  The hexagonal polygon model arises in a natural way via a transformation of
the 1-2 model on the hexagonal lattice, and it is related to the high
temperature expansion of the Ising model. There are three types of edge, and
three corresponding parameters $\alpha,\beta,\gamma>0$. By studying the
long-range order of a certain two-edge correlation function, it is shown that
the parameter space $(0,\infty)^3$ may be divided into subcritical and
supercritical regions, separated by critical surfaces satisfying an explicitly
known formula. This result complements earlier work on the Ising model and the
1-2 model. The proof uses the Pfaffian representation of Fisher, Kasteleyn, and
Temperley for the counts of dimers on planar graphs.
