  Recent work by Forsg{\aa}rd indicates that not every convex lattice polygon
arises as the characteristic polygon of an affine dimer or, equivalently, an
admissible oriented line arrangement on the torus in general position. We begin
the classification of convex lattice polygons arising as characteristic
polygons of affine dimers. We present several general constructions of new
affine dimers from old, and an algorithm for finding affine dimers with
prescribed polygon. With these tools we prove that all lattice triangles,
generalised parallelograms, and polygons of genus at most two admit an affine
dimer.
