From apollonian circle packings to fibonacci numbers je. With different people bringing different perspectives to the same mathematical structures, apollonian circle packings have quickly developed into a rich, promising field of study. This paper studies numbertheoretic properties of the set of integer curvatures appearing in such packings. The apollonian and dionysian is a philosophical and literary concept and dichotomydialectic, based on apollo and dionysus in greek mythology. The following result is fundamental in the study of apollonian circle packings and provides the. Root quadruples and number theory theorem glmwy 2003 1 for each n 1 there are. This has a similar number theoretic property as the original apollonian packing, namely if the three initial bends are integral and wis twice a square number then all bends will be integral.
Pdf irreducible apollonian configurations and packings. One of the best books i have ever read, by one of the smartest guys i know andy frisella, ceo of 1st phorm and s2 supplements third circle theory is not just a book about business and life, it is an analysis that will reshape the way you think about the path that your life is taking, why is it taking it, and what you can do to affect it. Oct 30, 2000 we show that up to scale there are exactly 8 different geometric strongly integral super packings, and that each contains a copy of every integral apollonian circle packing also up to scale. A theorem on apollonian circle packings for every integral apollonian circle packing there is a uniqueminimalquadrupleofintegercurvatures,a,b,c,d,satisfyinga. Who can date the most familiar of circle packings, the pennypacking seen in the background of figure1. Apollonian circle packings, i 549 2 2 3 15 38 71 114 167 110 107 38 71 114 110 107 35 63 198 198 99 143 195. Arithmetic properties of apollonian circle packings elena fuchs. This survey article is an expanded version of my lecture notes prepared for the th takagi lectures given at rims, kyoto in the fall of 20. Introduction to apollonian circle packings tangent baking.
By the following theorem of descartes, the curvatures of any four. Fractalsapollonian fractals wikibooks, open books for an. We show that the super apollonian group has finite volume in the group of all automorphisms of the parameter space of descartes configurations, which. From apollonian circle packings to fibonacci numbers. These packings and all this theory come from one dude, apollonius of perga, who wrote a bunch of math books back around 200 bc. Construction of apollonian circle packings beginning with4 mutually tangent circles, we can keep adding newer circles tangent to three of the previous circles, provided by the apollonius theorem. Positive curvature indicates that all other circles are externally. Apollonian definition, etymology and usage, examples and. Literally this math has been around for 2200 years. Abstractin this paper we study the integral properties of apollonian 3 circle packings, which are variants of the standard apollonian circle packings. The set of all such circle packings for which the curvature plus curvature times center is congruent to 1 modulo 2 is called the standard supergasket. Number theory and the circle packings of apollonius peter. Then it discusses integral apollonian packings those with all. Apollonian circle packings come from the apollonius problem.
This paper studies numbertheoretic properties of the set of. Apollonian circle packings arise by repeatedly filling the interstices between mutually tangent circles with further. Counting problems for apollonian circle packings 1 2. We show that up to scale there are exactly 8 different geometric strongly integral superpackings, and that each contains a copy of every integral apollonian circle packing also up to scale. Superapollonian group and integral packings article pdf available in discrete and computational geometry 351. It is possible for every circle in such a packing to have integer radius of curvature, and we call such a packing an \em integral apollonian circle packing. But acps encode fascinating information of an entirely different flavor. Curvature the curvature of a circle bend is defined to be the inverse of its radius. Given 4 mutually tangent circles in a packing whose curvatures are a1,a2,a3,a4. Conics introduced the terms parabola, ellipse, and hyperbola.
These are very challenging problems in number theory, graham says. The apollonian circles are defined in two different ways by a line segment denoted cd each circle in the first family the blue circles in the figure is associated with a positive real number r, and is defined as the locus of points x such that the ratio of distances from x to c and to d equals r. Into one of these we can pack three new circles, touching each other, with each new circle touching two of the original circles. It is possible for every circle in such a packing to have integer radius of curvature, and we call such a packing an integral apollonian circle packing. This survey article is an expanded version of my lecture notes prepared for the th takagi lec. His famous book on conics introduced the terms parabola, ellipse and hyperbola. This creates smaller gaps, which can be filled with smaller circles, and so on.
Zero curvature gives a line circle with infinite radius. This paper studies numbertheoretic properties of the set of integer curvatures appearing in such. Some western philosophical and literary figures have invoked this dichotomy in critical and creative works, most notably friedrich nietzsche and later followers. Even the apollonian gasket a has a history stretching across more than two.
Like many problems in number theory, the questions that arise from packing the plane with mutually tangent circles are easy to formulate but difficult to answer. Negative curvature indicates that all other circles are internally tangent to that circle. The institute is located at 17 gauss way, on the university of california, berkeley campus, close to grizzly peak, on the. Sep 11, 2000 apollonian circle packings arise by repeatedly filling the interstices between mutually tangent circles with further tangent circles. Apollonian circle packings arise by repeatedly filling the interstices between mutually tangent circles with further tangent circles. We show that the superapollonian group has finite volume in the group of all automorphisms of the parameter space of descartes configurations, which. One can study apollonian circle packings from many different angles various properties of the packings are investigated in a beautiful series of papers by graham, lagarias, mallows, wilkes, and yan see 24, 21, 22, 23. Bounded apollonian circle packings acps are constructed by repeatedly inscribing circles into the triangular interstices of a con. Abstractin this paper we study the integral properties of apollonian 3circle packings, which are variants of the standard apollonian circle packings. This paper studies number theoretic properties of the set of integer curvatures appearing in such packings. Apr 21, 2001 insights and proofs, however, are hard to come by.
Such packings are certainly of interest in classical geometry for. For theorem a, one needs the theory of automorphic representations for the full apollonian group, as hinted to at the end of 3. Integral apollonian packings michael cap khoury abstract for 24 sept 2009 begin with three circles inside a larger circle such that each pair of gaps, you can draw a circle which is tangent to three existing circles. The spectral geometry of the apollonian group controls the number theory of such integers. To begin our story about apollonian circle packings, we consider four mutually. Msri recent developments in analytic number theory.
Yan gives a joint characterization of the circle curvatures inverse radii and coordinates. That the circles tangent to one fixed circle in an apollonian circle packing had curvatures that were values of a translated quadratic form was observed in 10, 16,17 and used as a principle tool. We show that for a given bounded apollonian circle packing, there exists a constant such that the number of circles of curvature at most is asymptotic to as. Compared with previous operatorbased methods, we make two improvements in this paper. From apollonian circle packings to fibonacci numbers mathematics. Those centers in the unit square are in onetoone correspondence with. Aug 23, 2011 like many problems in number theory, the questions that arise from packing the plane with mutually tangent circles are easy to formulate but difficult to answer. Skein relations for spin networks, modified, journal of knot theory and its ramifications 27 7 2018 21 pages, arxiv.
Drawing the apollonian gasket with common lisp and vecto by luis diego fallas leibnitz packing by takaya iwamoto with program in autolisp apollonian gasket by paul bourke in basic and c how to pack n circles inside unit circle by erich friedman svg math animation example. The apollonian group a deeper study of an apollonian packing is facilitated by introducing the symmetry group awhich is called the apollonian group. We show that up to scale there are exactly 8 different geometric strongly integral super packings, and that each contains a copy of every integral apollonian circle packing also up to scale. Integral apollonian packings peter sarnak maa lecture. It is possible for every circle in such a packing to have integer radius of curvature, and we call such a packing an \\em integral apollonian circle packing. Specifically, we study the reduction theory, formulate a localglobal conjecture, and prove a density one version of this conjecture. We give an overview of various counting problems for apollonian circle packings, which turn out to be related to problems in dynamics and number theory for thin groups. The talk first discusses apollonian circle packings.
An apollonian circle packing is one of the most of beautiful circle packings. Counting problems in apollonian packings american mathematical. Notes from matt clays minicourse on the guirardel core in outer space the fault in our stars, cantors diagonalization. One can study an apollonian circle packing acp from many different angles. An apollonian circle packing is called strongly integral if every circle has curvature times center a gaussian integer. Given three objects, draw a circle mutually tangent to all three. Volume 1, number 1, 127, 2010 a generalization of apollonian packing of circles gerhard guettler and colin mallows three circles touching one another at distinct points form two curvilinear triangles. This survey article is an expanded version of my lecture notes prepared for the th takagi lectures given at rims, kyoto in. Journal of number theory 100 2003 145 apollonian circle packings. A transfer operator method is proposed to calculate d h, the hausdorff dimension of the apollonian gasket. In euclidean space it is possible for every circle in such a packing to have integer radius of curvature, and we call such a packing an integral apollonian circle packing.