Unavoidable Crossings in a Thinnest Plane Covering with Congruent Convex Disks

Two convex disks K and L in the plane are said to cross each other if the removal of their intersection causes each disk to fall into disjoint components. Almost all major theorems concerning the covering density of a convex disk were proved only for crossing-free coverings. This includes the classical theorem of L. Fejes Toth (Acta Sci. Math. Szeged 12/A:62–67, 1950) that uses the maximum area hexagon inscribed in the disk to give a significant lower bound for the covering density of the disk. From the early seventies, all attempts of generalizing this theorem were based on the common belief that crossings in a plane covering by congruent convex disks, being counterproductive for producing low density, are always avoidable. Partial success was achieved not long ago, first for “fat” ellipses (A. Heppes in Discrete Comput. Geom. 29:477–481, 2003) and then for “fat” convex disks (G. Fejes Toth in Discrete Comput. Geom. 34(1):129–141, 2005), where “fat” means of shape sufficiently close to a circle. A recently constructed example will be presented here, showing that, in general, all such attempts must fail. Three perpendiculars drawn from the center of a regular hexagon to its three nonadjacent sides partition the hexagon into three congruent pentagons. Obviously, the plane can be tiled by such pentagons. But a slight modification produces a (non-tiling) pentagon with an unexpected covering property: every thinnest covering of the plane by congruent copies of the modified pentagon must contain crossing pairs. The example has no bearing on the validity of Fejes Toth’s bound in general, but it shows that any prospective proof must take into consideration the existence of unavoidable crossings.