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arXiv · 2110.15831

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

Miek Messerschmidt
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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.

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Miek Messerschmidt. 2021-10-29. The number of configurations of radii that can occur in compact packings of the plane with discs of $n$ sizes is finite. https://doi.org/10.1007/s00454-022-00471-z

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