The Number of Configurations of Radii that Can Occur in Compact Packings of the Plane with Discs of n Sizes is Finite

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