Apollonian circle packings mark pollicott circle packings are a particularly elegant and simple way to construct quite complicated and elaborate sets in the plane. 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. Number theory and the circle packings of apollonius. Citeseerx document details isaac councill, lee giles, pradeep teregowda.
Table of contents 1 what is an apollonian circle packing. This paper studies generalizations of the classical apollonian circle packing, a beautiful geometric fractal that has a surprising underlying integral structure. Journal of number theory 100 2003 145 apollonian circle packings. Continuing this process indefinitely, we arrive at an infinite circle packing called an. Theorem fuchsstangez a density one theorem holds for all known integral circle packings. As this ratio is not rational, no integral apollonian circle packings possess this d3 symmetry, although many packings come close. Counting problems in apollonian packings elena fuchs abstract. The second in a series on apollonian circle packings beginning with math. I was led to these types of circle packings a couple of months ago after i learned about the descartes circle.
Root quadruples and number theory theorem glmwy 2003 1 for each n 1 there are. Lagarias university of michigan ann arbor, mi 48109 february 1, 2005 abstract apollonian circle packings arise by repeatedly. The ford circle associated with the fraction is denoted by or. In my talk, i will connect this question to the algebra of infinite root systems. In addition, the line is counted as a ford circle it can be thought of as the ford circle associated with infinity, which is the case. I am reading this paper apollonian circle packings. Euclidean apollonian circle packing in the plane is. Plus a simple algorithm for drawing the dedekind tessellation. As with many problems in number theory the basic questions here are easy to state but di. Introduction to number theory and apollonian circle packings. Number theory on bubbles evan warner stanford university may 22, 2014 evan warner stanford university apollonian circle packings may 22, 2014 1 59. Take three circles of any size for example, a quarter, a nickel, and a dime and arrange them so that each.
Although there are many problems in understanding all of the individual. This survey article is an expanded version of my lecture notes prepared for the th takagi lec. Computing the centers of apollonian circle packings. Apollonian packings and of kacmoody groups department. Alex kontorovich devora chait dimacs summer reu june 4, 2018. Two different ford circles are either disjoint or tangent to one another. An integral apollonian circle packing is an apollonian circle packing in which every circle has an integer curvature. Beginning with the construction of apollonian circle packings, we will discuss recent work on the asymptotic counting and distribution of circles in apollonian circle packings with respect to di erent conformal metrics. There are infinitely many different integral packings. Apollonian circle packings arise by repeatedly filling the interstices between mutually tangent circles. 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.
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. Moreover, if the four starting circles have integer curvatures, then every circle in the packing has an integer curvatures. Apollonian circle packing heres how we construct it. From apollonian circle packings to fibonacci numbers. 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 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. One systematically constructs a countable family of tangent circles whose radii tend to zero. The apollonian structure of integer superharmonic matrices. Preprint, 45 pages, submitted to a special volume for margulis pdf topological proof of benoistquints orbit closure theorem for sod, 1.
Fuchs pdf list of references for the lecture series on arithmetic of acps e. 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 four mutually tangent circles with further tangent. On a diophantine equation that generates all integral. Results on integer apollonian packings are joint work with ron graham, colin mallows, allan wilks, catherine yan, glmwy some of the work on fibonacci numbers is an ongoing joint project with jon bober. Skein relations for spin networks, modified, journal of knot theory and its ramifications 27 7 2018 21 pages, arxiv. 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. Apollonian circle packings and closed horospheres on hyperbolic 3manifolds. In this case the ratio of the curvature of the bounding circle to the three inner circles is 2 v 3. This paper studies number theoretic properties of the set of integer.
The apollonian structure of integer superharmonic matrices pages 167 from volume 186 2017. The question of which integers appear as curvatures in a given packing has inspired exciting work in number theory over the last 20 years. 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. Slides for my two talks given at yale, on the geometric and number theoretic aspects of circle packings, 2017 talk1,talk2 a short note on a theorem of mohammadioh, 2017 slides for the doob colloquium talk at uiuc, 2017 slides for the uc davis talk on the gap distribution of thin schottky group orbits, 2016. Symmetry of integral apollonian circle packings no symmetry. It turns out that there exist an infinite number of integral apollonian packings, where the curvature of every circle disk is an integer recall that four pairwise tangent circles are said to be in the descartes configuration. Spherical and hyperbolic packings nicholas eriksson university of california at berkeley berkeley, ca 94720 je. There is a ford circle associated with every rational number. If none of the curvatures are repeated within the first five, the gasket contains no symmetry, which is represented by symmetry group c 1. Theorem z, crelle 2015 for any primitive integral apollonian3 circle packing, the local obstruction is at 8, and almost every admissible integer is a curvature.
This paper studies number theoretic properties of the set of integer curvatures appearing in such packings. 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. In 1967 wilker showed that all osculatory circle packings p, which include all apollonian circle packings, have the same exponent ep, which we call the osculatory packing exponent e. We give an overview of various counting problems for apol. An authorcreated copy is available at archive math. Arithmetic properties of apollonian circle packings elena. Spherical and hyperbolic packings article pdf available in the ramanujan journal 143 april 2004 with 128 reads. Geometry and arithmetic of crystallographic sphere packings. Whenever two of the largest five circles in the gasket have the same curvature, that gasket will. The original publication will be available at springers website.
Integral apollonian packings peter sarnak maa lecture. Prime components of apollonian circle packings philadelphia area number theory seminar. 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. Number theory, graph theory and geometric statistics theorem bourgainkontorovich, inventiones, 2014 almost every integer in the admissible congruence classes mod. Apollonian circle packing this is an apollonian circle packing. Sphere packings and number theory alisa cui, devora chait, zachary stier mentor. 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.