On compact packings of the plane with circles of three radii

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 $S\in P$, there exists a maximal indexed set ${A_{0},\ldots,A_{n-1}}\subseteq P$sothat,forevery$i\in{0,\ldots,n-1}$,thecircle$A_{i}$ is tangent to both circles $S$ and $A_{i+1\mathrm{m}\mathrm{o}\mathrm{d}n}\mathrm{.}$ 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 \emph{all} these values on contemporary consumer hardware with the methods employed in this paper.