One systematically constructs a countable family of tangent circles whose radii tend to zero. Template of the apollonian window pdf for printing. It is named after greek mathematician apollonius of perga. There have been several generalizations of these packings.
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. 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. Preprint p9810029 of the department of physics, uiuc, urbana, il61801, 1998. X of distinct a proof of the positive density conjecture for integer apollonian circle packings 3. Alex kontorovich devora chait dimacs summer reu june 4, 2018. Integral apollonian packings mathematical association of. Symmetry of integral apollonian circle packings no symmetry. 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. For a bounded apollonian packing p, there are only nitely many circles of radius bigger than a given number.
Number theory and the circle packings of apollonius. Introduction to number theory and apollonian circle packings. 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. Spherical and hyperbolic packings nicholas eriksson university of california at berkeley berkeley, ca 94720 je.
A descartes configuration is a set of four mutually touching circles with distinct tangents. Sphere packings and number theory zachary stier mentor. Apollonian circle packing heres how we construct it. Number theory apollonian circle packings arise by repeatedly filling the interstices between mutually tangent circles with further tangent circles. Number theory apollonian circle packings arise by repeatedly filling the interstices between. Play with apollonian gasket of the first dozen circles. An apollonian circle packing is any packing of circles constructed recursively from an initial configuration of four mutually tangent circles by the. 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.
Apollonian packing can be transformed to a strongly integral one by a euclidean motion. Spherical and hyperbolic packings article pdf available in the ramanujan journal 143 april 2004 with 128 reads. 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. Sep 11, 2000 apollonian circle packings arise by repeatedly filling the interstices between mutually tangent circles with further tangent circles. Whenever two of the largest five circles in the gasket have the same curvature, that gasket will have d 1. She is currently the abraham robinson professor of. However, the diophantine questions are quite recent and are raised. Introduction to apollonian circle packings tangent baking. Number theory on bubbles evan warner stanford university may 22, 2014 evan warner stanford university apollonian circle packings may 22, 2014 1 59. As with many problems in number theory, the basic questions here are easy to state but dif.
An authorcreated copy is available at archive math. We give an overview of various counting problems for apol. Michigan, june, 2000, invited talk, \generalized parking functions, tree. As with many problems in number theory the basic questions here are easy to state but di. Apollonian packings, number theory, quadratic forms, sieve methods, circle method. 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. If none of the curvatures are repeated within the first five, the gasket contains no symmetry, which is represented by symmetry group c 1. A very nice property of the apollonian packings in the plane is that they can be modeled as iterated re ections on the initial.
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. Although there are many problems in understanding all of the individual. A good introduction to number theoretic questions related to apollonian packings can be found in 38. There are infinitely many different integral packings. I am reading this paper apollonian circle packings. Euclidean apollonian circle packing in the plane is. Geometry and arithmetic of crystallographic sphere packings. Integral apollonian circle packings any integral apollonian packing is either bounded or lies between two parallel lines. Draw your own apollonian window recipe plus the data for the first seven hundred circles. How many circles are there with curvatures bounded by t. Repeat this process for the six new curvilinear triangles, and so on.
Journal of number theory 100 2003 145 apollonian circle packings. Sphere packings and number theory alisa cui, devora chait, zachary stier mentor. By a theorem of apollonius, place unique circles in the lines. Ninth quadrennial international conference on graph theory, combinatorics, algorithms, and applications. She has made contributions to dynamics and its connections to number theory. Arithmetic properties of apollonian circle packings elena. Root quadruples and number theory theorem glmwy 2003 1 for each n 1 there are. Arithmetic properties of apollonian circle packings uc davis. One way to restate the condition of integral circle. The radii of an apollonian circle packing are computed from the initial curvatures e.
Fuchs pdf list of references for the lecture series on arithmetic of acps e. Indeed, several longstanding problems in analytic number theory. 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. The apollonian structure of integer superharmonic matrices. This particular packing has already played a role in number theory. She is a student of homogeneous dynamics and has worked extensively on counting and equidistribution for apollonian circle packings, sierpinski carpets and schottky dances. There are also other unbounded apollonian packings containing either only one line or no line at all. 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. Literally this math has been around for 2200 years. Skein relations for spin networks, modified, journal of knot theory and its ramifications 27 7 2018 21 pages, arxiv. 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.
Spherical and hyperbolic packings, nicholas eriksson and jeffrey c. Introduction place two tangent circles of radius 12 inside and tangent to a circle of radius 1. They include algebra through the apollonian group, analysis through enumeration and density questions, and number theory through questions on curvatures that are prime. Counting problems in apollonian packings american mathematical. It is possible for every circle in such a packing to have. I the precise value of is unknown, but approximately, 1. Integral apollonian packings peter sarnak maa lecture. Pdf andrew odlyzko web page the zeta function of the beta transformation, leopold flatto, jeffrey c. Plus a simple algorithm for drawing the dedekind tessellation. To begin our story about apollonian circle packings, we consider four mutually tangent circles, one. Spin structure of the apolonian window a poster, version 0. These packings define a new class of thin groups called k apollonian groups.
Hee oh, born 1969 is a south korean mathematician who works in dynamical systems. Citeseerx document details isaac councill, lee giles, pradeep teregowda. 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. A very nice property of the apollonian packings in the plane is that. From apollonian circle packings to fibonacci numbers. Pdf apollonian circle packings arise by repeatedly filling the interstices between mutually tangent circles with further tangent circles. Lagarias university of michigan ann arbor, mi 48109 february 1, 2005 abstract apollonian circle packings arise by repeatedly. Apollonian circle packing this is an apollonian circle packing. One can study apollonian circle packings from many different angles various properties of the packings are investigated in a beautiful.
The residual set dimension of the apollonian packing. There are many papers in the literature dealing with apollonian packings and their generalizations. Counting problems for apollonian circle packings 1 2. Jan 08, 2019 this paper studies generalizations of the classical apollonian circle packing, a beautiful geometric fractal that has a surprising underlying integral structure.
On the other hand, it is not immediately clear that fullplane packings exist. Take three circles of any size for example, a quarter, a nickel, and a dime and arrange them so that each. The original publication will be available at springers website. These packings and all this theory come from one dude, apollonius of perga, who wrote a bunch of math books back around 200 bc. Crystallographic packings give us a lot to work with, but we cant. Bianchi groups and apollonian circle packings mentor.
Number theory, graph theory and geometric statistics theorem bourgainkontorovich, inventiones, 2014 almost every integer in the admissible congruence classes mod. Table of contents 1 what is an apollonian circle packing. This paper studies number theoretic properties of the set of integer curvatures appearing in such. 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. 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. Apollonian circle packings mark pollicott circle packings are a particularly elegant and simple way to construct quite complicated and elaborate sets in the plane. This paper studies number theoretic properties of the set of integer curvatures appearing in such packings. Degrees and distances in random and evolving apollonian. We introduce super packings as orbits of the super apollonian group, starting from a given descartes con. This paper studies numbertheoretic properties of the set of integer curvatures appearing in such packings.
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