On compact packings of the plane with circles of three radii

Abstract

A compact circle-packing P of the Euclidean plane is a set of circles which bound mutually disjoint open discs with the property that, for every circle S∈P, there exists a maximal indexed set {A0,…,An−1}⊆P so that, for every i∈{0,…,n−1}, the circle Ai is tangent to both circles S and Ai+1modn.We show that there exist at most 13617 pairs (r,s) with 0<s<r<1 for which there exist a compact circle-packing of the plane consisting of circles with radii s, r and 1.We discuss computing the exact values of such 0<s<r<1 as roots of polynomials and exhibit a selection of compact circle-packings consisting of circles of three radii. We also discuss the apparent infeasibility of computing all these values on contemporary consumer hardware.