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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 implies that if the monotile is to exist, it cannot be convex.

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This recent preprint claims to find such a tile.

David Smith, Joseph Samuel Myers, Craig S. Kaplan, Chaim Goodman-Strauss, “An aperiodic monotile”, (2023-03-20) arXiv:2303.10798

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.

Chapter 1 of this paper gives a historical overview of previous results.

Akiva Weinberger 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.

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    $\begingroup$ Note that this is a 'mostly' solution to the problem — the question of whether there's such a tile using translations and rotations only (as opposed to translations, rotations and reflections) is still open. $\endgroup$
    – Steven Stadnicki
    Commented Mar 21, 2023 at 18:03
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    $\begingroup$ @StevenStadnicki that asterisk has been removed! They recently found a chiral monotile. cs.uwaterloo.ca/~csk/spectre $\endgroup$
    – Mark S
    Commented May 31, 2023 at 1:19
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In three dimensions there is a known aperiodic tile, the Schmitt–Conway–Danzer. I am pretty sure that this idea can easely be extended to any dimension $d \geq 3$, if anyone cares.

In one dimension, it is known that such tiling cannot exist.

The problem was open in two dimensions until few years ago. And it may be solved, depending on what you understand by aperiodic monotile.

The Socolar-Taylor tile is a tile which tiles the plane only aperiodically. BUT, depending on which version you pick, it is either disconnected or not the closure of its interior (you can make it connected by adding few curves to connect pieces).

If you want a tile which is connected and closure of its interior, the problem is still open as far as I know...

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Richard Kenyon showed that any topological disk which can tile by translates can also tile periodically.

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    $\begingroup$ Not exactly. That paper talks only about tilings of the plane with parallel traslates. These are MUCH easier than general tilings. $\endgroup$
    – Igor Pak
    Commented Oct 4, 2014 at 5:19
  • $\begingroup$ Ok, I fixed the statement. $\endgroup$
    – Ian Agol
    Commented Oct 4, 2014 at 6:02

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