Covering the Triangular Lattice with Non-Overlapping Unit Disks

For $d>0$, let Td be the triangular lattice with minimum distance d. We study for which d the set Td can be covered by closed unit disks with pairwise disjoint interiors, a question left open by Phetmak, Nuntasombat and Fakcharoenphol for 2-√3≤ d<2/√37 and on several gaps between their constructions. Writing N(d) for the largest number of points of Td in a closed unit disk, we show that Td cannot be covered whenever N(d)<4/d², and we determine this set of spacings exactly as a union of fifteen intervals; combined with their constructions, this shows that coverability is not monotone in d. At the right endpoints d=2/√ k of these intervals the density bound is attained, and we show that such a spacing is coverable if and only if k is the norm of an Eisenstein integer. We also give two new coverings, at d=1/√13 and d=2/√21, in which the disks form a hexagonal packing. Consequently, together with an earlier result of Okayama, Kiyomi and Uehara, the smallest coverable spacing dₘᵢₙ exists and satisfies $2/√55