The number of configurations of radii that can occur in compact packings of the plane with discs of $n$ sizes is finite

Abstract

By a compact packing of the plane by discs, $P$, we mean a collection of closed discs in the plane with pairwise disjoint interior so that, for every disc $C\in P$, there exists a sequence of discs $D_{0},\ldots,D_{m-1}\in P$ so that each $D_{i}$ is tangent to both $C$ and $D_{i+1\mod m}.$ We prove, for every $n\in\mathbb N$, that there exist only finitely many tuples $(r_{0},r_{1},\ldots,r_{n-1})\in\mathbb{R}^{n}$ with $0<r_{0}<r_{1}\ldots<r_{n-1}=1$ that can occur as the radii of the discs in any compact packing of the plane with $n$ distinct sizes of disc.