Compact packings of the plane with two sizes of discs (Q818685)

The maximal density \(\pi/\sqrt{12}\) of a packing of congruent discs in the Euclidean plane is well known. The ``next'' open question is to find the densest two-size packings, packings in the plane using discs of radius \(1\) and \(r\) \((0<r<1)\). For six special values of \(r\), \textit{A. Heppes} [Discrete Comput. Geom. 30, 241--262 (2003; Zbl 1080.52510)] has given the densest packing and has shown that the densest packing is compact. A disc packing is compact if each disc is tangent to a ring of other discs, each of which is tangent to two further ones of the ring. In the present paper it is proved that there are only nine values of \(r\) for which compact two-size packings are possible. For the three values not considered by Heppes, the author describes the possible compact packings.