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Is it possible to perform a local rearrangement of tiles in an aperiodic tiling (such as the Penrose tiling or certain sets of Wang tiles), such that all matching rules are maintained? By "local" I mean that the number of rearranged tiles is finite.

Is there a simple argument why this is (is not) possible? Maybe there is an obvious example I missed?

EDIT: as RavenclawPrefect shows in their answer, one can construct "trivial" examples of such tilings with allowed local rearrangements. Let me formulate a stricter question. Suppose, I define an aperiodic tiling to be "reducible" ("irreducible") if one can (cannot) construct a smaller set of aperiodic tiles by i) gluing one tile to another and ii) removing a tile from the set. Then, the example of RavenclawPrefect, which has 3 tiles in the set, is reducible, because we can glue two half-circles to each rhombus and then remove the half-circle from the set of tiles, thus reducing the tiling to the usual Penrose tiling with 2 tiles.

So, my question is: are local rearrangements in irreducible aperiodic tilings possible? In particular, can you perform local rearrangements in any of the tilings from this list?

Alehud
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    There are particular tilings in the hull of aperiodic cut and project sets that differ only in a compact region. These are known as the singular points in the hull (actually there are different kinds of singular points, some corresponding to compact local rearrangements, but others corresponding to 1 dimensional 'lines' of local rearrangements). If I remember right, the substitution-periodic points of the Penrose kite and dart substitution are equal everywhere except for a small ball at the origin that only includes 10 or so tiles. – Dan Rust May 26 '23 at 10:32
  • Where can I read about this? – Alehud May 27 '23 at 05:26
  • Any text that discusses the hull/tiling space of cut and project sets (aka model sets). – Dan Rust May 27 '23 at 21:11

1 Answers1

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Consider the Penrose tiles (with edge modifications as necessary to enforce the matching rules), but where we cut a small circular hole in one of the pieces and include an additional half-circle tile of which two copies fill the hole.

This three-piece set of tiles can only tile aperiodically, but within any circle we can rotate the two halves however we like.

RavenclawPrefect
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  • I see. You are technically correct, but I had in mind more non-trivial examples. Let me edit the question and formulate it more precisely. – Alehud May 26 '23 at 04:43
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    I suspect there still exist rearrangeable aperiodic tilings under your new constraints, although it might be true that for most of the "standard" ones you can't. In particular I believe you can't perform local rearrangements of the Penrose tiling, because it's determined via a pentagrid whose parameters should be inferable from a cofinite piece of the tiling. – RavenclawPrefect May 26 '23 at 19:46