  We prove strong crystallization results in two dimensions for an energy that
arises in the theory of block copolymers. The energy is defined on sets of
points and their weights, or equivalently on the set of atomic measures. It
consists of two terms; the first term is the sum of the square root of the
weights, and the second is the quadratic optimal transport cost between the
atomic measure and the Lebesgue measure.
  We prove that this system admits crystallization in several different ways:
(1) the energy is bounded from below by the energy of a triangular lattice
(called $\mathcal T$); (2) if the energy equals that of $\mathcal T$, then the
measure is a rotated and translated copy of $\mathcal T$; (3) if the energy is
close to that of $\mathcal T$, then locally the measure is close to a rotated
and translated copy of $\mathcal T$. These three results require the domain to
be a polygon with at most six sides. A fourth result states that the energy of
$\mathcal T$ can be achieved in the limit of large domains, for domains with
arbitrary boundaries.
  The proofs make use of three ingredients. First, the optimal transport cost
associates to each point a polygonal cell; the energy can be bounded from below
by a sum over all cells of a function that depends only on the cell. Second,
this function has a convex lower bound that is sharp at $\mathcal T$. Third,
Euler's polytope formula limits the average number of sides of the polygonal
cells to six, where six is the number corresponding to the triangular lattice.
