Newest apollonian circle packing questions feed subscribe to rss. Skein relations for spin networks, modified, journal of knot theory and its ramifications 27 7 2018 21 pages, arxiv. 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. This survey article is an expanded version of my lecture notes prepared for the th takagi lectures given at. 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. Number theory, graph theory and geometric statistics theorem bourgainkontorovich, inventiones, 2014 almost every integer in the admissible congruence classes mod. Newest apolloniancirclepacking questions mathematics. Apollonian circle packing this is an apollonian circle packing. His famous book conics introduced the terms parabola, ellipse, and hyperbola. The bibliography, of incredible quality, achieves to make the book a. Dynamics and number theory hee oh yale university icwm, 2014.
This is not my area of research at all, but i think its super cool. One can study an apollonian circle packing acp from many different angles. Sphere packings and number theory zachary stier mentor. So heres a quick write up background of the first third of. In the course of centuries, many exciting results have been obtained, ingenious methods created, related challenging problems proposed, and many surprising connections with other subjects found. 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. Integral apollonian packings mathematical association of. Sphere packing, lattices and groups, third edition, n. Yan gives a joint characterization of the circle curvatures inverse radii and coordinates. Sphere packings and number theory alisa cui, devora chait, zachary stier mentor. 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. Citeseerx document details isaac councill, lee giles, pradeep teregowda. 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.
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. It is also the only primitive integral apollonian circle packing that contains in. The spectral geometry of the apollonian group controls the number theory of such integers. Apollonius of perga i lived from about 262 bc to about 190 bc.
The book finish with a magnificent list of examples of haussdorff dimension computation. It was resolved completely relatively recently 2001 using the theory of automorphic forms. 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. This completely characterizes the pde that determines the continuum scaling limit of. 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. Fractalsapollonian fractals wikibooks, open books for. Intercity number theory seminar getaltheorie in het vlakke land. Apollonius of perga lived from about 262 bc to about 190 bc apollonius was known as the great geometer. The theory of discrete analytic functions 9780521823562. Apollonian circle packings what is an apollonian circle packing.
In this book the author solves the problem of maintaining the interest of students at both levels by offering a combinatorial approach to elementary number theory. Lagarias university of michigan ann arbor, mi 48109 february 1, 2005 abstract apollonian circle packings arise by repeatedly. Sphere packings is one of the most fascinating and challenging subjects in mathematics. This particular packing has already played a role in number theory. This paper studies numbertheoretic properties of the set of integer curvatures appearing in such packings. In my talk, i will connect this question to the algebra of infinite root systems. Such packings are certainly of interest in classical geometry. Apollonian circle packing heres how we construct it. In geometry, a sphere packing is an arrangement of nonoverlapping spheres within a containing space.
From apollonian circle packings to fibonacci numbers je. Im aware that this has become a field of number theory, so they are naturally interested in prime numbers. Take three circles of any size for example, a quarter, a nickel, and a dime and arrange them so that each. Effective circle count for apollonian packings and closed. Number theory and the circle packings of apollonius. Such packings can be described in terms of the descartes configurations they contain. We obtain this by proving an effective equidistribution of closed horospheres in the unit tangent bundle of a geometrically finite hyperbolic 3manifold, whose fundamental group has critical exponent bigger than 1. 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. However, sphere packing problems can be generalised to. 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. Apollonian circle packings arise by repeatedly filling the interstices between mutually tangent circles with further tangent circles. Conics introduced the terms parabola, ellipse, and hyperbola.
Apollonian packing david sheard explores the rich mathematics and history behind the apollonian packing, and the cover of issue 11 by david sheard. In general for forms in 3 variables the local to global is valid with. Table of contents 1 what is an apollonian circle packing. The spheres considered are usually all of identical size, and the space is usually three dimensional euclidean space. The main result of this paper is an effective count for apollonian circle packings that are either bounded or contain two parallel lines. The final chapter includes diverse examples of sets to which the general theory is applicable. This paper studies number theoretic properties of the set of integer curvatures appearing in such. Apollonius discovered that there are two other nonintersecting circles, c 4 and c 5, which have the property that they are tangent to all.
Spherical and hyperbolic packings nicholas eriksson university of california at berkeley berkeley, ca 94720 je. The apollonian structure of integer superharmonic matrices. Moreover, if the four starting circles have integer curvatures, then every circle in the packing has an integer curvatures. Integral apollonian packings 9 figure 9 in this con.
From apollonian circle packings to fibonacci numbers. Computing the centers of apollonian circle packings. I his famous book onconicsintroduced the terms parabola, ellipse and hyperbola. I start with three mutually tangent circles i draw two more circles, each of which is tangent to the. They include algebra through the apollonian group, analysis through enumeration and density questions, and number theory through questions on curvatures that are prime. Integral apollonian packings peter sarnak maa lecture. Buy an adventurers guide to number theory dover books on mathematics reprinted ed by richard friedberg isbn. 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 circle packings arise by repeatedly filling the interstices between four mutually tangent circles with further tangent circles. These types of results are also directly related to the gauss map and provide a recipe for using dynamical systems theory. This paper studies number theoretic properties of the set of integer curvatures appearing in such packings. Apollonian packings and of kacmoody groups department. 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. An apollonian gasket can be constructed as follows.
Counting problems for apollonian circle packings an apollonian circle packing is one of the most of beautiful circle packings whose construction can be described in a very simple manner based on an old theorem of apollonius of perga. 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. 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. In this talk, we investigate the arithmetic properties of the collection of integers appearing in prime components of apollonian circle packings. Sphere packing finds practical application in the stacking of oranges.
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 on bubbles evan warner stanford university may 22, 2014 evan warner stanford university apollonian circle packings may 22, 2014 1 59. Peter sarnak mahler lectures 2011 number theory and the circle packings of apollonius. The question of which integers appear as curvatures in a given packing has inspired exciting work in number theory over the last 20 years. The bibliography, of incredible quality, achieves to make the book a reference for anyone interested in fractals. But acps encode fascinating information of an entirely different flavor. Counting problems in apollonian packings elena fuchs abstract. Drawing the apollonian gasket with common lisp and vecto by luis diego fallas leibnitz packing by takaya iwamoto with program in autolisp apollonian gasket by paul bourke in basic and c how to pack n circles inside unit circle by erich friedman svg math animation example. How we measure reads a read is counted each time someone views a. An adventurers guide to number theory dover books on. Euclid devoted part of his elements to prime numbers and divisibility, topics that belong unambiguously to number theory and are basic to it books vii to ix of. Introduction to apollonian circle packings tangent.
Apollonian packing in the sense of conformal geometry. The new book of prime number records, 3rd edition, p. The theory is related by duality to kayeka sets sets of zero area containing lines in every direction. In any integral apollonian packing, there are in nitely many cs with ac prime and, better still, in nitely many pairs c.
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