Two metric triangulations are said to be discretely conformally equivalent if one can multiply the metric
by a constant at each vertex to convert one into the other. Springborn proved a discrete uniformization theorem on such triangulations, using a convex functional based on 3-dimensional hyperbolic volume. One can easily generalize to surfaces whose faces are cyclic quadrilaterals.
We show how this uniformization is related to the dimer model, and how the integrability in the dimer model provides an integrable structure for the underlying
circle patterns. One consequence is a geometric realization of the Hirota bilinear difference equation (a.k.a. the octahedron recurrence) using circle patterns.
This talk is based on joint work with Alexander Goncharov, Boris Springborn and Alexander Bobenko.