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As a curious bystander fascinated by aperiodic tilings, I skimmed the more informal parts of the (very nice!) paper about the hat monotile and have a few questions about the tilings it produces, specifically how the properties compare of those of Penrose tilings.

  1. Is it correct that both substitution systems (H7/H8 and the HTFP) both produce the same, unique tiling induced by the presented metatiles?

  2. Penrose tilings allow to construct infinitely many different aperiodic tilings. Is the aperiodic tiling presented for the hat tile unique? I.e. even if the presented constructions yield the same tiling - is it known if other tilings are possible (in principle), using some other method to obtain the tiling? If yes, how do they relate to each other?

  3. Penrose tilings contain every finite patch infinitely many times in every tiling. Does something similar hold for the tiling(s) with the hat tile?

I could imagine that these things are simply not known yet, but I might simply have overlooked something or lack some "known" general facts.

EDIT (2023-06-07):

@Quuxplusone:

Concerning point 2/3, I mean the fact discussed e.g. in this question: uncountable number of Penrose tiling

The general statement in this answer makes it look to me (if the statement is correct) that the answer should be probably yes(?), if the answer to question 3. is yes, because this would make the required assumption(s) true (repetitive).

apirogov
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  • Look at the tiling in blue, lite blue and white with the hat tile in the following article (link below). And look at the comments by Epsi Farik. The tiling in the article can be transformed into the aperiodic tiling of 3D blocks. However, Epsi Farik did not explain how to go from one tiling to his. – user25406 Mar 08 '24 at 21:38
  • (continue) In fact the article you mentioned in arxiv provides the explanation in Fig 2.3 on how to go from one tiling to another. Link: https://www.quantamagazine.org/hobbyist-finds-maths-elusive-einstein-tile-20230404/#comments – user25406 Mar 08 '24 at 21:47
  • (continue). In fact the author of the article posted a youtube video (mentioned in the article) that shows how to continuously go from one aperiodic tiling to another. Here's the video: https://www.youtube.com/watch?v=W-ECvtIA-5A – user25406 Mar 08 '24 at 21:53

1 Answers1

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I'm an utter non-expert, but I'll give a preliminary shot at an answer, or at least a guess.

1. I'm completely in the dark. From the structure of your question I bet the answer is "yes, they produce the same tiling," but I don't have any understanding of the facts. :)

2. "Penrose tilings allow to construct infinitely many different aperiodic tilings." Could you edit your question to explain (or even better, link to an explanation of) precisely what you mean? I think I understand, by analogy to the domino tile: I can make an aperiodic tiling of dominos by tiling the plane with squares and subdividing each square horizontally or vertically at random; or I can make an aperiodic tiling of dominos by starting at a "center" and spiraling outward forever. These two tilings are obviously "different," ergo it's possible to construct two different aperiodic tilings with dominos. You're asserting that it's possible to do the same with Penrose tiles,[citation needed] and asking if it's possible to do the same with hats.

One thing that might help is that it is known that you can tile various finite patches with hats in multiple different ways. For example:

If a patch like this appeared in the known tiling, then you could just "toggle" that patch, leaving everything else the same, and you'd have a different tiling. (Or would you?? Maybe you wouldn't!) But I doubt such a patch does appear anywhere in the known tiling, so, you're out of luck anyway. (I am very confident that the left-hand patch never appears in the known tiling, because it has a four-corners vertex with swastika arms, and I think I proved to myself that four-corners vertices appear only when one of the corners is a mirror-hat and thus never have swastika arms.)

3. You mean "contain every finite patch either zero times or an infinite number of times," right? There are certainly possible patches of hats that don't appear at all in the tiling (e.g. the six-hat example above). I believe the hierarchical structure of the tiling does, rather directly, imply that each patch that appears at least once somewhere, appears infinitely many times everywhere. Point to a patch — I'll hop up to $X$, the unique smallest superx-tile that contains your patch; hop over to one of its neighboring superx-tiles $Y$; and hop back down to the corresponding anatomical location in $Y$ to find an exactly congruent patch.

Quuxplusone
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    It is not the question, but same group has published new results about aperiodic tiling, without reflection (see https://arxiv.org/pdf/2305.17743.pdf ) – Lourrran Jun 06 '23 at 06:23
  • good hint concerning the even newer tile, I also just recently learned about it and wanted to check it out once I have the time :) and @Quuxplusone, I answered in an edit to clarify. – apirogov Jun 07 '23 at 11:58