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One of the more mind-boggling aspects of the Penrose tiles is that there are uncountably many distinct tilings of the plane, but every tiling contains every finite region that appears in another tiling.

I am wondering how much is known about whether the recently announced aperiodic monotile has these properties, either from closer inspection of the paper, or other means.

My main question is whether different tilings by the same monotile all share the same finite regions, as in the case of Penrose? Of course this question really only makes sense if there is more than one distinct tiling by the monotile.

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    $\begingroup$ @SamHopkins indeed, but do you really think I should ask them all separately? I don't want to spam up the place. $\endgroup$
    – Jim Conant
    Commented Mar 28, 2023 at 1:43
  • $\begingroup$ @SamHopkins I edited. $\endgroup$
    – Jim Conant
    Commented Mar 28, 2023 at 1:47
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    $\begingroup$ You're right, the questions seem more related to one another than I thought at first reading. $\endgroup$
    – Sam Hopkins
    Commented Mar 28, 2023 at 1:48

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The property of every tiling showing any finite portion of every other tiling is called repetitivity. Every primitive substitution tiling is repetitive (a substitution rule is primitive if its associated matrix $A$ is, meaning that there exist an $n$ such that every entry of $A^n$ is positive) because you'll find every proto-tile after some iterations of the substitution rule, every 2-tile patch after some more interatios, and so on. This is the case for Penrose tilings as many other known aperiodic tilings of the plane.

As per the new monotile, as far as I understand, there are infinitely many different tilings. They are constructed with a novel "fancy" method similar to a substitution rule. By fancy I mean that it is not a substitution rule in the usual way, but defined on some so-called "meta-tiles" instead of the usual tiles. However the intuition behind the argument from the previous paragraph should be similar for checking repetitivity.

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  • $\begingroup$ Is there a link between repetitivity of such aperiodic tilings and Voronin's universality theorem on the conceptual level? $\endgroup$
    – Sylvain JULIEN
    Commented Mar 29, 2023 at 20:01
  • $\begingroup$ @SylvainJULIEN I was wondering about the same thing :) some kind of "discrete" version. $\endgroup$
    – Wolfgang
    Commented May 5, 2023 at 14:54

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