Number theory on bubbles evan warner stanford university may 22, 2014 evan warner stanford university apollonian circle packings may 22, 2014 1 59. Introduction to apollonian circle packings tangent. One can study an apollonian circle packing acp from many different angles. Integral apollonian packings mathematical association of. This paper studies number theoretic properties of the set of integer curvatures appearing in such packings. The bibliography, of incredible quality, achieves to make the book a reference for anyone interested in fractals. Buy an adventurers guide to number theory dover books on mathematics reprinted ed by richard friedberg isbn. The main result of this paper is an effective count for apollonian circle packings that are either bounded or contain two parallel lines. Apollonian circle packings arise by repeatedly filling the interstices between mutually tangent circles with further tangent circles. But acps encode fascinating information of an entirely different flavor. 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. Take three circles of any size for example, a quarter, a nickel, and a dime and arrange them so that each. Introduction to number theory and apollonian circle packings.
These types of results are also directly related to the gauss map and provide a recipe for using dynamical systems theory. His famous book conics introduced the terms parabola, ellipse, and hyperbola. Lagarias university of michigan ann arbor, mi 48109 february 1, 2005 abstract apollonian circle packings arise by repeatedly. Newest apollonian circle packing questions feed subscribe to rss. An apollonian gasket can be constructed as follows. They include algebra through the apollonian group, analysis through enumeration and density questions, and number theory through questions on curvatures that are prime.
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. The theory of discrete analytic functions 9780521823562. Im aware that this has become a field of number theory, so they are naturally interested in prime numbers. This is not my area of research at all, but i think its super cool. 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. Conics introduced the terms parabola, ellipse, and hyperbola. Number theory and the circle packings of apollonius. Skein relations for spin networks, modified, journal of knot theory and its ramifications 27 7 2018 21 pages, arxiv. From apollonian circle packings to fibonacci numbers je.
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 survey article is an expanded version of my lecture notes prepared for the th takagi lectures given at. In general for forms in 3 variables the local to global is valid with. Newest apolloniancirclepacking questions mathematics. It was resolved completely relatively recently 2001 using the theory of automorphic forms.
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. Apollonian circle packing this is an apollonian circle packing. The apollonian structure of integer superharmonic matrices. Integral apollonian packings peter sarnak maa lecture. The question of which integers appear as curvatures in a given packing has inspired exciting work in number theory over the last 20 years. This paper studies numbertheoretic properties of the set of integer curvatures appearing in such 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 also the only primitive integral apollonian circle packing that contains in. 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. Such packings can be described in terms of the descartes configurations they contain. 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 apollonian packings 9 figure 9 in this con.
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. 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 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. Citeseerx document details isaac councill, lee giles, pradeep teregowda. Computing the centers of apollonian circle packings.
I his famous book onconicsintroduced the terms parabola, ellipse and hyperbola. Apollonius of perga i lived from about 262 bc to about 190 bc. However, sphere packing problems can be generalised to. The book finish with a magnificent list of examples of haussdorff dimension computation. In any integral apollonian packing, there are in nitely many cs with ac prime and, better still, in nitely many pairs c. 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. The theory is related by duality to kayeka sets sets of zero area containing lines in every direction. 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. Apollonian packing david sheard explores the rich mathematics and history behind the apollonian packing, and the cover of issue 11 by david sheard. Peter sarnak mahler lectures 2011 number theory and the circle packings of apollonius. In my talk, i will connect this question to the algebra of infinite root systems. 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. Table of contents 1 what is an apollonian circle packing. I start with three mutually tangent circles i draw two more circles, each of which is tangent to the.
Apollonian packing in the sense of conformal geometry. Apollonius of perga lived from about 262 bc to about 190 bc apollonius was known as the great geometer. So heres a quick write up background of the first third of. The final chapter includes diverse examples of sets to which the general theory is applicable. In geometry, a sphere packing is an arrangement of nonoverlapping spheres within a containing space. Apollonian packings and of kacmoody groups department. Yan gives a joint characterization of the circle curvatures inverse radii and coordinates.
An adventurers guide to number theory dover books on. Apollonian circle packing heres how we construct it. This completely characterizes the pde that determines the continuum scaling limit of. Fractalsapollonian fractals wikibooks, open books for. Apollonian circle packings arise by repeatedly filling the interstices between four mutually tangent circles with further tangent circles. Spherical and hyperbolic packings nicholas eriksson university of california at berkeley berkeley, ca 94720 je.
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. Sphere packings and number theory alisa cui, devora chait, zachary stier mentor. This particular packing has already played a role in number theory. 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. Sphere packing, lattices and groups, third edition, n. From apollonian circle packings to fibonacci numbers. 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. Sphere packings and number theory zachary stier mentor. 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.
The new book of prime number records, 3rd edition, p. How we measure reads a read is counted each time someone views a. The spectral geometry of the apollonian group controls the number theory of such integers. 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.
Moreover, if the four starting circles have integer curvatures, then every circle in the packing has an integer curvatures. Effective circle count for apollonian packings and closed. The bibliography, of incredible quality, achieves to make the book a. Intercity number theory seminar getaltheorie in het vlakke land. 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 paper studies number theoretic properties of the set of integer curvatures appearing in such. Sphere packings is one of the most fascinating and challenging subjects in mathematics. Dynamics and number theory hee oh yale university icwm, 2014.
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