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The notion of $SL_2$-tiling is a generalization of that of classicalCoxeter-Conway frieze pattern. We classify doubly antiperiodic $SL_2$-tilingsthat contain a rectangular domain of positive integers. Every such$SL_2$-tiling corresponds to a pair of frieze patterns and a unimodular$2\times2$-matrix with positive integer coefficients. We relate this notion totriangulated $n$-gons in the Farey graph.

$SL_2(\mathbb{Z})$-tilings of the torus, Coxeter-Conway friezes and Farey triangulations