Questions about aperiodic tilings with the hat tile

Question

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).

Answer 1

For your question 2: I think there are fundamentally different patterns. E.g. one with 3-fold symmetry, as shown here.

Similarly to Fig. 2.2 of the original paper, I have colored the chains of “stacked” hats similarly, which allows to see the structures quite clearly.
Certain ones of the “triangles” and/or “parallelograms” of different sizes can function as metatiles here to define this tiling recursively. (I think one should only consider those which are entirely delimited by 3 chains, without interruptions.)
I have noticed that there are parts of a certain size with a central symmetry, e.g. around the black square mark next to the lower left corner. The recursive definition should imply that in this tiling, there are parts with central symmetry of arbitrary size — but at this point, this is no more than a conjecture. (And of course, it should be easy to build a different tiling which has central symmetry altogether. Likewise, a still different tiling with 3-fold symmetry around a “triangle” of six gray hats, like the one just to the right of the black square mark.)

The fact that there are parts with central symmetry of arbitrary size should also hold for the pattern built with the “default” substitution rules as shown below. Note the slightly descending blue/black line of infinite length in both directions.
It would be interesting to find out whether the percentage of (complete) “triangles” and “parallelograms” of a given size is the same for both tilings. A priori, it seems smaller in the tiling with 3-fold symmetry than in this one...

Answer 2

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 super^x-tile that contains your patch; hop over to one of its neighboring super^x-tiles $Y$; and hop back down to the corresponding anatomical location in $Y$ to find an exactly congruent patch.