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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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    $\begingroup$ A quick internet search reveals much information, for example, the Socolar-Taylor tile. $\endgroup$ – Joel Reyes Noche Oct 3 '14 at 6:57
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    $\begingroup$ The wikipedia article on the einstein problem gives info on some more claimed solutions, such as the Schmitt-Conway-Tanzer tile. $\endgroup$ – Yoav Kallus Oct 3 '14 at 16:28
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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 Oct 4 '14 at 5:19
  • $\begingroup$ Ok, I fixed the statement. $\endgroup$ – Ian Agol Oct 4 '14 at 6:02

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