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A regular affine tiling of a flat (locally isometric to a euclidean plane) torus is defined to be the affine image of a tiling of a flat torus with congruent regular p-gons, adjacent ones sharing a side. Only triangular, hexagonal, and quadrangular affine tilings exist. Each tiling is determined up to a shift by its set of rotation numbers and its multiplicity. Criteria are given for two tilings to...