Compact packings of the plane with two sizes of discs

Abstract

We consider packings of the plane using discs of radius 1 and r. A packing is compact if every disc D is tangent to a sequence of discs D_1, D_2, ..., D_n such that D_i is tangent to D_{i+1}. We prove that there are only nine values of r with r<1 for which such packings are possible. For each of the nine values we describe the possible compact packings.