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 D1, D2, ..., Dn such that Di is tangent to Di+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.