Apollonian packings number theory pdf

Integral apollonian circle packings any integral apollonian packing is either bounded or lies between two parallel lines. Introduction to number theory and apollonian circle packings in this class, we will explore a problem inspired by an old greek construction which, in its simplicity, lends itself to surprisingly many still unanswered questions. It is possible for every circle in such a packing to have. Lagarias university of michigan ann arbor, mi 48109 february 1, 2005 abstract apollonian circle packings arise by repeatedly.

The original publication will be available at springers website. A very nice property of the apollonian packings in the plane is that they can be modeled as iterated re ections on the initial. From apollonian circle packings to fibonacci numbers. Spherical and hyperbolic packings article pdf available in the ramanujan journal 143 april 2004 with 128 reads. Bianchi groups and apollonian circle packings mentor. Alex kontorovich devora chait dimacs summer reu june 4, 2018. One way to restate the condition of integral circle. Integral apollonian packings mathematical association of. Citeseerx document details isaac councill, lee giles, pradeep teregowda. This paper studies number theoretic properties of the set of integer curvatures appearing in such. Crystallographic packings give us a lot to work with, but we cant. 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.

Counting problems for apollonian circle packings 1 2. They include algebra through the apollonian group, analysis through enumeration and density questions, and number theory through questions on curvatures that are prime. One of the most essential tools in number theory is the theory of automorphic forms and lfunctions see 45 and 31 for an elegant survey of various applications. Introduction to number theory and apollonian circle packings.

Pdf apollonian circle packings arise by repeatedly filling the interstices between mutually tangent circles with further tangent circles. I the precise value of is unknown, but approximately, 1. 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. Template of the apollonian window pdf for printing. Arithmetic properties of apollonian circle packings elena. Sep 11, 2000 apollonian circle packings arise by repeatedly filling the interstices between mutually tangent circles with further tangent circles. I was led to these types of circle packings a couple of months ago after i learned about the descartes circle theorem and the the descartes quadratic form. Degrees and distances in random and evolving apollonian. To begin our story about apollonian circle packings, we consider four mutually tangent circles, one. Euclidean apollonian circle packing in the plane is. We introduce super packings as orbits of the super apollonian group, starting from a given descartes con. 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. Preprint p9810029 of the department of physics, uiuc, urbana, il61801, 1998. Indeed, several longstanding problems in analytic number theory.

Apollonian circle packings mark pollicott circle packings are a particularly elegant and simple way to construct quite complicated and elaborate sets in the plane. There are many papers in the literature dealing with apollonian packings and their generalizations. This is the point at which this article takes off it leads to connections with several other areas of mathematics, and the author acquaints the reader with several of these. The residual set dimension of the apollonian packing. She is currently the abraham robinson professor of. As with many problems in number theory the basic questions here are easy to state but di. Number theory on bubbles evan warner stanford university may 22, 2014 evan warner stanford university apollonian circle packings may 22, 2014 1 59. Geometry and arithmetic of crystallographic sphere packings. On the one hand, infinitely many such generalized objects exist, but on the other, they may, in principle, be completely classified, as they fall into, only finitely, many families, all in bounded dimensions. Play with apollonian gasket of the first dozen circles. Skein relations for spin networks, modified, journal of knot theory and its ramifications 27 7 2018 21 pages, arxiv.

She is a student of homogeneous dynamics and has worked extensively on counting and equidistribution for apollonian circle packings, sierpinski carpets and schottky dances. 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. This paper studies number theoretic properties of the set of integer curvatures appearing in such packings. Ninth quadrennial international conference on graph theory, combinatorics, algorithms, and applications.

Introduction place two tangent circles of radius 12 inside and tangent to a circle of radius 1. 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. Repeat this process for the six new curvilinear triangles, and so on. Unbounded apollonian circle packings, selfsimilarity and residual points 5 a this is a bounded packing 1 1 1 1 4 4 4 4 4 4 4 4 9 9 9 9 9 9 9 9 b this is a strip packing c this is an unbounded packing d this is a half plane packing and doyle in 2. There are also other unbounded apollonian packings containing either only one line or no line at all. Number theory apollonian circle packings arise by repeatedly filling the interstices between mutually tangent circles with further tangent circles. These packings and all this theory come from one dude, apollonius of perga, who wrote a bunch of math books back around 200 bc. Jan 08, 2019 this paper studies generalizations of the classical apollonian circle packing, a beautiful geometric fractal that has a surprising underlying integral structure. If none of the curvatures are repeated within the first five, the gasket contains no symmetry, which is represented by symmetry group c 1. This paper studies numbertheoretic properties of the set of integer curvatures appearing in such packings. The radii of an apollonian circle packing are computed from the initial curvatures e. Introduction to apollonian circle packings tangent baking. It is named after greek mathematician apollonius of perga.

Table of contents 1 what is an apollonian circle packing. Take three circles of any size for example, a quarter, a nickel, and a dime and arrange them so that each. How many circles are there with curvatures bounded by t. Sphere packings and number theory zachary stier mentor. Apollonian circle packing this is an apollonian circle packing.

These packings define a new class of thin groups called k apollonian groups. Root quadruples and number theory theorem glmwy 2003 1 for each n 1 there are. Spin structure of the apolonian window a poster, version 0. There have been several generalizations of these packings. An apollonian circle packing is any packing of circles constructed recursively from an initial configuration of four mutually tangent circles by the. Number theory apollonian circle packings arise by repeatedly filling the interstices between.

Pdf andrew odlyzko web page the zeta function of the beta transformation, leopold flatto, jeffrey c. However, the diophantine questions are quite recent and are raised. Number theory, graph theory and geometric statistics theorem bourgainkontorovich, inventiones, 2014 almost every integer in the admissible congruence classes mod. Although there are many problems in understanding all of the individual.

Counting problems in apollonian packings american mathematical. Symmetry of integral apollonian circle packings no symmetry. As with many problems in number theory, the basic questions here are easy to state but dif. We give an overview of various counting problems for apol. Fuchs pdf list of references for the lecture series on arithmetic of acps e. Spherical and hyperbolic packings nicholas eriksson university of california at berkeley berkeley, ca 94720 je.

Literally this math has been around for 2200 years. Arithmetic properties of apollonian circle packings uc davis. This particular packing has already played a role in number theory. Hee oh, born 1969 is a south korean mathematician who works in dynamical systems. A very nice property of the apollonian packings in the plane is that. There are infinitely many different integral packings. By a theorem of apollonius, place unique circles in the lines. A descartes configuration is a set of four mutually touching circles with distinct tangents. Apollonian packing can be transformed to a strongly integral one by a euclidean motion. Draw your own apollonian window recipe plus the data for the first seven hundred circles.

Journal of number theory 100 2003 145 apollonian circle packings. An authorcreated copy is available at archive math. On the other hand, it is not immediately clear that fullplane packings exist. Sphere packings and number theory alisa cui, devora chait, zachary stier mentor. For a bounded apollonian packing p, there are only nitely many circles of radius bigger than a given number. Whenever two of the largest five circles in the gasket have the same curvature, that gasket will have d 1. We define certain natural subgroups whose orbits generalise apollonian circle packings, and show that sk, considered with orientations, is a disjoint union of all of these k apollonian packings. Apollonian packings are a wellknown object in mathematics formed by starting with a set of three mutually tangent circles and then repeatedly lling in the \holes with maximally sized circles. In mathematics, an apollonian gasket or apollonian net is a fractal generated starting from a triple of circles, each tangent to the other two, and successively filling in more circles, each tangent to another three. Number theory and the circle packings of apollonius. I am reading this paper apollonian circle packings. Apollonian circle packings arise by repeatedly filling the interstices between mutually tangent circles with further tangent circles. Plus a simple algorithm for drawing the dedekind tessellation.

One systematically constructs a countable family of tangent circles whose radii tend to zero. 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. She has made contributions to dynamics and its connections to number theory. X of distinct a proof of the positive density conjecture for integer apollonian circle packings 3. Michigan, june, 2000, invited talk, \generalized parking functions, tree. The apollonian structure of integer superharmonic matrices. A good introduction to number theoretic questions related to apollonian packings can be found in 38. We describe a reduction theory which multiplies descartes quadruples by elements of this group and uses it to find a quadruple of smallest size in a given packing. Apollonian circle packing heres how we construct it. One can study apollonian circle packings from many different angles various properties of the packings are investigated in a beautiful. Apollonian packings, number theory, quadratic forms, sieve methods, circle method.