Compact packings of the plane with three sizes of discs

08/31/2018

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by   Thomas Fernique, et al.

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Discs form a compact packing of the plane if they are interior disjoint and the graph which connects the center of mutually tangent discs is triangulated. There is only one compact packing by discs all of the same size, called hexagonal compact packing. It has been previously proven that there are exactly 9 values of r such that there exists a compact packing with discs of radius 1 and r. This paper shows that there are exactly 164 pairs (r,s) such that there exists a compact packing with discs of radius 1, r and s. In all these 164 cases, there exists a periodic packing.