Integral apollonian circle packings any integral apollonian packing is either bounded or lies between two parallel lines. Integral apollonian packings mathematical association of. Skein relations for spin networks, modified, journal of knot theory and its ramifications 27 7 2018 21 pages, arxiv. Bianchi groups and apollonian circle packings mentor. Spherical and hyperbolic packings, nicholas eriksson and jeffrey c. Apollonian circle packing this is an apollonian circle packing. Apollonian circle packing heres how we construct it. Alex kontorovich devora chait dimacs summer reu june 4, 2018.
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. Take three circles of any size for example, a quarter, a nickel, and a dime and arrange them so that each. This paper studies number theoretic properties of the set of integer curvatures appearing in such packings. This paper studies number theoretic properties of the set of integer curvatures appearing in such. Integral apollonian packings peter sarnak maa lecture. Geometry and arithmetic of crystallographic sphere packings. Preprint p9810029 of the department of physics, uiuc, urbana, il61801, 1998. Sep 11, 2000 apollonian circle packings arise by repeatedly filling the interstices between mutually tangent circles with further tangent circles.
A very nice property of the apollonian packings in the plane is that they can be modeled as iterated re ections on the initial. Table of contents 1 what is an apollonian circle packing. 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. Apollonian circle packings mark pollicott circle packings are a particularly elegant and simple way to construct quite complicated and elaborate sets in the plane. Indeed, several longstanding problems in analytic number theory. Pdf andrew odlyzko web page the zeta function of the beta transformation, leopold flatto, jeffrey c. 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.
The original publication will be available at springers website. 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. Repeat this process for the six new curvilinear triangles, and so on. There are many papers in the literature dealing with apollonian packings and their generalizations. Sphere packings and number theory alisa cui, devora chait, zachary stier mentor. An apollonian circle packing is any packing of circles constructed recursively from an initial configuration of four mutually tangent circles by the. It is named after greek mathematician apollonius of perga.
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. There are also other unbounded apollonian packings containing either only one line or no line at all. We introduce super packings as orbits of the super apollonian group, starting from a given descartes con. She is a student of homogeneous dynamics and has worked extensively on counting and equidistribution for apollonian circle packings, sierpinski carpets and schottky dances. We give an overview of various counting problems for apol. Counting problems in apollonian packings american mathematical. A good introduction to number theoretic questions related to apollonian packings can be found in 38. Number theory apollonian circle packings arise by repeatedly filling the interstices between. Template of the apollonian window pdf for printing. Literally this math has been around for 2200 years. From apollonian circle packings to fibonacci numbers. 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. Euclidean apollonian circle packing in the plane is.
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. Root quadruples and number theory theorem glmwy 2003 1 for each n 1 there are. The radii of an apollonian circle packing are computed from the initial curvatures e. Arithmetic properties of apollonian circle packings elena. Sphere packings and number theory zachary stier mentor. Fuchs pdf list of references for the lecture series on arithmetic of acps e. 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. There are infinitely many different integral packings. Introduction to number theory and apollonian circle packings.
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. These packings define a new class of thin groups called k apollonian groups. How many circles are there with curvatures bounded by t. Citeseerx document details isaac councill, lee giles, pradeep teregowda. Although there are many problems in understanding all of the individual. Journal of number theory 100 2003 145 apollonian circle packings.
Counting problems for apollonian circle packings 1 2. However, the diophantine questions are quite recent and are raised. 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 place two tangent circles of radius 12 inside and tangent to a circle of radius 1. Spherical and hyperbolic packings article pdf available in the ramanujan journal 143 april 2004 with 128 reads. Draw your own apollonian window recipe plus the data for the first seven hundred circles. One can study apollonian circle packings from many different angles various properties of the packings are investigated in a beautiful. Whenever two of the largest five circles in the gasket have the same curvature, that gasket will have d 1. The residual set dimension of the apollonian packing. 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. Crystallographic packings give us a lot to work with, but we cant. Play with apollonian gasket of the first dozen circles. To begin our story about apollonian circle packings, we consider four mutually tangent circles, one.
Introduction to apollonian circle packings tangent baking. Symmetry of integral apollonian circle packings no symmetry. Spin structure of the apolonian window a poster, version 0. She is currently the abraham robinson professor of. Lagarias university of michigan ann arbor, mi 48109 february 1, 2005 abstract apollonian circle packings arise by repeatedly. Number theory on bubbles evan warner stanford university may 22, 2014 evan warner stanford university apollonian circle packings may 22, 2014 1 59. Plus a simple algorithm for drawing the dedekind tessellation. The apollonian structure of integer superharmonic matrices. Ninth quadrennial international conference on graph theory, combinatorics, algorithms, and applications. Apollonian packing can be transformed to a strongly integral one by a euclidean motion. It is possible for every circle in such a packing to have. This paper studies numbertheoretic properties of the set of integer curvatures appearing in such packings. Degrees and distances in random and evolving apollonian.
They include algebra through the apollonian group, analysis through enumeration and density questions, and number theory through questions on curvatures that are prime. A very nice property of the apollonian packings in the plane is that. As with many problems in number theory, the basic questions here are easy to state but dif. I the precise value of is unknown, but approximately, 1. There have been several generalizations of these packings. These packings and all this theory come from one dude, apollonius of perga, who wrote a bunch of math books back around 200 bc. Hee oh, born 1969 is a south korean mathematician who works in dynamical systems. Apollonian packings, number theory, quadratic forms, sieve methods, circle method. X of distinct a proof of the positive density conjecture for integer apollonian circle packings 3. Number theory and the circle packings of apollonius. 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. If none of the curvatures are repeated within the first five, the gasket contains no symmetry, which is represented by symmetry group c 1.
Arithmetic properties of apollonian circle packings uc davis. An authorcreated copy is available at archive math. This particular packing has already played a role in number theory. Jan 08, 2019 this paper studies generalizations of the classical apollonian circle packing, a beautiful geometric fractal that has a surprising underlying integral structure. 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. 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. Michigan, june, 2000, invited talk, \generalized parking functions, tree.
Apollonian circle packings arise by repeatedly filling the interstices between mutually tangent circles with further tangent circles. For a bounded apollonian packing p, there are only nitely many circles of radius bigger than a given number. Number theory, graph theory and geometric statistics theorem bourgainkontorovich, inventiones, 2014 almost every integer in the admissible congruence classes mod. One way to restate the condition of integral circle. Number theory apollonian circle packings arise by repeatedly filling the interstices between mutually tangent circles with further tangent circles. One systematically constructs a countable family of tangent circles whose radii tend to zero. Spherical and hyperbolic packings nicholas eriksson university of california at berkeley berkeley, ca 94720 je. She has made contributions to dynamics and its connections to number theory. As with many problems in number theory the basic questions here are easy to state but di. 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. On the other hand, it is not immediately clear that fullplane packings exist. Pdf apollonian circle packings arise by repeatedly filling the interstices between mutually tangent circles with further tangent circles.
1464 949 634 626 1274 649 65 327 1029 1515 1153 251 1475 1246 1243 1326 1248 1483 1059 291 740 63 211 611 164 757 1507 371 235 1341 1532 269 1136 678 135 162 234 605 458 1040 15 509 238 180