  Reflexive polygons have attracted great interest both in mathematics and in
physics. This paper discusses a new aspect of the existing study in the context
of quiver gauge theories. These theories are 4d supersymmetric worldvolume
theories of D3 branes with toric Calabi-Yau moduli spaces that are conveniently
described with brane tilings. We find all 30 theories corresponding to the 16
reflexive polygons, some of the theories being toric (Seiberg) dual to each
other. The mesonic generators of the moduli spaces are identified through the
Hilbert series. It is shown that the lattice of generators is the dual
reflexive polygon of the toric diagram. Thus, the duality forms pairs of quiver
gauge theories with the lattice of generators being the toric diagram of the
dual and vice versa.
