Apollonian circle packings what is an apollonian circle packing. This survey article is an expanded version of my lecture notes prepared for the th takagi lectures given at. Computing the centers of apollonian circle packings. Peter sarnak mahler lectures 2011 number theory and the circle packings of apollonius a. It is named after greek mathematician apollonius of perga. Apollonian and dionysian are terms used by friedrich. A very nice property of the apollonian packings in the plane is that they can be modeled as iterated re ections on the initial con guration. The radii of an apollonian circle packing are computed from the initial curvatures e. Selbergs 14 conjecture and implications for expansion constants of the relevant families of graphs. The bibliography, of incredible quality, achieves to make the book a reference for anyone interested in fractals.
His famous book on conics introduced the terms parabola, ellipse and hyperbola. Counting problems in apollonian packings elena fuchs abstract. For example, we might wish to count the number of points in a ball in this set, or determine for which r this set is large. Plus a simple algorithm for drawing the dedekind tessellation. This paper studies number theoretic properties of the set of integer curvatures appearing in such packings. Prime number conjecture in some experiments with integral apollonian packings by fuchssanden. Apollonian packings are not unique in the sense of. Pdf irreducible apollonian configurations and packings. An apollonian circle packing is a classical construction which is made by repeatedly inscribing circles into the triangular interstices in a descartes con guration of four mutually tangent circles. An apollonian circle packing is any packing of circles constructed recursively from an initial configuration of four mutually tangent circles by the. Apollonian packings and of kacmoody groups department. The first three in the tables happen to produce the fourth, but how were these first 3 obtained to begin with. This particular packing has already played a role in number theory.
A tisket, a tasket, an apollonian gasket american scientist. All types of form or structure are apollonian, thus, sculpture is the most apollonian of the arts, since it relies entirely on form for its effect. Introduction to apollonian circle packings tangent. Lagarias university of michigan ann arbor, mi 48109 february 1, 2005 abstract apollonian circle packings arise by repeatedly. Number theory, graph theory and geometric statistics theorem bourgainkontorovich, inventiones, 2014 almost every integer in the admissible congruence classes mod. 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. As with many problems in number theory the basic questions here are easy to state but di. Sphere packings and number theory zachary stier mentor.
Effective bisector estimate with application to apollonian. The essence of the proof of the prime number theorem is to analyse the. There are many papers in the literature dealing with apollonian packings and their generalizations. So when we want to study properties of apollonian packings, we can focus on.
Apollonian circle packing this is an apollonian circle packing. 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. Newest apolloniancirclepacking questions mathematics. 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. Dedekind tessellation interactive see also some pictures. It observed there exist infinitely many types of integral apollonian packings in which all circles had integer curvatures, with the integral structure being related to the. The theory is related by duality to kayeka sets sets of zero area containing lines in every direction. Integer appollonian packings show and tell the observable. Apollonian circle packing heres how we construct it. Root quadruples and number theory theorem glmwy 2003 1 for each n 1 there are.
Sep 11, 2000 apollonian circle packings arise by repeatedly filling the interstices between mutually tangent circles with further tangent circles. Irreducible apollonian configurations and packings. In this talk, we investigate the arithmetic properties of the collection of integers appearing in prime components of apollonian circle packings. They include algebra through the apollonian group, analysis through enumeration and density questions, and number theory through questions on curvatures that are prime. In my talk, i will connect this question to the algebra of infinite root systems. Number theory and the circle packings of apollonius peter.
We study nkg with new methods from combinatorial number theory. An apollonian configuration of circles is a collection of circles in the plane with disjoint interiors such that the complement of the interiors of the circles consists of curvilinear triangles. Peter sarnak mahler lectures 2011 number theory and the circle packings of apollonius. Apollonian sphere packing is the threedimensional equivalent of the apollonian gasket. Apollonius of perga i lived from about 262 bc to about 190 bc. I looked in some detail at your paper apollonian circle packings. Journal of number theory 100 2003 145 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. 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. An apollonian gasket can be constructed as follows. A beautifull image of the regular apollonian gasket by jos leys.
The spectral geometry of the apollonian group controls the number theory of such integers. Such packings can be described in terms of the descartes configurations they contain. Effective bisector estimate with application to apollonian circle packings ilya vinogradov. Apollonian packing can be transformed to a strongly integral one by a euclidean motion. The final chapter includes diverse examples of sets to which the general theory is applicable. Citeseerx document details isaac councill, lee giles, pradeep teregowda. So heres a quick write up background of the first third of. 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.
One systematically constructs a countable family of tangent circles whose radii tend to zero. Also, dce solutions 2,3,6,23 yield the same packing as 1,2,2,3. These types of results are also directly related to the gauss map and provide a recipe for using dynamical systems theory. These types of results are also directly related to the gauss map and provide a recipe for using dynamical systems theory to study continued fraction expansions. To obtain an integer packing one needs four integer curvatures that satisfy dce, not just three. Integral apollonian packings mathematical association of. As i read the literature on apollonian packings, an interesting cultural difference emerged between physicists and mathematicians. I start with three mutually tangent circles i draw two more circles, each of which is tangent to the. Newest apollonian circle packing questions feed subscribe to rss. From apollonian circle packings to fibonacci numbers je.
In any integral apollonian packing, there are in nitely many cs with ac prime and, better still, in nitely many pairs c. It is a matter for considerable regret that fermat, who cultivated the theory of numbers with so much success, did not leave us with the proofs of the theorems he discovered. I his famous book onconicsintroduced the terms parabola, ellipse and hyperbola. This completely characterizes the pde that determines the continuum scaling limit of. The apollonian and dionysian is a philosophical and literary concept and dichotomydialectic, based on apollo and dionysus in greek mythology. Some western philosophical and literary figures have invoked this dichotomy in critical and creative works, most notably friedrich nietzsche and later followers. Rational thought is also apollonian since it is structured and makes distinctions.
Apollonian circle packings arise by repeatedly filling the interstices between mutually tangent circles with further tangent circles. Number theory on bubbles evan warner stanford university may 22, 2014 evan warner stanford university apollonian circle packings may 22, 2014 1 59. This paper studies numbertheoretic properties of the set of integer curvatures appearing in such packings. An apollonian circle packing acp is an ancient greek construction which is made by repeatedly inscribing circles into the triangular interstices in a descartes con. But for the residual set of an apollonian circle packing, the. We introduce super packings as orbits of the super apollonian group, starting from a given descartes con. This survey article is an expanded version of my lecture notes prepared for the th takagi lec. Number theory and the circle packings of apollonius. Table of contents 1 what is an apollonian circle packing. From apollonian circle packings to fibonacci numbers. Start with three circles c 1, c 2 and c 3, each one of which is tangent to the other two in the general construction, these three circles have to be different sizes, and they must have a common tangent.
If we started with an odd number of tangent circles, then. The book finish with a magnificent list of examples of haussdorff dimension computation. Sphere packings and number theory alisa cui, devora chait, zachary stier mentor. From apollonian circle packings to fibonacci numbers mathematics. The thetafunction has a functional equation which is based on this rule, and this rule forms the basis for the integral apollonian circle packing method. The bibliography, of incredible quality, achieves to make the book a. Apollonian circle packings mark pollicott circle packings are a particularly elegant and simple way to construct quite complicated and elaborate sets in the plane. It appears that peter sarnak has proved in 2007 that there are infinitely many primes in an apollonian packing, and even crazier infinitely many pairs of adjacent prime circles. Spherical and hyperbolic packings nicholas eriksson university of california at berkeley berkeley, ca 94720 je. Apollonian packing david sheard explores the rich mathematics and history behind the apollonian packing, and the cover of issue 11 by david sheard. It observed there exist infinitely many types of integral apollonian packings in which all circles had integer curvatures, with the integral structure being related to the integral nature of the apollonian group. This paper studies numbertheoretic properties of the set of. This is not my area of research at all, but i think its super cool.
Dynamics and number theory hee oh yale university icwm, 2014. Apollonian circle packings arise by repeatedly filling the interstices between four mutually tangent circles with further tangent circles. See also its center, a big picture, and a thirdkind encounter click on the corresponding image. Aug 23, 2011 in this video, professor peter sarnak explains the fundamental features of such packings and how modern tools from number theory to algebra are being used to answer some of these old questions. This paper studies number theoretic properties of the set of integer curvatures appearing in such. It is also the only primitive integral apollonian circle packing that contains in. Although there are many problems in understanding all of the individual.
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. Such packings are certainly of interest in classical geometry. Conics introduced the terms parabola, ellipse, and hyperbola. Moreover, if the four starting circles have integer curvatures, then every circle in the packing has an integer curvatures. Skein relations for spin networks, modified, journal of knot theory and its ramifications 27 7 2018 21 pages, arxiv. One can study an apollonian circle packing acp from many different angles. We extend fundamental results concerning apollonian packings, which constitute a major object of study in number theory, to certain homogeneous sets that arise naturally in complex dynamics and. 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. Integral apollonian packings peter sarnak maa lecture. The apollonian structure of integer superharmonic matrices. Im aware that this has become a field of number theory, so they are naturally interested in prime numbers. Some western philosophical and literary figures have invoked this dichotomy in critical and creative works, most notably friedrich nietzsche and later followers in greek mythology, apollo and dionysus are both sons of zeus. The first time i heard of apollonian circle packings was at that conference i went to in marseille last month, during the first lecture of hee ohs minicourse.
Here are some comments and answers to some of the open questions raised at the end of the paper, as well as my take on the diophantine properties of such. But acps encode fascinating information of an entirely different flavor. Results on integer apollonian packings are joint work with. The question of which integers appear as curvatures in a given packing has inspired exciting work in number theory over the last 20 years. It will be in our usual classroom and will cover everything from the beginning of the course till the end, with no intended bias towards the first or second half of the course. We introduce superpackings as orbits of the superapollonian group, starting from a given descartes con. The geometry of fractal sets cambridge tracts in mathematics.
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