On translating one polyomino to tile the plane

Given a polyomino, we prove that we can decide whether translated copies of the polyomino can tile the plane. Copies that are rotated, for example, are not allowed in the tilings we consider. If such a tiling exists the polyomino is called anexact polyomino. Further, every such tiling of the plane by translated copies of the polyomino is half-periodic. Moreover, all the possible surroundings of an exact polyomino are described in a simple way.