# How many fixed polyominoes does it take to force an aperiodic tiling of the plane?

A longstanding open problem asks whether there is a single connected tile that only tiles the plane aperiodically. As far as I know, this is still open even in the case of polyominoes: we do not know if there is a single polyomino that forces an aperiodic tiling of the plane.

I am curious about the case where we treat the polyominoes as fixed, i.e., when we consider rotated/reflected copies of a tile as distinct, and only permit translations. In this situation, we do know that more than one tile is needed; as shown in this paper, a single fixed polyomino tiles the plane periodically if it forms any tessellation at all. (In fact, we can ensure the tiling is isohedral!)

Conversely, there exist finite sets of fixed polyominoes which tile the plane, but not in a periodic manner. If we use Wang tiles to generate polyominoes, we can obtain a solution with $$11$$ tiles (which is minimal), but there are smaller solutions. For instance, consider Matthew Cook's set of three polyominoes which force an aperiodic tiling (with rotations and reflections allowed):

We observe that the tiling only makes use of $$4$$ orientations each of two of the tiles, and a single orientation of the third tile, so only $$9$$ fixed polyominoes are needed in total.

The answer is therefore somewhere between $$2$$ and $$9$$ inclusive; I am interested in learning of any improvements to either the upper or lower bounds, or pointers to discussion of this problem in the literature.

One potential avenue for a smaller upper bound is the aperiodic set of polyominoes described at this Wolfram MathWorld page as being announced by Roger Penrose in 1994, but it gives no further details, and I have been unable to track down a reference.

• The Penrose set was described in Shadows of the Mind (R. Penrose, 1994, p. 32), and it looks like it uses at least 16 orientations (maybe even more). There are also images of the tiling in this Spanish paper "Los mosaicos de Penrose por poliominós" (P.G. Sequeiros) that you can download here: casanchi.org/mat/mosaicospenrose01.pdf Commented Jun 3, 2021 at 18:36
• Just a note that Nicolas Ollinger claims that Ammann et al. found a set of $8$ fixed polyominoes which tile aperiodically, citing this paper, but I can't actually find a claim to this effect or any examples in the paper - they all seem to require a ninth "key" tile if we don't have edge annotations. Commented Sep 1, 2022 at 6:21