Compact Packings of the Plane with Two Sizes of Discs

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 D₁, D₂, ..., 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.