We study the $T$-system of type $A$, also known as the octahedron recurrence/equation, viewed as a $2+1$-dimensional discrete evolution equation. The solution is expressed algebraically in terms of the stepped surface over which the initial data are specified as "slanted." We showed that the corresponding combinatorial model for with respect to this initial data is the pinecone dimer model introduced by Propp and West. This allows us to explore the thermodynamic limit of the corresponding dimer models and to derive exact "arctic" curves separating the various phases of the system.
