To begin our story about apollonian circle packings, we consider four mutually. Such packings are certainly of interest in classical geometry for. For each operation we compute the same formula twice, once for the bends and once for the bendcenter products. Number theory, graph theory and geometric statistics theorem bourgainkontorovich, inventiones, 2014 almost every integer in the admissible congruence classes mod.
Rational thought is also apollonian since it is structured and makes distinctions. 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. The geometry of fractal sets cambridge tracts in mathematics book 85 kindle edition by falconer, k. The final chapter includes diverse examples of sets to which the general theory is applicable. Apollonian definition of apollonian by merriamwebster. 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. This survey article is an expanded version of my lecture notes prepared for the th takagi lec.
Peter sarnak mahler lectures 2011 number theory and the circle packings of apollonius a. 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. Citeseerx document details isaac councill, lee giles, pradeep teregowda. Apollonius of perga i lived from about 262 bc to about 190 bc.
From apollonian circle packings to fibonacci numbers. The first few of these integral apollonian gaskets are listed in the following table. Apollonian devised by or named after apollonius of perga, an ancient greek geometer, celebrated for his original investigations in conic sections. Such packings are certainly of interest in classical geometry. There are many papers in the literature dealing with apollonian packings and their generalizations. Apollonian sphere packing is the threedimensional equivalent of the apollonian gasket. The mathematical sciences research institute msri, founded in 1982, is an independent nonprofit mathematical research institution whose funding sources include the national science foundation, foundations, corporations, and more than 90 universities and institutions. Pdf irreducible apollonian configurations and packings. But acps encode fascinating information of an entirely different flavor. Some western philosophical and literary figures have invoked this dichotomy in critical and creative works, most notably friedrich nietzsche and later followers. Use features like bookmarks, note taking and highlighting while reading the geometry of fractal sets cambridge tracts in mathematics book 85. This paper studies numbertheoretic properties of the set of integer curvatures appearing in such. 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. Arithmetic properties of apollonian circle packings elena.
Perhaps the most astounding fact about the apollonian gasket is that if the first four circles have integer bends, then every other circle in the packing does too. In analogy with apollonian circle packings, we call the collection of curves p fc kg k2nf0g, or, interchangeably, the collection of the corresponding. His famous book on conics introduced the terms parabola, ellipse and hyperbola. 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. 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. 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. Aug 25, 2014 alexander gatherer cardiff university the opposing concepts of the dionysian hereon di and apollonian hereon ap are central themes within nietzsches first major work, the birth of tragedy h. Computing the centers of apollonian circle packings. Apollonian packing can be transformed to a strongly integral one by a euclidean motion. One of the most essential tools in number theory is the theory of automorphic forms and lfunctions see 45 and 31 for an elegant survey of various applications. 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. The institute is located at 17 gauss way, on the university of california, berkeley campus, close to grizzly peak, on the. The theory is related by duality to kayeka sets sets of zero area containing lines in every direction. If we started with an odd number of tangent circles, then. Apollonian circle packings arise by repeatedly filling the interstices between mutually tangent circles with further tangent circles. The sizes of the new circles are determined by descartes theorem. Number theory and the circle packings of apollonius peter sarnak johannesburg, july 2014 1. 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. Journal of number theory vol 100, issue 1, pages 1202. We introduce superpackings as orbits of the superapollonian group, starting from a given descartes con. Apollonian and dionysian are terms used by friedrich. The existence of an integral matrix group explaining. The apollonian and dionysian is a philosophical and literary concept and dichotomydialectic, based on apollo and dionysus in greek mythology.
Abelian l adic representations and elliptic curves, 2nd ed. Friedrich nietzsche used the term in his book the birth of tragedy to describe one of the two opposing tendencies or elements in greek tragedy. Dynamics and number theory hee oh yale university icwm, 2014. 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. Apollonian packings is enumerated by the integer solutions to the descartes equation. A tisket, a tasket, an apollonian gasket american scientist. In a sense, there is therefore only one apollonian gasket, which can be thought of as a tessellation of the hyperbolic plane by circles and hyperbolic triangles. Integral apollonian packings mathematical association of. The geometry of fractal sets cambridge tracts in mathematics. 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. Apollonian gaskets and descartes theorem ii the math. Apollonian definition is of, relating to, or resembling the god apollo. This paper studies number theoretic properties of the set of integer curvatures appearing in such.
This paper studies number theoretic properties of the set of integer curvatures appearing in such packings. The book finish with a magnificent list of examples of haussdorff dimension computation. Journal of number theory vol 100, issue 1, pages 1202 may. His contemplation of such opposing forces of nature are primarily used to analyse greek culture in general, and greek art in particular, stating that. No one can predict when what seems to be a most obscure theorem may suddenly be called upon to play some vital and hitherto unsuspected role. In mathematics, an apollonian gasket or apollonian net is a fractal generated starting from a.
We show that for a given bounded apollonian circle packing, there exists a constant such that the number of circles of curvature at most is asymptotic to as. From apollonian circle packings to fibonacci numbers mathematics. Apollonian, of, relating to, or resembling the god apollo. It is named after greek mathematician apollonius of perga. There are infinitely many different integral packings. Number theory and the circle packings of apollonius peter sarnak johannesburg. They include algebra through the apollonian group, analysis through enumeration and density questions, and number theory through questions on curvatures that are prime. Introduction to apollonian circle packings tangent baking. Journal of number theory 100 2003 145 apollonian circle packings. Introduction to apollonian circle packings tangent. Arithmetic properties of apollonian circle packings elena fuchs.
Most of number theory has very few practical applications. Skein relations for spin networks, modified, journal of knot theory and its ramifications 27 7 2018 21 pages, arxiv. Plus a simple algorithm for drawing the dedekind tessellation. 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. Abelian ladic representations and elliptic curves, 2nd ed. Indeed, several longstanding problems in analytic number theory. One of the most studied objects in number theory which continues to intrigue. Lagarias university of michigan ann arbor, mi 48109 february 1, 2005 abstract apollonian circle packings arise by repeatedly. On the localglobal principle for integral apollonian 3. Spherical and hyperbolic packings nicholas eriksson university of california at berkeley berkeley, ca 94720 je. Apollonian definition, etymology and usage, examples and.
So heres a quick write up background of the first third of. Irreducible apollonian configurations and packings. This paper studies numbertheoretic properties of the set of integer curvatures appearing in such packings. The apollonian gasket is the limit set of a group of mobius transformations. In this video, professor peter sarnak explains the fundamental features of such packings and how modern tools from number theory to algebra are. 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. The apollonian group article pdf available in discrete and computational geometry 344. Along the way, we prove a uniform spectral gap for the congruence towers of the symmetry group. Integral apollonian packings peter sarnak maa lecture. 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. Abstractin this paper we study the integral properties of apollonian 3circle packings, which are variants of the standard apollonian circle packings.
Root quadruples and number theory theorem glmwy 2003 1 for each n 1 there are. If you are given the first three circles of an apollonian gasket, the bend of the fourth is found as explained above by solving a quadratic equation. According to nietzsche, the apollonian attributes are reason, culture, harmony, and restraint. Alexander gatherer cardiff university the opposing concepts of the dionysian hereon di and apollonian hereon ap are central themes within nietzsches first major work, the birth of tragedy hereon bt. Peter sarnak mahler lectures 2011 number theory and the circle packings of apollonius. Counting problems in apollonian packings american mathematical. J download it once and read it on your kindle device, pc, phones or tablets. As with many problems in number theory the basic questions here are easy to state but di.
The observation that there are apollonian packings having all curvatures integral is an old one, and was noted by soddy 37, 38, who also gave an extension to three dimensions, the bowl of integers. For integral, let denote the number of circles with prime curvature less than. It is also the only primitive integral apollonian circle packing that contains in. Results on integer apollonian packings are joint work with. The apollonian gasket is the limit set of a group of mobius transformations known as a kleinian group. This is not my area of research at all, but i think its super cool. From apollonian packings to homogeneous sets 7 set is the standard sierpinski carpet s 3 see figure 3 below. That does not reduce its importance, and if anything it enhances its fascination. We observe that there exist apollonian packings which have strong integrality properties, in which all circles in. More to the point, the were both gods in the ancient greek pantheon, despite representing nearly opposing values and orientations. Apollonian circle packings arise by repeatedly filling the interstices between four mutually tangent circles with further tangent circles. So for each circle we just keep track of the signed bend as well as the product of the signed bend and the center. Number theory and the circle packings of apollonius.
Sep 11, 2000 apollonian circle packings arise by repeatedly filling the interstices between mutually tangent circles with further tangent circles. I his famous book onconicsintroduced the terms parabola, ellipse and hyperbola. An apollonian circle packing is any packing of circles constructed recursively from an initial configuration of four mutually tangent circles by the. Apollo and dionysus were gods in ancient creek religion.
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