Compact packings of the plane with three sizes of discs

TL;DR

This paper classifies all possible arrangements of three different disc sizes in a plane where the packing is both non-overlapping and fills the space efficiently, revealing exactly 164 such configurations.

Contribution

The paper extends previous work by identifying all pairs of disc sizes that permit compact packings with three sizes, providing a complete enumeration of such configurations.

Findings

Exactly 164 pairs (r,s) allow compact packings with three disc sizes.

Only one compact packing exists with a single disc size.

Nine specific values of r enable packings with two sizes.

Abstract

A compact packing is a set of non-overlapping discs where all the holes between discs are curvilinear triangles. There is only one compact packing by discs of size $1$. There are exactly $9$ values of $r$ which allow a compact packing by discs of sizes $1$ and $r$. We prove here that there are exactly $164$ pairs $(r,s)$ allowing a compact packing by discs of sizes $1$, $r$ and $s$.