A triangular lattice is a two-dimensional point lattice generated by two vectors of equal length
meeting at an angle of 
. It is the root
lattice 
up to scale and rotation, and is the set
of vertices of a triangular
grid. Its Voronoi cells are regular
hexagons, so the same point lattice is sometimes
called a hexagonal lattice in packing contexts.
For each 
, the density-one triangular lattice uniquely
minimizes the Gaussian interaction energy with pair potential 
among planar periodic configurations of density
one, up to Euclidean motion, where 
is the distance between two points. Faulhuber and Petersen (2026) proved the equality
characterization using the auxiliary functions of the
earlier Gaussian energy bound.
Let 
have minimum distance 
between lattice points. Kittipassorn et al. (2026)
constructed coverings of 
by closed unit disks with pairwise
disjoint interiors at 
and 
. The disks
cover the lattice points, rather than the entire plane.
A smallest coverable spacing exists and satisfies
 |
Coverability by such disks is not monotone in 
.
