Integral Apollonian Packings
نویسنده
چکیده
We review the construction of integral Apollonian circle packings. There are a number of Diophantine problems that arise in the context of such packings. We discuss some of them and describe some recent advances. 1. AN INTEGRAL PACKING. The quarter, nickel, and dime in Figure 1 are placed so that they are mutually tangent. This configuration is unique up to rigid motions. As far as I can tell there is no official exact size for these coins, but the diameters of 24, 21, Figure 1. and 18 millimeters are accurate to the nearest millimeter and I assume henceforth that these are the actual diameters. Let C be the unique (see below) circle that is tangent to the three coins as shown in Figure 2. It is a small coincidence that its diameter is rational, as indicated. C d = diameter d2 = 21 mm d3 = 24 mm d1 = 18 mm d4 = 504 mm 157 RATIONAL! Figure 2. doi:10.4169/amer.math.monthly.118.04.291 April 2011] INTEGRAL APOLLONIAN PACKINGS 291 What is more remarkable is that if we continue to place circles in the resulting regions bounded by three circles as described next, then all the diameters are rational. Since the circles become very small, so do their radii, and it is more convenient to work with their curvatures, which are the reciprocals of the radii. In fact in this example it is natural to scale everything further by 252, so let us take 252 mm as our unit of measurement, and then for each circle C let a(C) be the curvature of C in these units. With this rescaling, all of the curvatures turn out to be integers. In Figure 3 our three tangent circles are displayed together with the unique outer mutually tangent circle. The a(C) for each circle is depicted inside the circle. Note that the outer circle has a negative sign indicating that the other circles are in its interior (it is the only circle with a negative sign).
منابع مشابه
Apollonian Circle Packings: Number Theory II. Spherical and Hyperbolic Packings
Apollonian circle packings arise by repeatedly filling the interstices between mutually tangent circles with further tangent circles. 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. There are infinitely many different integral packings; these were studied in the paper [8]....
متن کاملOn the Local-global Principle for Integral Apollonian 3-circle Packings
In this paper we study the integral properties of Apollonian-3 circle packings, which are variants of the standard Apollonian circle packings. Specifically, we study the reduction theory, formulate a local-global conjecture, and prove a density one version of this conjecture. Along the way, we prove a uniform spectral gap for the congruence towers of the symmetry group.
متن کاملThe Local-global Principle for Integral Bends in Orthoplicial Apollonian Sphere Packings
We introduce an orthoplicial Apollonian sphere packing, which is a sphere packing obtained by successively inverting a configuration of 8 spheres with 4-orthplicial tangency graph. We will show that there are such packings in which the bends of all constituent spheres are integral, and establish the asymptotic local-global principle for the set of bends in these packings.
متن کاملApollonian Circle Packings: Geometry and Group Theory II. Super-Apollonian Group and Integral Packings
A Descartes configuration is a set of four mutually tangent circles in the Riemann sphere, having disjoint interiors. Apollonian circle packings arise by repeatedly filling the interstices between four mutually tangent circles with further tangent circles. Such packings can be described in terms of the Descartes configurations they contain. Part I shoewed there is a natural group action on Desc...
متن کاملThe sensual Apollonian circle packing
The curvatures of the circles in integral Apollonian circle packings, named for Apollonius of Perga (262-190 BC), form an infinite collection of integers whose Diophantine properties have recently seen a surge in interest. Here, we give a new description of Apollonian circle packings built upon the study of the collection of bases of Z[i], inspired by, and intimately related to, the ‘sensual qu...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- The American Mathematical Monthly
دوره 118 شماره
صفحات -
تاریخ انتشار 2011