Integral Apollonian Packings

نویسنده

Peter Sarnak

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).

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