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?
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.