The aperiodic monotile problem asks whether there exists a single tile that every tiling of the plane made with it results non-periodic. What is known about this problem? If this tile exists, how can it be/not be? how could it (not) be constructed? 


EDIT: I'm thinking about the most restrictive case where the single tile should be homeomorphic to an euclidean closed disk. 


EDIT: 
The kind of answer I'm looking for is, for example, [this result](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.84.5262&rep=rep1&type=pdf) implies that if the monotile is to exist, it cannot be convex. 



(non-)existence of the aperiodic monotile
This recent preprint claims to find such a tile.


[David Smith, Joseph Samuel Myers, Craig S. Kaplan, Chaim Goodman-Strauss, “An aperiodic monotile”,](https://arxiv.org/abs/2303.10798) (2023-03-20) arXiv:2303.10798



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> A longstanding open problem asks for an aperiodic monotile, also known as an "einstein": a shape that admits tilings of the plane, but never periodic tilings. We answer this problem for topological disk tiles by exhibiting a continuum of combinatorially equivalent aperiodic polygons. We first show that a representative example, the "hat" polykite, can form clusters called "metatiles", for which substitution rules can be defined. Because the metatiles admit tilings of the plane, so too does the hat. We then prove that generic members of our continuum of polygons are aperiodic, through a new kind of geometric incommensurability argument. Separately, we give a combinatorial, computer-assisted proof that the hat must form hierarchical -- and hence aperiodic -- tilings.
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Chapter 1 of this paper gives a historical overview of previous results.


[Akiva Weinberger](https://mathoverflow.net/q/182440/5340) points to this preprint in a comment to this question, though I actually first heard about it from another source and then searched MathOverflow for the right question.


