arXiv subjects
Publications and source records attributed to Olga Sizova.
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 $s$.
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.