# Are there any aperiodic tilings of an infinite cylinder?

Aperiodic tilings for the plane are quite popular. I can't find any papers on aperiodic tilings on infinite surfaces like a cylinder. One reason might be that these are no longer called tiles but something else (a curved finite surface).

Do references for this exist?

Periodic tilings of cylinder can easily be accomplished with a machine: https://en.wikipedia.org/wiki/Knurling

A lathe machine has parts that rotate and easily create a periodic pattern that has one tile type. Not sure if there exists a machine that would do it with multiple tile types.

My end goal is to find a similar method to aperiodically tile a cylinder. At least one part of rotation would have to exhibit aperiodic behavior, but I leave that outside of the scope of this question.

• It is easy to come up with aperiodic tilings of the cylinder. For example, if the circumference of the cylinder is $C$, then you can start with a periodic tiling by squares, where each square has side length $C/n$. Then you modify this to be aperiodic by shifting each row of $n$ squares by a random amount. Is this what you want? Or are you looking for a set of tiles that only tile the cylinder aperiodically, similar to the Penrose tiles for the plane? Commented Jun 23, 2023 at 18:41
• @MikeEarnest Wouldn't the tiling you propose be periodic in the shift direction, with period the size of the square? Commented Jun 23, 2023 at 19:32

Tilings of the recently discovered aperiodic monotile are "embedded" (not sure if that is the correct word) in a periodic hexagonal tiling. Figure 1.1 from that paper shows that it is possible to cut such an aperiodic tiling into an infinite strip and wrap the strip into a cylinder (for example, I have added two red lines to show a possible strip).

Since the original tile is of finite size, consisting of exactly 16 half-equilateral triangles joined together, and the strip width is chosen in harmony with the periodic tiling, after wrapping (as long as the strip is wide enough), there will only be a finite set of additional tile shapes created by this process, even if in locations where on both side of the strip there are partial tiles which are merged (into tiles of up to 30 half-equilateral triangles). All the tiles after wrapping will also be collections of half-equilateral triangles. This means that yes, such tilings exist. (I admit it's not elegant, but it works, as far as I can see. However, since extra tile shapes have been added, the proof from the original paper that the tiling is necessarily aperiodic no longer applies, i.e., the new set of tiles may very well also allow a periodic tiling.)

Actually, based on Dan Rust's answer, it is impossible that there would be a set of tiles which could only tile aperiodically, because the cylinder can effectively be "reduced" to a one-dimensional line (see the discussion, there).

Interestingly, if the $$y=0$$ line (perpendicular to the cut lines) is also chosen in harmony with the hexagonal tiling, this technique also produces an aperiodic tiling of the infinite "Moebius strip" produced when one of the two sides of the strip is inverted through the $$y=0$$ line before joining the two sides. (For almost the same reason as for the cylinder, there cannot be an aperiodic set of tiles, because the infinite Moebius strip can be "reduced" to $$\mathbb{R}^{+}$$.)

• @C-RAM As Mike Earnest commented on the question, it's not exactly clear what the OP wants from his tilings. Since my answer creates extra tile shapes, the proof from that paper that the original tile can only tile aperiodically no longer applies, so I have not answered the question if the OP desired a set of tiles which can only tile aperiodically. Commented Jun 23, 2023 at 19:38
• Fair enough, I missed that you weren't going for a monotile. Commented Jun 23, 2023 at 19:53
• Nice property. This answer points to a better question: if there exist interesting aperiodic tilings of an infinite strip (that can be wrapped around a cylinder). There must be interesting cylinder tilings that cannot tile an infinite strip. Commented Jun 29, 2023 at 17:59
• @Looft Thanks for the comment, because I think it means my answer was not exactly clear. In the case that the "cuts" to make the strip actually divided tiles on both sides of the strip, I meant for the tile remnants to "merge" after wrapping, so my answer produces a tiling which does not tile the strip (and can contain tiles which are made of up to 30 half-equilateral triangles). But there are many such tilings. My comment on Dan Rust's answer also proposes a tiling based on his answer which does not tile any strip. Commented Jun 29, 2023 at 20:00
• @RonKaminsky Your answer was completely clear. I know that the proposed cut above uses more than 1 tile type. The original question allows for a lot of freedom. But from the answers, it sounds like there is work that can be applied but will require more study from myself. Commented Jun 29, 2023 at 20:52

[EDIT: I realise it's important to give a definition of aperiodic as there are many, especially for spaces that are not Euclidean. Below, I take aperiodic to mean non-periodic and 'repetitive' with respect to the group action, which in this case would be by the group $$\mathbb{R}\times S^1$$. Repetitive means that the orbit-closure of the tiling under the group action has no non-trivial proper invariant subspaces (aka minimality of the dynamical system). Or equivalently, for every radius $$r >0$$, there is an $$R>0$$ such that every patch of radius $$r$$ is contained in every patch of radius $$R$$ up to an $$\mathbb{R}\times S^1$$-translation. This is probably the most common definition of an aperiodic tiling with respect to an action by a locally-compact abelian group. Though, another common (weaker) definition just asks that the orbit-closure of the tiling does not contain any periodic tilings, and that would work here too.]

1 dimensional tilings are essentially symbolic, as it's just the order of the tiles that matters. Likewise, a tiling of the cylinder with a finite number of prototiles can (mostly) be considered to be a sequence of symbols with an additional rotation label for each letter in the sequence. There has been some recent work on substitutions on such compact alphabets, including a couple of my own papers: https://arxiv.org/abs/2204.07516 ; https://arxiv.org/abs/2108.01762 There are some examples of substitutions on circular alphabets in these papers, but the process is easily generalised.

Now, what I said isn't quite true (which is why I said 'mostly' above), because tiles don't need to 'wrap around' the whole cylinder. But if you collect tiles which do wrap around into a patch, then if the tiling has finite local complexity, there will be finitely many of these patches up to some notion of 'minimal patch'. You can then use these patches as symbols of a new alphabet and then you are in the above situation. This process is not one-to-one though unfortunately, as there could be multiple ways of grouping tiles into these wrapping patches.

Anyway, this goes a decent way to reducing the study of 'nice' tilings of the cylinder to instead studying sequences on compact alphabets, of which there are plenty of interesting aperiodic examples.

• So if I understand correctly, included in your proposal would be a tiling where all the tiles are based on $[0,x]\times S_{1}$ for a small set of rationally independent values for $x$, where the upper and lower edges have matching perturbations so that rotational periodicity is broken and the tiles can only "fit" into each other in a specific way. Those would be the 1-d symbols and we now only have to be careful that the sequence of these symbols is not periodic. Commented Jun 24, 2023 at 16:29
• If I understand you correctly, then yes I think it's included. Except it would not necessarily be "aperiodic" by my above definition because 1 dimensional matching rules always admit periodic tiling. Of course, the set of all such tilings would include aperiodic tilings. Commented Jun 24, 2023 at 19:26
• Since 1-d matching rules always admit periodic tilings, your observation that any tiling can be mapped into a new tiling with tiles which are "bands" means that there cannot be a set of tiles which allow only an aperiodic tiling, no? Commented Jun 28, 2023 at 16:51
• Sure, but an aperiodic set of tiles is a different notion to an aperiodic tiling. Commented Jun 28, 2023 at 20:34