Such packings can be described in terms of the descartes configurations they contain. Apollonian circle packings arise by repeatedly filling the interstices between mutually tangent circles with further tangent circles. Sphere packings is one of the most fascinating and challenging subjects in mathematics. Peter sarnak mahler lectures 2011 number theory and the circle packings of apollonius.
The final chapter includes diverse examples of sets to which the general theory is applicable. Apollonian packing david sheard explores the rich mathematics and history behind the apollonian packing, and the cover of issue 11 by david sheard. How we measure reads a read is counted each time someone views a. 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. But acps encode fascinating information of an entirely different flavor. Buy an adventurers guide to number theory dover books on mathematics reprinted ed by richard friedberg isbn. 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. Apollonian packing in the sense of conformal geometry. Apollonian circle packing this is an 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 \\em integral apollonian circle packing. The question of which integers appear as curvatures in a given packing has inspired exciting work in number theory over the last 20 years.
Sphere packing finds practical application in the stacking of oranges. 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 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.
Integral apollonian packings peter sarnak maa lecture. 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. The spheres considered are usually all of identical size, and the space is usually three dimensional euclidean space. Yan gives a joint characterization of the circle curvatures inverse radii and coordinates. They include algebra through the apollonian group, analysis through enumeration and density questions, and number theory through questions on curvatures that are prime. This paper studies number theoretic properties of the set of integer curvatures appearing in such packings. Intercity number theory seminar getaltheorie in het vlakke land. 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. An adventurers guide to number theory dover books on. This completely characterizes the pde that determines the continuum scaling limit of. Fractalsapollonian fractals wikibooks, open books for. Lagarias university of michigan ann arbor, mi 48109 february 1, 2005 abstract apollonian circle packings arise by repeatedly. Apollonian circle packings arise by repeatedly filling the interstices between four mutually tangent circles with further tangent circles. The book finish with a magnificent list of examples of haussdorff dimension computation.
Newest apollonian circle packing questions feed subscribe to rss. From apollonian circle packings to fibonacci numbers. Integral apollonian packings 9 figure 9 in this con. In geometry, a sphere packing is an arrangement of nonoverlapping spheres within a containing space. An apollonian gasket can be constructed as follows. 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. Such packings are certainly of interest in classical geometry. Apollonian packings and of kacmoody groups department. 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. This particular packing has already played a role in number theory.
Newest apolloniancirclepacking questions mathematics. 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. Apollonius discovered that there are two other nonintersecting circles, c 4 and c 5, which have the property that they are tangent to all. This survey article is an expanded version of my lecture notes prepared for the th takagi lectures given at. It was resolved completely relatively recently 2001 using the theory of automorphic forms. One can study an apollonian circle packing acp from many different angles.
Integral apollonian packings mathematical association of. 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. Introduction to number theory and apollonian circle packings. Sphere packing, lattices and groups, third edition, n. Im aware that this has become a field of number theory, so they are naturally interested in prime numbers. It is also the only primitive integral apollonian circle packing that contains in.
This is not my area of research at all, but i think its super cool. This paper studies numbertheoretic properties of the set of integer curvatures appearing in such packings. I start with three mutually tangent circles i draw two more circles, each of which is tangent to the. Apollonius of perga i lived from about 262 bc to about 190 bc. In any integral apollonian packing, there are in nitely many cs with ac prime and, better still, in nitely many pairs c. The theory is related by duality to kayeka sets sets of zero area containing lines in every direction. From apollonian circle packings to fibonacci numbers je. Conics introduced the terms parabola, ellipse, and hyperbola. The new book of prime number records, 3rd edition, p. The spectral geometry of the apollonian group controls the number theory of such integers. Number theory, graph theory and geometric statistics theorem bourgainkontorovich, inventiones, 2014 almost every integer in the admissible congruence classes mod.
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. The bibliography, of incredible quality, achieves to make the book a reference for anyone interested in fractals. Moreover, if the four starting circles have integer curvatures, then every circle in the packing has an integer curvatures. Apollonius of perga lived from about 262 bc to about 190 bc apollonius was known as the great geometer. Computing the centers of apollonian circle packings. The apollonian structure of integer superharmonic matrices. Dynamics and number theory hee oh yale university icwm, 2014. The main result of this paper is an effective count for apollonian circle packings that are either bounded or contain two parallel lines. 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. 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. In my talk, i will connect this question to the algebra of infinite root systems. So heres a quick write up background of the first third of.
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. In this talk, we investigate the arithmetic properties of the collection of integers appearing in prime components of apollonian circle packings. I his famous book onconicsintroduced the terms parabola, ellipse and hyperbola. Apollonian circle packing heres how we construct it. His famous book conics introduced the terms parabola, ellipse, and hyperbola. Introduction to apollonian circle packings tangent. Spherical and hyperbolic packings nicholas eriksson university of california at berkeley berkeley, ca 94720 je. In general for forms in 3 variables the local to global is valid with.
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. 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. Number theory on bubbles evan warner stanford university may 22, 2014 evan warner stanford university apollonian circle packings may 22, 2014 1 59. Apollonian circle packings what is an apollonian circle packing. Sphere packings and number theory zachary stier mentor. 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. Number theory and the circle packings of apollonius. The theory of discrete analytic functions 9780521823562. Counting problems in apollonian packings elena fuchs abstract. Table of contents 1 what is an apollonian circle packing.
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