From the classification of monohedral tilings with convex pentagons, we know that all convex pentagons which tile the plane can do so periodically; I'd like to know whether the same result is known to extend to non-convex pentagons. That is, if congruent copies of a pentagon $P$ tile the plane, is there necessarily one such tiling which is composed of a single patch of finitely many tiles, translated periodically in a lattice?
There are unlikely to be any known counterexamples, since the Einstein problem remains open, but I am curious whether there exists a proof of this statement - if so, a reference would be appreciated.
Einstein problem is open for the pentagons (Check Rao article), he only 'supposedly' closed it for the case of convex pentagons.