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arXiv · 1709.03487

On compact packings of the plane with circles of three radii

Miek Messerschmidt
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Abstract

A compact circle-packing $P$ of the Euclidean plane is a set of circles which bound mutually disjoint open discs with the property that, for every circle $S\in P$, there exists a maximal indexed set $\{A_{0},\ldots,A_{n-1}\}\subseteq P$ so that, for every $i\in\{0,\ldots,n-1\}$, the circle $A_{i}$ is tangent to both circles $S$ and $A_{i+1\mod n}.$ We show that there exist at most $13617$ pairs $(r,s)$ with $0<s<r<1$ for which there exist a compact circle-packing of the plane consisting of circles with radii $s$, $r$ and $1$. We discuss computing the exact values of such $0<s<r<1$ as roots of polynomials and exhibit a selection of compact circle-packings consisting of circles of three radii. We also discuss the apparent infeasibility of computing \emph{all} these values on contemporary consumer hardware with the methods employed in this paper.

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Miek Messerschmidt. 2019-07-28. On compact packings of the plane with circles of three radii. https://doi.org/10.1016/j.comgeo.2019.05.002

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