Compact Packings of the Plane with Three Sizes of Discs

摘要

A compact packing is a set of non-overlapping discs where all the holes between discs are curvilinear triangles. There is only one compact packing by discs of radius 1. There are exactly 9 values of r which allow a compact packing with discs of radius 1 and r. It has been proven that at most 11462 pairs (r, s) allow a compact packing with discs of radius 1, r and s. We prove that there are exactly 164 such pairs.