SPATIAL STATISTICS of APOLLONIAN GASKETS 1. Introduction Apollonian Gaskets, Named After the Ancient Greek Mathematician, Apollo
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
On Dynamical Gaskets Generated by Rational Maps, Kleinian Groups, and Schwarz Reflections
ON DYNAMICAL GASKETS GENERATED BY RATIONAL MAPS, KLEINIAN GROUPS, AND SCHWARZ REFLECTIONS RUSSELL LODGE, MIKHAIL LYUBICH, SERGEI MERENKOV, AND SABYASACHI MUKHERJEE Abstract. According to the Circle Packing Theorem, any triangulation of the Riemann sphere can be realized as a nerve of a circle packing. Reflections in the dual circles generate a Kleinian group H whose limit set is an Apollonian- like gasket ΛH . We design a surgery that relates H to a rational map g whose Julia set Jg is (non-quasiconformally) homeomorphic to ΛH . We show for a large class of triangulations, however, the groups of quasisymmetries of ΛH and Jg are isomorphic and coincide with the corresponding groups of self- homeomorphisms. Moreover, in the case of H, this group is equal to the group of M¨obiussymmetries of ΛH , which is the semi-direct product of H itself and the group of M¨obiussymmetries of the underlying circle packing. In the case of the tetrahedral triangulation (when ΛH is the classical Apollonian gasket), we give a piecewise affine model for the above actions which is quasiconformally equivalent to g and produces H by a David surgery. We also construct a mating between the group and the map coexisting in the same dynamical plane and show that it can be generated by Schwarz reflections in the deltoid and the inscribed circle. Contents 1. Introduction 2 2. Round Gaskets from Triangulations 4 3. Round Gasket Symmetries 6 4. Nielsen Maps Induced by Reflection Groups 12 5. Topological Surgery: From Nielsen Map to a Branched Covering 16 6. Gasket Julia Sets 18 arXiv:1912.13438v1 [math.DS] 31 Dec 2019 7. -
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. -
Geometry and Arithmetic of Crystallographic Sphere Packings
Geometry and arithmetic of crystallographic sphere packings Alex Kontorovicha,b,1 and Kei Nakamuraa aDepartment of Mathematics, Rutgers University, New Brunswick, NJ 08854; and bSchool of Mathematics, Institute for Advanced Study, Princeton, NJ 08540 Edited by Kenneth A. Ribet, University of California, Berkeley, CA, and approved November 21, 2018 (received for review December 12, 2017) We introduce the notion of a “crystallographic sphere packing,” argument leading to Theorem 3 comes from constructing circle defined to be one whose limit set is that of a geometrically packings “modeled on” combinatorial types of convex polyhedra, finite hyperbolic reflection group in one higher dimension. We as follows. exhibit an infinite family of conformally inequivalent crystallo- graphic packings with all radii being reciprocals of integers. We (~): Polyhedral Packings then prove a result in the opposite direction: the “superintegral” Let Π be the combinatorial type of a convex polyhedron. Equiv- ones exist only in finitely many “commensurability classes,” all in, alently, Π is a 3-connectedz planar graph. A version of the at most, 20 dimensions. Koebe–Andreev–Thurston Theorem§ says that there exists a 3 geometrization of Π (that is, a realization of its vertices in R with sphere packings j crystallographic j arithmetic j polyhedra j straight lines as edges and faces contained in Euclidean planes) Coxeter diagrams having a midsphere (meaning, a sphere tangent to all edges). This midsphere is then also simultaneously a midsphere for the he goal of this program is to understand the basic “nature” of dual polyhedron Πb. Fig. 2A shows the case of a cuboctahedron Tthe classical Apollonian gasket. -
A Note on Unbounded Apollonian Disk Packings
A note on unbounded Apollonian disk packings Jerzy Kocik Department of Mathematics Southern Illinois University, Carbondale, IL62901 [email protected] (version 6 Jan 2019) Abstract A construction and algebraic characterization of two unbounded Apollonian Disk packings in the plane and the half-plane are presented. Both turn out to involve the golden ratio. Keywords: Unbounded Apollonian disk packing, golden ratio, Descartes configuration, Kepler’s triangle. 1 Introduction We present two examples of unbounded Apollonian disk packings, one that fills a half-plane and one that fills the whole plane (see Figures 3 and 7). Quite interestingly, both are related to the golden ratio. We start with a brief review of Apollonian disk packings, define “unbounded”, and fix notation and terminology. 14 14 6 6 11 3 11 949 9 4 9 2 2 1 1 1 11 11 arXiv:1910.05924v1 [math.MG] 14 Oct 2019 6 3 6 949 9 4 9 14 14 Figure 1: Apollonian Window (left) and Apollonian Belt (right). The Apollonian disk packing is a fractal arrangement of disks such that any of its three mutually tangent disks determine it by recursivly inscribing new disks in the curvy-triangular spaces that emerge between the disks. In such a context, the three initial disks are called a seed of the packing. Figure 1 shows for two most popular examples: the “Apollonian Window” and the “Apollonian Belt”. The num- bers inside the circles represent their curvatures (reciprocals of the radii). Note that the curvatures are integers; such arrangements are called integral Apollonian disk pack- ings; they are classified and their properties are still studied [3, 5, 6]. -
2014 AMS Elections---Biographies of Candidates
Biographies of Candidates 2014 Biographical information about the candidates has been supplied and verified by the candidates. Candidates have had the opportunity to make a statement of not more than 200 words (400 words for presidential candidates) on any subject matter without restriction and to list up to five of their research papers. Candidates have had the opportunity to supply a photograph to accompany their biographical information. Candidates with an asterisk (*) beside their names were nominated in response to a petition. Abbreviations: American Association for the Advancement of Science (AAAS); American Mathematical Society (AMS); American Statistical Association (ASA); Association for Computing Machinery (ACM); Association for Symbolic Logic (ASL); Association for Women in Mathematics (AWM); Canadian Mathematical Society, Société Mathématique du Canada (CMS); Conference Board of the Mathematical Sciences (CBMS); Institute for Advanced Study (IAS), Insti- tute of Mathematical Statistics (IMS); International Mathematical Union (IMU); London Mathematical Society (LMS); Mathematical Association of America (MAA); Mathematical Sciences Research Institute (MSRI); National Academy of Sciences (NAS); National Academy of Sciences/National Research Council (NAS/NRC); National Aeronautics and Space Administration (NASA); National Council of Teachers of Mathematics (NCTM); National Science Foundation (NSF); Society for Industrial and Applied Mathematics (SIAM). Vice President Mathematics, Princeton, 2004–2010; Dean of Natural Sci- ences, Duke, 2010–2012; Director, Information Initiative, Robert Calderbank Duke, 2012–present. Boards: Institute for Mathematics Professor of Mathematics, Direc- and Applications, Board of Governors, 1996–1999; NSF tor of the Information Initiative, Committee of Visitors, Division of Mathematical Sciences, Duke University (iiD). 2001; American Institute of Mathematics, Scientific Advi- Born: December 28, 1954, Bridg- sory Board, 2006–present; Institute for Pure and Applied water, Somerset, UK. -
Origin of Exponential Growth in Nonlinear Reaction Networks
Origin of exponential growth in nonlinear reaction networks Wei-Hsiang Lina,b,c,d,e,1, Edo Kussellf,g, Lai-Sang Youngh,i,j, and Christine Jacobs-Wagnera,b,c,d,e,k,1 aMicrobial Science Institute, Yale University, West Haven, CT 06516; bDepartment of Molecular, Cellular and Developmental Biology, Yale University, New Haven, CT 06511; cHoward Hughes Medical Institute, Yale University, New Haven, CT 06510; dDepartment of Biology, Stanford University, Palo Alto, CA 94305; eChemistry, Engineering & Medicine for Human Health (ChEM-H) Institute, Stanford University, Palo Alto, CA 94305; fCenter of Genomics and Systems Biology, Department of Biology, New York University, New York, NY 10003; gDepartment of Physics, New York University, New York, NY 10003; hCourant Institute of Mathematical Science, New York University, New York, NY 10012; iSchool of Mathematics, Institute for Advanced Study, Princeton, NJ 08540; jSchool of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540; and kDepartment of Microbial Pathogenesis, Yale School of Medicine, New Haven, CT 06510 Contributed by Christine Jacobs-Wagner, September 10, 2020 (sent for review June 24, 2020; reviewed by Jeff Gore and Ned S. Wingreen) Exponentially growing systems are prevalent in nature, spanning For example, the levels of metabolites and proteins can oscillate all scales from biochemical reaction networks in single cells to food during cell growth (13–15). The abundance of species in eco- webs of ecosystems. How exponential growth emerges in nonlin- systems can also oscillate (6) or exhibit nonperiodic fluctuations ear systems is mathematically unclear. Here, we describe a general (7). Yet, the biomass of these systems increases exponentially theoretical framework that reveals underlying principles of long- (6, 7, 16). -
Non-Euclidean Geometry and Indra's Pearls
Non!Euclidean geometry and Indra's pearls © 1997!2004, Millennium Mathematics Project, University of Cambridge. Permission is granted to print and copy this page on paper for non!commercial use. For other uses, including electronic redistribution, please contact us. June 2007 Features Non!Euclidean geometry and Indra's pearls by Caroline Series and David Wright Many people will have seen and been amazed by the beauty and intricacy of fractals like the one shown on the right. This particular fractal is known as the Apollonian gasket and consists of a complicated arrangement of tangent circles. [Click on the image to see this fractal evolve in a movie created by David Wright.] Few people know, however, that fractal pictures like this one are intimately related to tilings of what mathematicians call hyperbolic space. One such tiling is shown in figure 1a below. In contrast, figure 1b shows a tiling of the ordinary flat plane. In this article, which first appeared in the Proceedings of the Bridges conference held in London in 2006, we will explore the maths behind these tilings and how they give rise to beautiful fractal images. Non!Euclidean geometry and Indra's pearls 1 Non!Euclidean geometry and Indra's pearls Figure 1b: A Euclidean tiling of the plane by Figure 1a: A non!Euclidean tiling of the disc by regular regular hexagons. Image created by David heptagons. Image created by David Wright. Wright. Round lines and strange circles In hyperbolic geometry distances are not measured in the usual way. In the hyperbolic metric the shortest distance between two points is no longer along a straight line, but along a different kind of curve, whose precise nature we'll explore below. -
Arxiv:1705.06212V2 [Math.MG] 18 May 2017
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 pair correlation, electrostatic energy, and nearest neighbor spacing of centers of circles from Apollonian 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 iteratively 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 significant results have been obtained. Figure 1. Construction of an Apollonian gasket arXiv:1705.06212v2 [math.MG] 18 May 2017 For example, it has been known since Soddy [22] 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 [11], [9], 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. -
The Apollonian Gasket by Eike Steinert & Peter Strümpel Structure 2
Technische Universität Berlin Institut für Mathematik Course: Mathematical Visualization I WS12/13 Professor: John M. Sullivan Assistant: Charles Gunn 18.04.2013 The Apollonian Gasket by Eike Steinert & Peter Strümpel structure 2 1. Introduction 2. Apollonian Problem 1. Mathematical background 2. Implementation 3. GUI 3. Apollonian Gasket 1. Mathematical background 2. Implementation 3. GUI 4. Future prospects Introduction – Apollonian Gasket 3 • It is generated from construct the two Take again 3 tangent triples of circles Apollonian circles which circles touches the given ones: • Each cirlce is tangent to the other two • internally and • externally Constuct again cirlces which touches the given ones Calculation of the 4 Apollinian circles Apollonian Problem Apollonian Gasket Apollonian problem - 5 mathematical background Apollonius of Perga ca. 200 b.c.: how to find a circle which touches 3 given objects? objects: lines, points or circles limitation of the problem: only circles first who found algebraic solution was Euler in the end of the 18. century Apollonian problem - 6 mathematical background algebraic solution based on the fact, that distance between centers equal with the sum of the radii so we get system of equations: +/- determines external/internal tangency 2³=8 combinations => 8 possible circles Apollonian problem - 7 mathematical background subtracting two linear equations: solving with respect to r we get: Apollonian problem - 8 mathematical background these results in first equation quadratic expression for r -
Fractal Images from Multiple Inversion in Circles
Bridges 2019 Conference Proceedings Fractal Images from Multiple Inversion in Circles Peter Stampfli Rue de Lausanne 1, 1580 Avenches, Switzerland; [email protected] Abstract Images resulting from multiple inversion and reflection in intersecting circles and straight lines are presented. Three circles and lines making a triangle give the well-known tilings of spherical, Euclidean or hyperbolic spaces. Four circles and lines can form a quadrilateral or a triangle with a circle around its center. Quadrilaterals give tilings of hyperbolic space or fractal tilings with a limit set that resembles generalized Koch snowflakes. A triangle with a circle results in a Poincaré disc representation of tiled hyperbolic space with a fractal covering made of small Poincaré disc representations of tiled hyperbolic space. An example is the Apollonian gasket. Other such tilings can simultaneously be decorations of hyperbolic, elliptic and Euclidean space. I am discussing an example, which is a self-similar decoration of both a sphere with icosahedral symmetry and a tiled hyperbolic space. You can create your own images and explore their geometries using public browser apps. Introduction Inversion in a circle is nearly the same as a mirror image at a straight line but it can magnify or reduce the image size. This gives much more diverse images. I am presenting some systematic results for multiple inversion in intersecting circles. For two circles we get a distorted rosette with dihedral symmetry and three circles give periodic decorations of elliptic, Euclidean or hyperbolic space. This is already well-known, but what do we get for four or more circles? Iterative Mapping Procedure for Creating Symmetric Images The color c(p) of a pixel is simply a function of its position p. -
Uniform Exponential Mixing and Resonance Free Regions for Convex Cocompact Congruence Subgroups of Sl2(Z)
UNIFORM EXPONENTIAL MIXING AND RESONANCE FREE REGIONS FOR CONVEX COCOMPACT CONGRUENCE SUBGROUPS OF SL2(Z) HEE OH AND DALE WINTER Abstract. Let Γ < SL2(Z) be a non-elementary finitely generated sub- group and Γ(q) be its congruence subgroup of level q for each q 2 N. We obtain an asymptotic formula for the matrix coefficients of 2 L (Γ(q)n SL2(R)) with a uniform exponential error term for all square- free q with no small prime divisors. As an application we establish a uniform resonance-free half plane for the resolvent of the Laplacian on 2 Γ(q)nH over q as above. Our approach is to extend Dolgopyat's dynam- ical proof of exponential mixing of the geodesic flow uniformly over con- gruence covers, by establishing uniform spectral bounds for congruence transfer operators associated to the geodesic flow. One of the key ingre- dients is the `2-flattening lemma due to Bourgain-Gamburd-Sarnak. 1. Introduction 1.1. Uniform exponential mixing. Let G = SL2(R) and Γ be a non- elementary finitely generated subgroup of SL2(Z). We will assume that Γ contains the negative identity −e but no other torsion elements. In other words, Γ is the pre-image of a torsion-free subgroup of PSL2(Z) under the canonical projection SL2(Z) ! PSL2(Z). For each q ≥ 1, consider the congruence subgroup of Γ of level q: Γ(q) := fγ 2 Γ: γ ≡ e mod qg: For t 2 R, let et=2 0 a = : t 0 e−t=2 As is well known, the right translation action of at on ΓnG corresponds to the geodesic flow when we identify ΓnG with the unit tangent bundle of a 2 hyperbolic surface ΓnH . -
Apollonian Circles Patterns in Musical Scales Posing Problems Triangles
Summer/Autumn 2017 A Problem Fit for a PrincessApollonian Apollonian Circles Gaskets Polygons and PatternsPrejudice in Exploring Musical Social Scales Issues Daydreams in MusicPosing Patterns Problems in Scales ProblemTriangles, Posing Squares, Empowering & Segregation Participants A NOTE FROM AIM #playwithmath Dear Math Teachers’ Circle Network, In this issue of the MTCircular, we hope you find some fun interdisciplinary math problems to try with your Summer is exciting for us, because MTC immersion MTCs. In “A Problem Fit for a Princess,” Chris Goff workshops are happening all over the country. We like traces the 2000-year history of a fractal that inspired his seeing the updates in real time, on Twitter. Your enthu- MTC’s logo. In “Polygons and Prejudice,” Anne Ho and siasm for all things math and problem solving is conta- Tara Craig use a mathematical frame to guide a con- gious! versation about social issues. In “Daydreams in Music,” Jeremy Aikin and Cory Johnson share a math session Here are some recent tweets we enjoyed from MTC im- motivated by patterns in musical scales. And for those mersion workshops in Cleveland, OH; Greeley, CO; and of you looking for ways to further engage your MTC San Jose, CA, respectively: participants’ mathematical thinking, Chris Bolognese and Mike Steward’s “Using Problem Posing to Empow- What happens when you cooperate in Blokus? er MTC Participants” will provide plenty of food for Try and create designs with rotational symmetry. thought. #toocool #jointhemath – @CrookedRiverMTC — Have MnMs, have combinatorial games Helping regions and states build networks of MTCs @NoCOMTC – @PaulAZeitz continues to be our biggest priority nationally.