We show that for every surface of positive genus, there exist many quadrilateral manifold meshes that can be texturemapped with locally translated copies of a single square-texture pattern. This implies, for instance, that every positivegenus surface can be covered seamlessly with any of the 17 plane symmetric wallpaper patterns. We identify sufficient conditions for meshes to be classified as "quad-pattern-coverable", and we present several methods to construct such meshes. Moreover, we identify some mesh operations that preserve the quad-pattern-coverability property. For instance, since vertex insertion remeshing, which is the remeshing operation behind Catmull-Clark subdivision, preserves quad-pattern-coverability, it is possible to cover any surface of positive genus with iteratively finer versions of the same texture.Keywords: Modeling