  A lattice is called well-rounded if its minimal vectors span the
corresponding Euclidean space. In this paper we study the similarity classes of
well-rounded sublattices of ${\mathbb Z}^2$. We relate the set of all such
similarity classes to a subset of primitive Pythagorean triples, and prove that
it has structure of a noncommutative infinitely generated monoid. We discuss
the structure of a given similarity class, and define a zeta function
corresponding to each similarity class. We relate it to Dedekind zeta of
${\mathbb Z}[i]$, and investigate the growth of some related Dirichlet series,
which reflect on the distribution of well-rounded lattices. Finally, we
construct a sequence of similarity classes of well-rounded sublattices of
${\mathbb Z}^2$, which gives good circle packing density and converges to the
hexagonal lattice as fast as possible with respect to a natural metric we
define.
