Abstract: We study the T-system, also known as the octahedron relation, which solution corresponds to the partition function for dimer coverings of the Aztec Diamond graph. After studying the solutions for the uniform initial data, we find exact solutions for a particular (and more general) class of periodic initial conditions. We show that the density function, that measures average dimer occupation of a face of the Aztec graph, obeys a linear system of equations with periodic coefficients. We explore the thermodynamic limit of the dimer models and derive exact "arctic" curves (generalizing the arctic circle for domino tilings on the aztec diamond) separating various phases. We finish with a discussion of possible generalizations and future applications to other recursion relations (joint work with P. Di Francesco).