Dimers and circle patterns

We establish a correspondence between the dimer model on bipartite graph and circle pattern with combinatorics of that graph, which holds for graphs are either planar or embedded torus. The set positive face weights gives global coordinates space patterns dual. Under this correspondence, extends previously known isoradial case, urban renewal (local move models) is equivalent to Miquel patterns). As consequence dynamics governed by octahedron recurrence. special cases these embeddings, we recover harmonic embeddings resistor networks s-embeddings Ising model.