Counting problems in Apollonian packings
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چکیده
منابع مشابه
Counting Problems in Apollonian Packings
An Apollonian circle packing is a classical construction which is made by repeatedly inscribing circles into the triangular interstices in a Descartes configuration of four mutually tangent circles. Remarkably, if the original four circles have integer curvature, all of the circles in the packing will have integer curvature, making the packings of interest from a number theoretic point of view....
متن کاملApollonian Circle Packings
Figure 1: An Apollonian Circle Packing Apollonius’s Theorem states that given three mutually tangent circles, there are exactly two circles which are tangent to all three. Apollonian circle packings are produced by repeating the construction of mutually tangent circles to fill all remaining spaces. A remarkable consequence of Descartes’ Theorem is if the initial four tangent circles have integr...
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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 t...
متن کاملApollonian Circle Packings
Circle packings are a particularly elegant and simple way to construct quite complicated and elaborate sets in the plane. One systematically constructs a countable family of tangent circles whose radii tend to zero. Although there are many problems in understanding all of the individual values of their radii, there is a particularly simple asymptotic formula for the radii of the circles, origin...
متن کاملStationary Apollonian packings
The notion of stationary Apollonian packings in the d-dimensional Euclidean space is introduced as a mathematical formalization of so-called random Apollonian packings and rotational random Apollonian packings, which constitute popular grain packing models in physics. Apart from dealing with issues of existence and uniqueness in the entire Euclidean space, asymptotic results are provided for th...
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ژورنال
عنوان ژورنال: Bulletin of the American Mathematical Society
سال: 2013
ISSN: 0273-0979,1088-9485
DOI: 10.1090/s0273-0979-2013-01401-0