Packing unequal disks in the Euclidean plane

Abstract

A packing of disks in the plane is a set of disks with disjoint interiors. This paper is a survey of some open questions about such packings. It is organized into five themes: compacity, conjugacy, density, uniformity and computability.

Figures (11)

• Figure 1: An example of a triangulated binary packing for each of the $9$ possible ratios. Each packing is periodic and the black parallelogram depicts a fundamental domain.
• Figure 2: The tilings defined by each of these tile sets correspond to the sets of all the possible triangulated binary packings (Fig. \ref{['fig:1']}). Can you visualize them?
• Figure 3: Cases $1$ and $3$ of Fig. \ref{['fig:2']}: the two tiles freely alternate and form an infinite horizontal stripe which can then be repeated in the vertical direction.
• Figure 4: Case $2$ of Fig. \ref{['fig:2']}: either two mirrored infinite stripes can be alternated (left) or arbitrarily large "crested triangles" can be used to tile as in the triangular grid (center and right).
• Figure 5: A triangulated ternary packing that can be seen as a tiling by three tiles (the two depicted polygons and the triangle between the centers of three large disks, which does not appear here).

Theorems & Definitions (3)

• Conjecture 1
• Conjecture 2
• Conjecture 3