A parallelogram tile fills the plane by translation in at most two distinct ways

We consider the tilings by translation of a single polyomino or tile on the square grid Z2Z2. It is well-known that there are two regular tilings of the plane, namely, parallelogram and hexagonal tilings. Although there exist tiles admitting an arbitrary number of distinct hexagon tilings, it has been conjectured that no polyomino admits more than two distinct parallelogram tilings. In this paper, we prove this conjecture.