On compact packings of the plane with circles of three radii

TL;DR

This paper investigates the limited configurations of compact circle packings in the plane with three specific radii, providing bounds, exact value computations, and discussing computational challenges.

Contribution

It establishes an upper bound on the number of possible radius pairs for such packings and explores methods to compute their exact values.

Findings

At most 13,617 radius pairs allow such packings.

Exact radii values can be found as roots of polynomials.

Computational methods face significant feasibility challenges.

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 $S\in P$, there exists a maximal indexed set $\{A_{0},\ldots,A_{n-1}\}\subseteq P$ so that, for every $i\in\{0,\ldots,n-1\}$, the circle $A_{i}$ is tangent to both circles $S$ and $A_{i+1\mod n}.$ 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.