Yesterday on the arXiv appeared a preprint by Socolar and Taylor presenting an aperiodic tiling of the plane with one tile. I haven’t had time to check in any depth yet but that certainly looks interesting indeed, it had been a bit of a holy grail for a long time.
This subject of aperiodic tilings is fascinating: some tilings are aperiodic such that any finite patch will not be enough to say so. Even worse: others are aperiodic but it is impossible to actually prove it (undecidability of the domino problem).