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 $D_0,\dots ,D_{m-1}\in P$ so that each $D_i$ is tangent to both $C$ and $D_{i+1\mathrm{mod}m}.$ We prove, for every $n\in \mathbb{N}$, that there exist only finitely many tuples $(r_0,r_1,\dots ,r_{n-1})\in {\mathbb{R}}^n$ with $0<r_0<r_1\dots <r_{n-1}=1$ that can occur as the radii of the discs in any compact packing of the plane with $n$ distinct sizes of disc.

Sections 3-6 are revised so as to align with a better definition of 'fundamental set'

Cited by in corpus (1)

• Packing unequal disks in the Euclidean plane