel lines. In the configuration of g(C1), g(C2), g(C3) (see Fig. 1), it is clear that there 0 are precisely two circles, say, D and D tangent to all three g(Ci), 1 ≤ i ≤ 3. Using g−1, which is again a M¨obiustransformation, it follows that g−1(D) −1 0 and g (D ) are precisely those two circles tangent to C1,C2,C3. In order to construct an Apollonian circle packing, we begin with four mutually tangent circles in the plane (see Figure 2 for possible configura- tions) and keep adding newer circles tangent to three of the previous circles provided by Theorem 1.1. Continuing this process indefinitely, we arrive at an infinite circle packing, called an Apollonian circle packing. APOLLONIAN PACKINGS 3 38 27 27 86 131 95 54 54 110 18 18 167 155 182 63 63 138 74 134 74 195 231 170 11 30 11 11 266 30 59 95 95 251 191 59 194 98 183 98 98 294 314 215 267 110 110 39 39 299 242 254 83 83 143 107 66 66 194 215 234 35 179 290 35 90 314 279 90 111 111 111350 111 6 6 6 6 338 6 227 171 455 174 59 423 59 174 506 347 162 162486 315 255 131 354 354 131 179 179471 378 522 299 371 150 47 98 294 194 167 98 47 62 507 62 363 23 119 23 23 362 119 290 147 147 170 438 170447 14 14 375 330 330 14 75 75 122 122231 231 443 186 150 150302 50 50 383 275 251 182 87 87 87 278 87 26 134 218 26 131 2 2 2 2 143 42 42 62 167 114 371 114 374 546 219 219 71 447 71 654 554 222 659 222 531 455 279 738 279 846 623 210 210 347 911 38 38 107 662 107 38 318 842 318 635 546 287 867 287 323 642 323927 762 338 858 110 203 599 203 110 647 699 683 218 638 218 263 498 263 102 798 102 362 407 803 863 302 198 491 198 302 602 326 590 99 143 99 63 326 63 590 35 35 35 602 302 198 491 198 302 863 407 803 362 798 263 498 263 218 638 218 15 15 15 15 699 15 683 102 102 647 203 599 203 338 858 323 323 642 762 110 927 110 287 867 287 546 635 318 842 318 347 662 210 210 911 623 107 107846 279 738 279 455 38 38 531 38 222 222 659 554 654 219 447 219 71 546 71 374 371 114 114 167 3 3 3 3 3 3 3 62 42 143 42 131 134 218 87 87 87 278 182 87 26 251 26 275 383 150 150302 50 50 443 186 231 231 330 330 375 75 122 170 438 170447 122 75 147 147 362 290 363 507 194 167 98 294 98 150 119 371119 47 62 62 299 47 522 179 179471 378 23 131 23 23 354 354 131 255 315 162 162486 506 347 14 14 423 14 174 174 59 59 455 338 227 171 350 314 279 90 111 111 111 111 90 290 35 179 35 234 215 66 66 194 107 143 83 83 254 6 6 6 6 242 6 39 39 299 267 110 314 110215 98 98 294 183 194 251 191 98 59 95 95 59 266 30 30 170 231 74 134 74 195 63 63 138 2 2 2 11 11 2 155 182 11 167 110 54 95 54 18 18 131 27 27 86 38 Figure 3. First few generations 169 144 144 169 196 169 225 324 289 169 196 361 441 169324 289144361 256 256 361144289 324169441 361 196 169 289 324 225 169 196 169 144 144 169 516 516 81 121 100 121 81 81 121 100 121 81 121 81 121372 100 121388 436 81121 12181 436 388121 100 121372 81 121 81 121 100 121 81 81 121 100 121 81 316 316 268 268 64 64 64 64 244 64561 64561 244 64 64 64 64 49 49 49 49 49 49 49 49 49 49 513 49 49 513 49 49 49 49 49 49 49 49 49 49 196 196 196 456 456 196 196 196 417 417 36 156 156 36 36 396 396 156393 393 156 396 396 36 36 156 156 36 148 148 148 148 148 148 321 321 297 297 25 25 25 25 25 25 25 25 25 25 12425249 25 25 24925124 25 25 25 25 25 25 25 25 25 25 424 424 216 216 216625 625 216 216 216 600 384 384 600 201 201 204 577 201 201 577 204 201 201 84 84 84 537 537 84 84 84 76 76 76 76 177 76 76 177 76 76 76 76 304 472 472 304 16 16 16 16 16 16 16 457 457 16 16 16 16 16 16 16 153 153 153 153 433 153 153 153 153433 153 153 153 153 412 256 256 412 393 393 240 240 129 129 129348 348129 129 129 52 52 52 52 52337 337 52 52 52 52 52 313 313 108 108 108184 184108 108 108 265 265 160 160 496 160 160496 160 160 481 481 9 9 9 9 9 9 9 9 9 616 616 9 9 9 9 9 9 9 9 9 232 232 457232 232457 232 232 81 81 81 81 673 81 81673 81 81 81 81 576 576 220 568 436 648 648 436 568 220 136 217 217 217 217 136 136433 217 217 217 217 433136 136 217 217 217 217 136 625 417 417 625 72 72 72 72 409 72 72409 72 72 72 72 585 585 193 193 504 193 193504 193 193 372 372 28 28 28 28 28 28 28361 36128 28 28 28 28 28 28 177 177 520 177 177520 177 177 448 448 168 168 337168505 505 168337 168 168 424 424 57 57 57 57 57313 31357 57 57 57 57 96 96 304 96 96304 96 96 145 145 156 156 4 4 4 4 4 345 4 345 4 4 4 4 4 105 105 105336 505 505 105336 105 105 201 201 201412 412201 201 201 592 592 489 489 256 256 256681 681 256 256 256 64 64 64 64 64 576 576 64 64 64 64 64 192 193 193 192 481192577 768 193321 321193 768 577192481 192 193 193 192 600 600 97 97 97 97 388 808 97 97808 388 97 97 97 97 280 280 280745 745 280 280 280 33 33 33 33 33 33 33553 553 33 33 33 33 33 33 33 444 532 532 444 232 265 265 232 232705 760 265 265760 705 232 232 265 265 232 88 88 88 88 609 88697 697 88609 88 88 88 88 169 169 504 169537 537169504 169 169 276 276 12 12 12 12 12249 249 12 12 12 12 148 148 148465 465148 148 148 432 580 580 432 217 217 217433 433 217 217 217 73 73 73 73 73616 616 73 73 73 73 73 184 184 492184561 561 184492 184 184 24 24 24 345 24345 24 24 24 268 268 385 385 40 12940129 129316 40129316 12940129 40 201 201 60 60 60 84 84 60 20160201 60 40 40 316 40316 40 40 385 385 129 129 268129 129268 129 129 345 345 184 184 184 184 184 184 24 24 24 492 561 24561 492 24 24 24 616 616 217 217 217433 433 217 217 217 73 73 73 73 432 73580 580 73432 73 73 73 73 148 148 148465 465148 148 148 249 249 276 276 169 169 169 169 169 169 504 537 537 504 88 88 88 88 609 88697 697 88609 88 88 88 88 232 265 265 232 232705 760 265 265760 705 232 232 265 265 232 444 532 532 444 553 553 280 280 280 280 280 280 12 12 12 12 12745 745 12 12 12 12 388 808 808 388 192 192 192 600 600 192 192 192 193 193 481 577 768 193321 321193 768 577 481 193 193 97 97 25697 97256 25697576 576 97256 25697 97256 97 97 33 33 33 33 33 33 33489 681 681 48933 33 33 33 33 33 33 592 592 412 412 201 201 336 201 201 336 201 201 64 64 64 64 64505 505 64 64 64 64 64 345 345 105 105 156105 105156 105 105 145 145 96 96 304 96 96304 96 96 313 313 424 424 168 168 168 168 168 168 337 505 505 337 57 57 57 57 57448 44857 57 57 57 57 177 177 520 177 177520 177 177 361 361 372 372 193 193 504 193 193504 193 193 585 585 409 409 28 28 7228 28 72 7228 28 72 72625 28417 41728625 72 7228 28 72 7228 28 72 28 28 136 217 217 217 217 136 136 217 217 217 217 136 136 217 217 217 217 136 433 568 436 648 648 436 568 433 220 576 576 220 673 673 232 232 457232 232457 232 232 616 616 81 81 81 81 81481 481 81 81 81 81 81 160 160 496 160 160496 160 160 265 265 184 184 313 313 108 108 108337 108337 108 108 348 348 240 240 393 393 52 52 12952 52129 12952412 256 256 41252129 12952 52129 52 52 433 433 153 153 153 153 153 153457 457 153 153 153 153 153 153 1 1 1 1 1 1 1 304 472 472 304 1 1 1 1 1 1 1 177 177 537 537 76 76 76201 20176 577 76201 20176577 76201 20176 76 76 204 600 384 384 600 204 84 216 216 84 84 216625 625 216 84 84 216 216 84 424 424 249 249 297 297 321 321 148 148 124 148 393 393 148 124 148 148 156 156 417 396 396 156 156 396 396 417 156 156 456 456 196 196 513 196 196513 196 196 4 4 4 4 4 244 561 268 4268 561 244 4 4 4 4 4 316 316 372 372 388 436 436 388 9 9 16 9 9 16 16 9 9 16 16 9 9 16 16 9 516 516 9 16 16 9 9 16 16 9 9 16 16 9 9 16 9 9 25 25 25 49 2549 4925 49 25 36 25 49 2549 4925 49 2536 4936 25 49 2549 4925 49 2536 49 36 25 49 2549 4925 49 2536 25 49 2549 4925 49 25 25 25 81 169 144 81 81 144 169 81 19681169 225 169 196 169 144 25681 81256 144 169 196 169 225 16981196 81 169 144 81 81 144 169 81 64 64 121 100 12164 64121 100 121 121 121324 289100 12136164441 324 289 361 121 121 361 289 324 44164361121 100289 324121 121 121 100 12164 64121 100 121 64 64 Figure 4.
Load more

Recommended publications
  • Lecture 10: Impulse and Momentum
    ME 230 Kinematics and Dynamics Wei-Chih Wang Department of Mechanical Engineering University of Washington Kinetics of a particle: Impulse and Momentum Chapter 15 Chapter objectives • Develop the principle of linear impulse and momentum for a particle • Study the conservation of linear momentum for particles • Analyze the mechanics of impact • Introduce the concept of angular impulse and momentum • Solve problems involving steady fluid streams and propulsion with variable mass W. Wang Lecture 10 • Kinetics of a particle: Impulse and Momentum (Chapter 15) - 15.1-15.3 W. Wang Material covered • Kinetics of a particle: Impulse and Momentum - Principle of linear impulse and momentum - Principle of linear impulse and momentum for a system of particles - Conservation of linear momentum for a system of particles …Next lecture…Impact W. Wang Today’s Objectives Students should be able to: • Calculate the linear momentum of a particle and linear impulse of a force • Apply the principle of linear impulse and momentum • Apply the principle of linear impulse and momentum to a system of particles • Understand the conditions for conservation of momentum W. Wang Applications 1 A dent in an automotive fender can be removed using an impulse tool, which delivers a force over a very short time interval. How can we determine the magnitude of the linear impulse applied to the fender? Could you analyze a carpenter’s hammer striking a nail in the same fashion? W. Wang Applications 2 Sure! When a stake is struck by a sledgehammer, a large impulsive force is delivered to the stake and drives it into the ground.
    [Show full text]
  • Impact Dynamics of Newtonian and Non-Newtonian Fluid Droplets on Super Hydrophobic Substrate
    IMPACT DYNAMICS OF NEWTONIAN AND NON-NEWTONIAN FLUID DROPLETS ON SUPER HYDROPHOBIC SUBSTRATE A Thesis Presented By Yingjie Li to The Department of Mechanical and Industrial Engineering in partial fulfillment of the requirements for the degree of Master of Science in the field of Mechanical Engineering Northeastern University Boston, Massachusetts December 2016 Copyright (©) 2016 by Yingjie Li All rights reserved. Reproduction in whole or in part in any form requires the prior written permission of Yingjie Li or designated representatives. ACKNOWLEDGEMENTS I hereby would like to appreciate my advisors Professors Kai-tak Wan and Mohammad E. Taslim for their support, guidance and encouragement throughout the process of the research. In addition, I want to thank Mr. Xiao Huang for his generous help and continued advices for my thesis and experiments. Thanks also go to Mr. Scott Julien and Mr, Kaizhen Zhang for their invaluable discussions and suggestions for this work. Last but not least, I want to thank my parents for supporting my life from China. Without their love, I am not able to complete my thesis. TABLE OF CONTENTS DROPLETS OF NEWTONIAN AND NON-NEWTONIAN FLUIDS IMPACTING SUPER HYDROPHBIC SURFACE .......................................................................... i ACKNOWLEDGEMENTS ...................................................................................... iii 1. INTRODUCTION .................................................................................................. 9 1.1 Motivation ........................................................................................................
    [Show full text]
  • Post-Newtonian Approximation
    Post-Newtonian gravity and gravitational-wave astronomy Polarization waveforms in the SSB reference frame Relativistic binary systems Effective one-body formalism Post-Newtonian Approximation Piotr Jaranowski Faculty of Physcis, University of Bia lystok,Poland 01.07.2013 P. Jaranowski School of Gravitational Waves, 01{05.07.2013, Warsaw Post-Newtonian gravity and gravitational-wave astronomy Polarization waveforms in the SSB reference frame Relativistic binary systems Effective one-body formalism 1 Post-Newtonian gravity and gravitational-wave astronomy 2 Polarization waveforms in the SSB reference frame 3 Relativistic binary systems Leading-order waveforms (Newtonian binary dynamics) Leading-order waveforms without radiation-reaction effects Leading-order waveforms with radiation-reaction effects Post-Newtonian corrections Post-Newtonian spin-dependent effects 4 Effective one-body formalism EOB-improved 3PN-accurate Hamiltonian Usage of Pad´eapproximants EOB flexibility parameters P. Jaranowski School of Gravitational Waves, 01{05.07.2013, Warsaw Post-Newtonian gravity and gravitational-wave astronomy Polarization waveforms in the SSB reference frame Relativistic binary systems Effective one-body formalism 1 Post-Newtonian gravity and gravitational-wave astronomy 2 Polarization waveforms in the SSB reference frame 3 Relativistic binary systems Leading-order waveforms (Newtonian binary dynamics) Leading-order waveforms without radiation-reaction effects Leading-order waveforms with radiation-reaction effects Post-Newtonian corrections Post-Newtonian spin-dependent effects 4 Effective one-body formalism EOB-improved 3PN-accurate Hamiltonian Usage of Pad´eapproximants EOB flexibility parameters P. Jaranowski School of Gravitational Waves, 01{05.07.2013, Warsaw Relativistic binary systems exist in nature, they comprise compact objects: neutron stars or black holes. These systems emit gravitational waves, which experimenters try to detect within the LIGO/VIRGO/GEO600 projects.
    [Show full text]
  • Arxiv:2008.00328V3 [Math.DS] 28 Apr 2021 Uniformly Hyperbolic and Topologically Mixing
    ERGODICITY AND EQUIDISTRIBUTION IN STRICTLY CONVEX HILBERT GEOMETRY FENG ZHU Abstract. We show that dynamical and counting results characteristic of negatively-curved Riemannian geometry, or more generally CAT(-1) or rank- one CAT(0) spaces, also hold for geometrically-finite strictly convex projective structures equipped with their Hilbert metric. More specifically, such structures admit a finite Sullivan measure; with respect to this measure, the Hilbert geodesic flow is strongly mixing, and orbits and primitive closed geodesics equidistribute, allowing us to asymptotically enumerate these objects. In [Mar69], Margulis established counting results for uniform lattices in constant negative curvature, or equivalently for closed hyperbolic manifolds, by means of ergodicity and equidistribution results for the geodesic flows on these manifolds with respect to suitable measures. Thomas Roblin, in [Rob03], obtained analogous ergodicity and equidistribution results in the more general setting of CAT(−1) spaces. These results include ergodicity of the horospherical foliations, mixing of the geodesic flow, orbital equidistribution of the group, equidistribution of primitive closed geodesics, and, in the geometrically finite case, asymptotic counting estimates. Gabriele Link later adapted similar techniques to prove similar results in the even more general setting of rank-one isometry groups of Hadamard spaces in [Lin20]. These results do not directly apply to manifolds endowed with strictly convex projective structures, considered with their Hilbert metrics and associated Bowen- Margulis measures, since strictly convex Hilbert geometries are in general not CAT(−1) or even CAT(0) (see e.g. [Egl97, App. B].) Nevertheless, these Hilbert geometries exhibit substantial similarities to Riemannian geometries of pinched negative curvature. In particular, there is a good theory of Busemann functions and of Patterson-Sullivan measures on these geometries.
    [Show full text]
  • [Math.GT] 9 Oct 2020 Symmetries of Hyperbolic 4-Manifolds
    Symmetries of hyperbolic 4-manifolds Alexander Kolpakov & Leone Slavich R´esum´e Pour chaque groupe G fini, nous construisons des premiers exemples explicites de 4- vari´et´es non-compactes compl`etes arithm´etiques hyperboliques M, `avolume fini, telles M G + M G que Isom ∼= , ou Isom ∼= . Pour y parvenir, nous utilisons essentiellement la g´eom´etrie de poly`edres de Coxeter dans l’espace hyperbolique en dimension quatre, et aussi la combinatoire de complexes simpliciaux. C¸a nous permet d’obtenir une borne sup´erieure universelle pour le volume minimal d’une 4-vari´et´ehyperbolique ayant le groupe G comme son groupe d’isom´etries, par rap- port de l’ordre du groupe. Nous obtenons aussi des bornes asymptotiques pour le taux de croissance, par rapport du volume, du nombre de 4-vari´et´es hyperboliques ayant G comme le groupe d’isom´etries. Abstract In this paper, for each finite group G, we construct the first explicit examples of non- M M compact complete finite-volume arithmetic hyperbolic 4-manifolds such that Isom ∼= G + M G , or Isom ∼= . In order to do so, we use essentially the geometry of Coxeter polytopes in the hyperbolic 4-space, on one hand, and the combinatorics of simplicial complexes, on the other. This allows us to obtain a universal upper bound on the minimal volume of a hyperbolic 4-manifold realising a given finite group G as its isometry group in terms of the order of the group. We also obtain asymptotic bounds for the growth rate, with respect to volume, of the number of hyperbolic 4-manifolds having a finite group G as their isometry group.
    [Show full text]
  • New Rotational Dynamics- Inertia-Torque Principle and the Force Moment the Character of Statics Guagsan Yu* Harbin Macro, Dynamics Institute, P
    Computa & tio d n ie a l l p M Yu, J Appl Computat Math 2015, 4:3 p a Journal of A t h f e o m l DOI: 10.4172/2168-9679.1000222 a a n t r ISSN: 2168-9679i c u s o J Applied & Computational Mathematics Research Article Open Access New Rotational Dynamics- Inertia-Torque Principle and the Force Moment the Character of Statics GuagSan Yu* Harbin Macro, Dynamics Institute, P. R. China Abstract Textual point of view, generate in a series of rotational dynamics experiment. Initial research, is wish find a method overcome the momentum conservation. But further study, again detected inside the classical mechanics, the error of the principle of force moment. Then a series of, momentous the error of inside classical theory, all discover come out. The momentum conservation law is wrong; the newton third law is wrong; the energy conservation law is also can surpass. After redress these error, the new theory namely perforce bring. This will involve the classical physics and mechanics the foundation fraction, textbooks of physics foundation part should proceed the grand modification. Keywords: Rigid body; Inertia torque; Centroid moment; Centroid occurrence change? The Inertia-torque, namely, such as Figure 3 the arm; Statics; Static force; Dynamics; Conservation law show, the particle m (the m is also its mass) is a r 1 to O the slewing radius, so it the Inertia-torque is: Introduction I = m.r1 (1.1) Textual argumentation is to bases on the several simple physics experiment, these experiments pass two videos the document to The Inertia-torque is similar with moment of force, to the same of proceed to demonstrate.
    [Show full text]
  • On Eigenvalues of Geometrically Finite Hyperbolic Manifolds With
    On Eigenvalues of Geometrically Finite Hyperbolic Manifolds with Infinite Volume Xiaolong Hans Han April 6, 2020 Abstract Let M be an oriented geometrically finite hyperbolic manifold of dimension n ≥ 3 with infinite volume. Then for all k ≥ 0, we provide a lower bound on the k-th eigenvalue of the Laplacian operator acting on functions on M by a constant and the k-th eigenvalue of some neighborhood Mf of the thick part of the convex core. 1. Introduction As analytic data of a manifold, the spectrum of the Laplacian operator acting on func- tions contains rich information about the geometry and topology of a manifold. For example, by results of Cheeger and Buser, with a lower bound on Ricci curvature, the first eigenvalue is equivalent to the Cheeger constant of a manifold, which depends on the geometry of the manifold. The rigidity and richness of hyperbolic geometry make the connection even more intriguing: by Canary [4], for infinite volume, topologically tame hyperbolic 3-manifolds, the 2 bottom of the L -spectrum of −∆, denoted by λ0, is zero if and only if the manifold is not geometrically finite. If λ0 of a geometrically finite, infinite volume hyperbolic manifold is 1, by Canary and Burger [2], the manifold is either the topological interior of a handlebody or an R-bundle over a closed surface, which is a very strong topological restriction. Conversely, if we have data about the geometry of a manifold, we can convert it into data about its spectrum, like lower bounds on eigenvalues, dimensions of eigenspaces, existence of embedded eigenvalues, etc.
    [Show full text]
  • ISOMETRY TRANSFORMATIONS of HYPERBOLIC 3-MANIFOLDS Sadayoshi KOJIMA 0. Introduction by a Hyperbolic Manifold, We Mean a Riemanni
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Topology and its Applications 29 (1988) 297-307 297 North-Holland ISOMETRY TRANSFORMATIONS OF HYPERBOLIC 3-MANIFOLDS Sadayoshi KOJIMA Department of Information Science, Tokyo Institute of Technology, Ohokayama, Meguro, Tokyo, Japan Received 1 December 1986 Revised 10 June 1987 The isometry group of a compact hyperbolic manifold is known to be finite. We show that every finite group is realized as the full isometry group of some compact hyperbolic 3-manifold. AMS (MOS) Subj. Class.: 57S17, 53C25, 57M99 3-manifold hyperbolic manifold isometry 0. Introduction By a hyperbolic manifold, we mean a Riemannian manifold with constant sectional curvature -1. It is known by several ways that the isometry group of a hyperbolic manifold of finite volume is finite. On the other hand, Greenberg has shown in [3] that every finite group is realized as the full conformal automorphism group of some Riemann surface of hyperbolic type. Since such a Riemann surface, which is a 2-manifold with a conformal structure, inherits a unique conformally equivalent hyperbolic structure through the uniformization theorem, there are no limitations on the group structure of the full isometry group of a hyperbolic manifold in dimension 2. In this paper, we first give a quick proof of the theorem of Greenberg quoted above, and then along the same line we show, raising the dimension by one. Theorem. Every jinite group is realized as the full isometry group of some closed hyperbolic 3 -manifold. An immediate corollary to this is the following.
    [Show full text]
  • Hyperbolic Geometry
    Flavors of Geometry MSRI Publications Volume 31,1997 Hyperbolic Geometry JAMES W. CANNON, WILLIAM J. FLOYD, RICHARD KENYON, AND WALTER R. PARRY Contents 1. Introduction 59 2. The Origins of Hyperbolic Geometry 60 3. Why Call it Hyperbolic Geometry? 63 4. Understanding the One-Dimensional Case 65 5. Generalizing to Higher Dimensions 67 6. Rudiments of Riemannian Geometry 68 7. Five Models of Hyperbolic Space 69 8. Stereographic Projection 72 9. Geodesics 77 10. Isometries and Distances in the Hyperboloid Model 80 11. The Space at Infinity 84 12. The Geometric Classification of Isometries 84 13. Curious Facts about Hyperbolic Space 86 14. The Sixth Model 95 15. Why Study Hyperbolic Geometry? 98 16. When Does a Manifold Have a Hyperbolic Structure? 103 17. How to Get Analytic Coordinates at Infinity? 106 References 108 Index 110 1. Introduction Hyperbolic geometry was created in the first half of the nineteenth century in the midst of attempts to understand Euclid’s axiomatic basis for geometry. It is one type of non-Euclidean geometry, that is, a geometry that discards one of Euclid’s axioms. Einstein and Minkowski found in non-Euclidean geometry a This work was supported in part by The Geometry Center, University of Minnesota, an STC funded by NSF, DOE, and Minnesota Technology, Inc., by the Mathematical Sciences Research Institute, and by NSF research grants. 59 60 J. W. CANNON, W. J. FLOYD, R. KENYON, AND W. R. PARRY geometric basis for the understanding of physical time and space. In the early part of the twentieth century every serious student of mathematics and physics studied non-Euclidean geometry.
    [Show full text]
  • SPATIAL STATISTICS of APOLLONIAN GASKETS 1. Introduction Apollonian Gaskets, Named After the Ancient Greek Mathematician, Apollo
    SPATIAL STATISTICS OF APOLLONIAN GASKETS WEIRU CHEN, MO JIAO, CALVIN KESSLER, AMITA MALIK, AND XIN ZHANG Abstract. Apollonian gaskets are formed by repeatedly filling the interstices between four mutually tangent circles with further tangent circles. We experimentally study the nearest neighbor spacing, pair correlation, and electrostatic energy of centers of circles from Apol- lonian gaskets. Even though the centers of these circles are not uniformly distributed in any `ambient' space, after proper normalization, all these statistics seem to exhibit some interesting limiting behaviors. 1. introduction Apollonian gaskets, named after the ancient Greek mathematician, Apollonius of Perga (200 BC), are fractal sets obtained by starting from three mutually tangent circles and iter- atively inscribing new circles in the curvilinear triangular gaps. Over the last decade, there has been a resurgent interest in the study of Apollonian gaskets. Due to its rich mathematical structure, this topic has attracted attention of experts from various fields including number theory, homogeneous dynamics, group theory, and as a consequent, significant results have been obtained. Figure 1. Construction of an Apollonian gasket For example, it has been known since Soddy [23] that there exist Apollonian gaskets with all circles having integer curvatures (reciprocal of radii). This is due to the fact that the curvatures from any four mutually tangent circles satisfy a quadratic equation (see Figure 2). Inspired by [12], [10], and [7], Bourgain and Kontorovich used the circle method to prove a fascinating result that for any primitive integral (integer curvatures with gcd 1) Apollonian gasket, almost every integer in certain congruence classes modulo 24 is a curvature of some circle in the gasket.
    [Show full text]
  • Apollonius of Pergaconics. Books One - Seven
    APOLLONIUS OF PERGACONICS. BOOKS ONE - SEVEN INTRODUCTION A. Apollonius at Perga Apollonius was born at Perga (Περγα) on the Southern coast of Asia Mi- nor, near the modern Turkish city of Bursa. Little is known about his life before he arrived in Alexandria, where he studied. Certain information about Apollonius’ life in Asia Minor can be obtained from his preface to Book 2 of Conics. The name “Apollonius”(Apollonius) means “devoted to Apollo”, similarly to “Artemius” or “Demetrius” meaning “devoted to Artemis or Demeter”. In the mentioned preface Apollonius writes to Eudemus of Pergamum that he sends him one of the books of Conics via his son also named Apollonius. The coincidence shows that this name was traditional in the family, and in all prob- ability Apollonius’ ancestors were priests of Apollo. Asia Minor during many centuries was for Indo-European tribes a bridge to Europe from their pre-fatherland south of the Caspian Sea. The Indo-European nation living in Asia Minor in 2nd and the beginning of the 1st millennia B.C. was usually called Hittites. Hittites are mentioned in the Bible and in Egyptian papyri. A military leader serving under the Biblical king David was the Hittite Uriah. His wife Bath- sheba, after his death, became the wife of king David and the mother of king Solomon. Hittites had a cuneiform writing analogous to the Babylonian one and hi- eroglyphs analogous to Egyptian ones. The Czech historian Bedrich Hrozny (1879-1952) who has deciphered Hittite cuneiform writing had established that the Hittite language belonged to the Western group of Indo-European languages [Hro].
    [Show full text]
  • Newton Euler Equations of Motion Examples
    Newton Euler Equations Of Motion Examples Alto and onymous Antonino often interloping some obligations excursively or outstrikes sunward. Pasteboard and Sarmatia Kincaid never flits his redwood! Potatory and larboard Leighton never roller-skating otherwhile when Trip notarizes his counterproofs. Velocity thus resulting in the tumbling motion of rigid bodies. Equations of motion Euler-Lagrange Newton-Euler Equations of motion. Motion of examples of experiments that a random walker uses cookies. Forces by each other two examples of example are second kind, we will refer to specify any parameter in. 213 Translational and Rotational Equations of Motion. Robotics Lecture Dynamics. Independence from a thorough description and angular velocity as expected or tofollowa userdefined behaviour does it only be loaded geometry in an appropriate cuts in. An interface to derive a particular instance: divide and author provides a positive moment is to express to output side can be run at all previous step. The analysis of rotational motions which make necessary to decide whether rotations are. For xddot and whatnot in which a very much easier in which together or arena where to use them in two backwards operation complies with respect to rotations. Which influence of examples are true, is due to independent coordinates. On sameor adjacent joints at each moment equation is also be more specific white ellipses represent rotations are unconditionally stable, for motion break down direction. Unit quaternions or Euler parameters are known to be well suited for the. The angular momentum and time and runnable python code. The example will be run physics examples are models can be symbolic generator runs faster rotation kinetic energy.
    [Show full text]