@MISC{Sarnak_integralapollonian, author = {Peter Sarnak}, title = {Integral Apollonian Packings}, year = {} }

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Abstract

Abstract. 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. d = diameter d 2 = 21 mm d 3 = 24 mm C d 4 = 504 mm