Empilements compacts avec trois tailles de disque
Thomas Fernique, Amir Hashemi, Olga Sizova

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 values of such that there exists a compact packing with discs of radius and . This paper shows that there are exactly pairs such that there exists a compact packing with discs of radius , and . In all these cases, there exists a periodic packing.
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Taxonomy
TopicsPoint processes and geometric inequalities · Geometric and Algebraic Topology · Mathematics and Applications
