# Are local rearrangements in aperiodic tilings possible?

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?

• 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. Commented May 26, 2023 at 10:32