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.