  An isoradial graph is a planar graph in which each face is inscribable into a
circle of common radius. We study the 2-dimensional perfect matchings on a
bipartite isoradial graph, obtained from the union of an isoradial graph and
its interior dual graph. Using the isoradial graph to approximate a
simply-connected domain bounded by a simple closed curve, by letting the mesh
size go to zero, we prove that in the scaling limit, the distribution of height
is conformally invariant and converges to a Gaussian free field.
