Composite models of polygons

A composite polygon is composed of a lattice polygon in the square lattice, which contains in its interior an internal structure, which may also be a lattice polygon, or a lattice tree or a lattice animal, or a lattice disc ( or a collection of these). The properties of composite polygons are conside red in this manuscript. In particular, I shall consider the growth constant s and generating functions of these models, as well as the statistical mech anics of interacting models of composite polygons. It is shown that there i s an adsorption transition of the internal structure on the containing poly gon, and a transition which corresponds to the inflation of the containing polygon (by the internal structure).