In response to this CompSciTheory ([cstheory](https://cstheory.stackexchange.com/)) question,
[A simple problem whose decidability is not known
](https://cstheory.stackexchange.com/q/18846/337),
I [posted](https://cstheory.stackexchange.com/a/32797/337) that:

> It is unknown whether or not
[it is decidable to determine if a given shape can tile the plane](https://cstheory.stackexchange.com/q/32538/337),

referring to an earlier cstheory question.
This is even open for polyomino tiles.
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[![Tiling][1]][1]
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<sup>
(Image from [Wikipedia](https://en.wikipedia.org/wiki/Polyomino#Tiling_the_plane_with_copies_of_a_single_polyomino).)
</sup>
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  [1]: https://i.sstatic.net/284oP.jpg