Regular affine tilings and regular maps on a flat torus Original Research Article

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 be affine images of each other. The usual codes are calculated from the rotation numbers and multiplicity. The results are extended to regular toroidal maps.