DOCSLIB.ORG

Apollonian Gasket

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Apollonian Gasket A American Scientist Previous Page|||| Contents Zoom in Zoom out Front Cover | Search Issue | Next Page BEF MaGS Computing Science A Tisket, a Tasket, an Apollonian Gasket Dana Mackenzie n the spring of 2007 I had the it onto T-shirts. And in a book about I good fortune to spend a semester Fractals made fractals with the lovely title Indra’s at the Mathematical Sciences Research Pearls, mathematician David Wright Institute in Berkeley, an institution of compared the gasket to Dr. Seuss’s The higher learning that takes “higher” to of circles do Cat in the Hat: a whole new extreme. Perched precari- The cat takes off his hat to reveal ously on a ridge far above the Univer- funny things to Little Cat A, who then removes sity of California at Berkeley campus, his hat and releases Little Cat B, the building offers postcard-perfect mathematicians who then uncovers Little Cat C, vistas of the San Francisco Bay, 1,200 and so on. Now imagine there are feet below. That’s on the west side. not one but three cats inside each Rather sensibly, the institute assigned cat’s hat. That gives a good im- me an office on the east side, with a sequence was 1-squared, 2-squared, pression of the explosive prolifer- view of nothing much but my com- 3-squared, and so on. ation of these tiny ideal triangles. puter screen. Otherwise I might not Before I became a full-time writer, have gotten any work done. I used to be a mathematician. Seeing However, there was one flaw in the those numbers awakened the math geek Getting the Bends plan: Someone installed a screen-saver in me. What did they mean? And what Even the first step of drawing an Apol- program on the computer. Of course, it did they have to do with the fractal on lonian gasket is far from straightfor- had to be mathematical. The program the screen? Quickly, before the screen- ward. Given three circles, how do you drew an endless assortment of fractals saver image vanished into the ether, I draw a fourth circle that is exactly tan- of varying shapes and ingenuity. Every sketched it on my notepad, making a gent to all three? couple minutes the screen would go resolution to find out someday. Apparently the first mathematician blank and refresh itself with a complete- As it turned out, the picture on the to seriously consider this question was ly different fractal. I have to confess that screen was a special case of a more gen- Apollonius of Perga, a Greek geom- I spent a few idle minutes watching the eral construction. Start with three circles eter who lived in the third century b.c. fractals instead of writing. of any size, with each one touching the He has been somewhat overshadowed One day, a new design popped up other two. Draw a new circle that fits by his predecessor Euclid, in part be- on the screen (see the first figure). It was snugly into the space between them, cause most of his books have been lost. different from all the other fractals. It and another around the outside enclos- However, Apollonius’s surviving book was made up of simple shapes—cir- ing all the circles. Now you have four Conic Sections was the first to system- cles, in fact—and unlike all the other roughly triangular spaces between the atically study ellipses, hyperbolas and screen-savers, it had numbers! My circles. In each of those spaces, draw a parabolas—curves that have remained attention was immediately drawn new circle that just touches each side. central to mathematics ever since. to the sequence of numbers running This creates 12 triangular pores; insert One of Apollonius’s lost manu- along the bottom edge: 1, 4, 9, 16 … a new circle into each one of them, just scripts was called Tangencies. Accord- They were the perfect squares! The touching each side. Keep on going for- ing to later commentators, Apollonius ever, or at least until the circles become apparently solved the problem of too small to see. The resulting foam-like drawing circles that are simultane- Dana Mackenzie received his doctorate in math- structure is called an Apollonian gasket ously tangent to three lines, or two ematics from Princeton University in 1983 and (see the second figure). lines and a circle, or two circles and a taught at Duke University and Kenyon College. Something about the Apollonian line, or three circles. The hardest case Since 1996 he has been a freelance writer special- gasket makes ordinary, sensible math- of all was the case where the three izing in math and science, and he has frequently ematicians get a little bit giddy. It circles are tangent. edited articles on mathematical topics for Ameri- inspired a Nobel laureate to write a No one knows, of course, what can Scientist. His published books include The Big Splat, or How Our Moon Came to Be (Wiley, poem and publish it in the journal Na- Apollonius’ solution was, or whether it 2003), and volumes 6 and 7 of What’s Happening ture. An 18th-century Japanese samu- was correct. After many of the writings in the Mathematical Sciences (American Math- rai painted a similar picture on a tablet of the ancient Greeks became available ematical Society, 2007 and 2009). Email: ____scribe@ and hung it in front of a Buddhist tem- again to European scholars of the Re- danamackenzie.com ple. Researchers at AT&T Labs printed naissance, the unsolved “problem of 10 American Scientist, Volume 98 A American Scientist Previous Page|||| Contents Zoom in Zoom out Front Cover | Search Issue | Next Page BEF MaGS A American Scientist Previous Page|||| Contents Zoom in Zoom out Front Cover | Search Issue | Next Page BEF MaGS 81 64 49 36 25 81 49 64 25 49 81 81 49 25 64 49 81 25 36 49 64 81 81 64 49 36 25 81 49 64 25 49 81 81 49 25 64 49 81 25 36 49 64 81 Apollonius” became a great challenge. 84 16 76 76 16 84 84 16 76 76 16 84 52 9 9 52 52 9 9 52 81 81 81 81 In 1643, in a letter to Princess Elizabeth 72 28 28 72 72 28 28 72 57 57 57 57 of Bohemia, the French philosopher 96 4 96 96 4 96 64 64 64 64 and mathematician René Descartes cor- 33 97 97 33 33 97 97 33 rectly stated (but incorrectly proved) a 88 12 88 88 12 88 73 73 73 73 24 24 beautiful formula concerning the radii 40 40 60 60 of four mutually touching circles. If the 84 84 radii are r, s, t and u, then Descartes’s formula looks like this: 84 84 60 60 40 40 24 24 2 73 73 73 73 1⁄r2+ 1⁄s2+ 1⁄t2+ 1⁄u2= 1⁄2 (1⁄r+ 1⁄s+ 1⁄t+ 1⁄u) . 88 12 88 88 12 88 33 97 97 33 33 97 97 33 All of these reciprocals look a little1 64 64 1 64 64 1 96 96 96 96 bit extravagant, so the formula is usu- 57 57 57 57 72 28 28 72 72 28 28 72 ally simplified by writing it in terms 81 81 81 81 52 4 52 52 4 52 76 76 76 76 84 9 9 84 84 9 9 84 of the curvatures or the bends of the 25 16 25 25 16 25 25 16 25 25 16 25 81 64 49 36 81 49 64 49 81 81 49 64 49 81 36 49 64 81 81 64 49 36 81 49 64 49 81 81 49 64 49 81 36 49 64 81 circles. The curvature is simply de- fined as the reciprocal of the radius. Numbers in an Apollonian gasket correspond to the curvatures or “bends” of the circles, with Thus, if the curvatures are denoted by larger bends corresponding to smaller circles. The entire gasket is determined by the first four a, b, c and d, then Descartes’s formula mutually tangent circles; in this case, two circles with bend 1 and two “circles” with bend 0 reads as follows: (and therefore infinite radius). The circles with a bend of zero look, of course, like straight lines. (Image courtesy of Alex Kontorovich.) a2+b2+c2+d2=(a+b+c+d)2/2. As the third figure shows, Des- 1990s by Allan Wilks and Colin Mal- the circle centers as complex num- cartes’s formula greatly simplifies the lows of AT&T Labs, and Wilks used bers. Imaginary and complex num- task of finding the size of the fourth it to write a very efficient computer bers were not widely accepted by circle, assuming the sizes of the first program for plotting Apollonian gas- mathematicians until a century and a three are known. It is much less obvi- kets. One such plot went on his office half after Descartes died. ous that the very same equation can door and eventually got made into the In spite of its relative simplicity, be used to compute the location of the aforementioned T-shirt. Descartes’s formula has never become fourth circle as well, and thus com- Descartes himself could not have widely known, even among mathema- pletely solve the drawing problem. discovered this procedure, because it ticians. Thus, it has been rediscovered This fact was discovered in the late involves treating the coordinates of over and over through the years.
Load more

Recommended publications
  • Configurations on Centers of Bankoff Circles 11
    CONFIGURATIONS ON CENTERS OF BANKOFF CIRCLES ZVONKO CERINˇ Abstract. We study configurations built from centers of Bankoff circles of arbelos erected on sides of a given triangle or on sides of various related triangles. 1. Introduction For points X and Y in the plane and a positive real number λ, let Z be the point on the segment XY such that XZ : ZY = λ and let ζ = ζ(X; Y; λ) be the figure formed by three mj utuallyj j tangenj t semicircles σ, σ1, and σ2 on the same side of segments XY , XZ, and ZY respectively. Let S, S1, S2 be centers of σ, σ1, σ2. Let W denote the intersection of σ with the perpendicular to XY at the point Z. The figure ζ is called the arbelos or the shoemaker's knife (see Fig. 1). σ σ1 σ2 PSfrag replacements X S1 S Z S2 Y XZ Figure 1. The arbelos ζ = ζ(X; Y; λ), where λ = jZY j . j j It has been the subject of studies since Greek times when Archimedes proved the existence of the circles !1 = !1(ζ) and !2 = !2(ζ) of equal radius such that !1 touches σ, σ1, and ZW while !2 touches σ, σ2, and ZW (see Fig. 2). 1991 Mathematics Subject Classification. Primary 51N20, 51M04, Secondary 14A25, 14Q05. Key words and phrases. arbelos, Bankoff circle, triangle, central point, Brocard triangle, homologic. 1 2 ZVONKO CERINˇ σ W !1 σ1 !2 W1 PSfrag replacements W2 σ2 X S1 S Z S2 Y Figure 2. The Archimedean circles !1 and !2 together.
    [Show full text]
  • Kaleidoscopic Symmetries and Self-Similarity of Integral Apollonian Gaskets
    Kaleidoscopic Symmetries and Self-Similarity of Integral Apollonian Gaskets Indubala I Satija Department of Physics, George Mason University , Fairfax, VA 22030, USA (Dated: April 28, 2021) Abstract We describe various kaleidoscopic and self-similar aspects of the integral Apollonian gaskets - fractals consisting of close packing of circles with integer curvatures. Self-similar recursive structure of the whole gasket is shown to be encoded in transformations that forms the modular group SL(2;Z). The asymptotic scalings of curvatures of the circles are given by a special set of quadratic irrationals with continued fraction [n + 1 : 1; n] - that is a set of irrationals with period-2 continued fraction consisting of 1 and another integer n. Belonging to the class n = 2, there exists a nested set of self-similar kaleidoscopic patterns that exhibit three-fold symmetry. Furthermore, the even n hierarchy is found to mimic the recursive structure of the tree that generates all Pythagorean triplets arXiv:2104.13198v1 [math.GM] 21 Apr 2021 1 Integral Apollonian gaskets(IAG)[1] such as those shown in figure (1) consist of close packing of circles of integer curvatures (reciprocal of the radii), where every circle is tangent to three others. These are fractals where the whole gasket is like a kaleidoscope reflected again and again through an infinite collection of curved mirrors that encodes fascinating geometrical and number theoretical concepts[2]. The central themes of this paper are the kaleidoscopic and self-similar recursive properties described within the framework of Mobius¨ transformations that maps circles to circles[3]. FIG. 1: Integral Apollonian gaskets.
    [Show full text]
  • 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.
    [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]
  • Strophoids, a Family of Cubic Curves with Remarkable Properties
    Hellmuth STACHEL STROPHOIDS, A FAMILY OF CUBIC CURVES WITH REMARKABLE PROPERTIES Abstract: Strophoids are circular cubic curves which have a node with orthogonal tangents. These rational curves are characterized by a series or properties, and they show up as locus of points at various geometric problems in the Euclidean plane: Strophoids are pedal curves of parabolas if the corresponding pole lies on the parabola’s directrix, and they are inverse to equilateral hyperbolas. Strophoids are focal curves of particular pencils of conics. Moreover, the locus of points where tangents through a given point contact the conics of a confocal family is a strophoid. In descriptive geometry, strophoids appear as perspective views of particular curves of intersection, e.g., of Viviani’s curve. Bricard’s flexible octahedra of type 3 admit two flat poses; and here, after fixing two opposite vertices, strophoids are the locus for the four remaining vertices. In plane kinematics they are the circle-point curves, i.e., the locus of points whose trajectories have instantaneously a stationary curvature. Moreover, they are projections of the spherical and hyperbolic analogues. For any given triangle ABC, the equicevian cubics are strophoids, i.e., the locus of points for which two of the three cevians have the same lengths. On each strophoid there is a symmetric relation of points, so-called ‘associated’ points, with a series of properties: The lines connecting associated points P and P’ are tangent of the negative pedal curve. Tangents at associated points intersect at a point which again lies on the cubic. For all pairs (P, P’) of associated points, the midpoints lie on a line through the node N.
    [Show full text]
  • 9 · the Growth of an Empirical Cartography in Hellenistic Greece
    9 · The Growth of an Empirical Cartography in Hellenistic Greece PREPARED BY THE EDITORS FROM MATERIALS SUPPLIED BY GERMAINE AUJAe There is no complete break between the development of That such a change should occur is due both to po­ cartography in classical and in Hellenistic Greece. In litical and military factors and to cultural developments contrast to many periods in the ancient and medieval within Greek society as a whole. With respect to the world, we are able to reconstruct throughout the Greek latter, we can see how Greek cartography started to be period-and indeed into the Roman-a continuum in influenced by a new infrastructure for learning that had cartographic thought and practice. Certainly the a profound effect on the growth of formalized know­ achievements of the third century B.C. in Alexandria had ledge in general. Of particular importance for the history been prepared for and made possible by the scientific of the map was the growth of Alexandria as a major progress of the fourth century. Eudoxus, as we have seen, center of learning, far surpassing in this respect the had already formulated the geocentric hypothesis in Macedonian court at Pella. It was at Alexandria that mathematical models; and he had also translated his Euclid's famous school of geometry flourished in the concepts into celestial globes that may be regarded as reign of Ptolemy II Philadelphus (285-246 B.C.). And it anticipating the sphairopoiia. 1 By the beginning of the was at Alexandria that this Ptolemy, son of Ptolemy I Hellenistic period there had been developed not only the Soter, a companion of Alexander, had founded the li­ various celestial globes, but also systems of concentric brary, soon to become famous throughout the Mediter­ spheres, together with maps of the inhabited world that ranean world.
    [Show full text]
  • 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.
    [Show full text]
  • 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].
    [Show full text]
  • 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.
    [Show full text]
  • Volume 6 (2006) 1–16
    FORUM GEOMETRICORUM A Journal on Classical Euclidean Geometry and Related Areas published by Department of Mathematical Sciences Florida Atlantic University b bbb FORUM GEOM Volume 6 2006 http://forumgeom.fau.edu ISSN 1534-1178 Editorial Board Advisors: John H. Conway Princeton, New Jersey, USA Julio Gonzalez Cabillon Montevideo, Uruguay Richard Guy Calgary, Alberta, Canada Clark Kimberling Evansville, Indiana, USA Kee Yuen Lam Vancouver, British Columbia, Canada Tsit Yuen Lam Berkeley, California, USA Fred Richman Boca Raton, Florida, USA Editor-in-chief: Paul Yiu Boca Raton, Florida, USA Editors: Clayton Dodge Orono, Maine, USA Roland Eddy St. John’s, Newfoundland, Canada Jean-Pierre Ehrmann Paris, France Chris Fisher Regina, Saskatchewan, Canada Rudolf Fritsch Munich, Germany Bernard Gibert St Etiene, France Antreas P. Hatzipolakis Athens, Greece Michael Lambrou Crete, Greece Floor van Lamoen Goes, Netherlands Fred Pui Fai Leung Singapore, Singapore Daniel B. Shapiro Columbus, Ohio, USA Steve Sigur Atlanta, Georgia, USA Man Keung Siu Hong Kong, China Peter Woo La Mirada, California, USA Technical Editors: Yuandan Lin Boca Raton, Florida, USA Aaron Meyerowitz Boca Raton, Florida, USA Xiao-Dong Zhang Boca Raton, Florida, USA Consultants: Frederick Hoffman Boca Raton, Floirda, USA Stephen Locke Boca Raton, Florida, USA Heinrich Niederhausen Boca Raton, Florida, USA Table of Contents Khoa Lu Nguyen and Juan Carlos Salazar, On the mixtilinear incircles and excircles,1 Juan Rodr´ıguez, Paula Manuel and Paulo Semi˜ao, A conic associated with the Euler line,17 Charles Thas, A note on the Droz-Farny theorem,25 Paris Pamfilos, The cyclic complex of a cyclic quadrilateral,29 Bernard Gibert, Isocubics with concurrent normals,47 Mowaffaq Hajja and Margarita Spirova, A characterization of the centroid using June Lester’s shape function,53 Christopher J.
    [Show full text]
  • 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.
    [Show full text]
  • 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
    [Show full text]