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In these articles "Mathematicians Excited About New 13-Sided Shape Called 'the Hat'" (Gizmodo), "An 'einstein' tile? Mathematicians discover pattern that never repeats" (Interesting Engineering), from the paper An aperiodic monotile (arXiv) from Smith, et al., they make a claim

Researchers identified a shape that was previously only theoretical: a 13-sided configuration called “the hat” that can tile a surface without repeating."

If we look at the image, we can see at least three places where the pattern repeats

enter image description here

What do they mean exactly when they say that? Is it that they can repeat but not touch or something else?

What exactly are they proving to show this given we can see repeating patterns?

Blue
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Moo
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    It's not that there isn't any particular piece that doesn't repeat itself (I mean, the single tile itself repeats over and over!). It's that the entire thing is not obtained by repeating a single pattern over and over. A periodic tiling is one in which you can find a parallelogram (usually larger than the tiles) that is repeated to form the same overall tiling. See e.g., here. An aperiodic tiling is one that is not periodic. Here you do not have a single pattern that repeats ad infinitum to obtain the overall layout. – Arturo Magidin Mar 28 '23 at 19:54
  • @ArturoMagidin: Thank you for that clarification! – Moo Mar 28 '23 at 19:55

2 Answers2

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A tiling is aperiodic if it does not have any (global) translational symmetry. However, in many cases, every finite portion of an aperiodic tiling will repeat infinitely many times (as an example, you can see the same property in an irrational number's decimal expansion). This property is called repetitivity.

The novelty of the newly discovered "Hat" monotile is not only that it is possible to construct aperiodic tilings with it (it is also possible with 1-2 right triangles as in the pinwheel tiling), but also that it is impossible to construct a periodic tiling out of it.

Melquíades Ochoa
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A periodic tiling of the plane is one where you can translate the entire pattern to another point and overlay it exactly with itself. For example, if you look at a hexagonal grid then if you move any one hexagon to overlap another, the entire pattern overlaps.

An aperiodic tiling, such as the one discovered using the Einstein tile, has the property that no translation of the pattern will overlap perfectly with itself - although you've found three sections that are the same, notice that if you overlay two of them together then the rest of the pattern doesn't match. Aperiodic tilings had previously been found using multiple shapes - the Penrose tiling uses two different tiles to cover the plane, and it also has rotational symmetry meaning that if you rotate the pattern 72 degrees you get a perfect overlap - but this is the first time that an aperiodic tiling has been found that uses only one shape.

ConMan
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    Aperiodicity is more than just non-periodicity. One generally also requires repetitivity or some other indicator of 'low complexity'. For example, a random tiling would not be considered to be aperiodic. Non-periodic tilings using a single tile are easy to produce. Just take a load of $2\times1$ rectangles and put them together in a non-periodic way. One can even make a repetitive tiling this way (see the Table tiling). The real importance of the hat monotile is that the tile itself can tile the plane on its own, but cannot be used to make a periodic tiling. – Dan Rust Mar 07 '25 at 15:09