Empilements compacts avec trois tailles de disque

TL;DR

This paper characterizes all possible compact packings of the plane with three different disc sizes, showing exactly 164 such configurations, all of which are periodic.

Contribution

It extends previous work by identifying all pairs of radii allowing compact packings with three disc sizes, revealing a finite set of 164 configurations.

Findings

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

All these packings are periodic.

The unique single-size packing is the hexagonal packing.

Abstract

Discs form a compact packing of the plane if they are interior disjoint and the graph which connects the center of mutually tangent discs is triangulated. There is only one compact packing by discs all of the same size, called hexagonal compact packing. It has been previously proven that there are exactly $9$ values of $r$ such that there exists a compact packing with discs of radius $1$ and $r$. This paper shows that there are exactly $164$ pairs $(r,s)$ such that there exists a compact packing with discs of radius $1$, $r$ and $s$. In all these $164$ cases, there exists a periodic packing.