Does there exist a closed curve, with finite area and finite circumference, of which it is undecidable (in an axiomatic system where it is constructable) whether it can tile the plane?


I know the general problem of a set of polygons is undecidable, but I haven't found any information on the single tile case.



Decidability of tiling R^2
If I remember right any Turing machine can be translated into a set of Wang tiles, so that the tiling problem for this set of tiles is equivalent to the decidability of the turing machine.


Assuming the non-existence of an aperiodic set of tiles, Wang provided the Decidability of any tiling problem in R^2. Since this is not possible, it basically proves the existence of aperiodic tiles.


So the general decidability problem turned out to be equivalent to the non-existence of an aperiodic set of tiles , which suggests that the decidability problem for 1 tile might be equivalent to the non-existence of an aperiodic 1 tile. I wonder if the Wang theorem can be changed to prove this, could be a starting point.


Such a tile is called an Einstein tile, and if I remember right the problem of its existence was open until few months ago, but it was reported recently that one was found...


