Talk:Aperiodic tiling

message: The article is looking better; still room for many useful edits. Can we now clean out this discussion page? Getting a little overgrown with old debates, I think. I would like only to keep (a) a new discussion of the perenial trouble with terminology, to head this off in the future. (b) a discussion of future additions -- other spaces, better discussion of the cut and project method, perhaps a full list of all known examples.--C Goodman-Strauss 20:17, 7 October 2007 (UTC)
thread_title:  Time to clean up this page!

message: no need to keep this note here forever; just wanted to mention an edit and its rationale. The Constructions section began  with two paras of patent nonsense which has been deleted. There was a sort of befuddled intro to  Amman bars though, that should be sorted out and stuck back in.--C G Strauss 21:10, 6 October 2007 (UTC)
thread_title:  removed some nonsense 

message: The section on symmetry has some underlying merit; the point is that there are forbidden symmetries in periodic tilings and some aperiodic sets of tiles admit only  tilings with some  interesting symmetries. But the discussion was not clear as written, confusing (yet again) the distinction between nonperiodicity and aperiodicity. Furthermore, non-periodic tilings with 'forbidden' symmetries are trivial to concoct.  I don't know how to rewrite it, or that it should be, but took it out --C Goodman-Strauss 21:31, 7 October 2007 (UTC)
thread_title:  removed more nonsense 

message: On the tiling listserve, recently, there was a discussion of a comprehensive list of all known 2D examples; this might be interesting to incorporate into this article, and expand into a list of every known example in every setting. There really aren't that many! If this goes forward, I would suggest merging and weaving together the Constructions and history sections. --C G Strauss 21:14, 6 October 2007 (UTC)
thread_title:  An absolutely comprehensive list?

message: This article does not define what "periodic", "aperiodic" and "non-periodic" means, making most of it rather incomprehensible unless you are already accustomed to the subject. --OpenFuture (talk) 20:07, 2 November 2008 (UTC)
replies: text: Sure it does, in the second paragraph. First it defines "tiling", then "periodic tiling". "Non-periodic tiling" is simply a tiling that is not periodic, that's what the common English "non-" prefix is for. Finally, it defines an "aperiodic set of tiles" as being a set of tiles that can only be used to create non-periodic tilings. Anomie⚔ 23:43, 2 November 2008 (UTC)
thread_title:  Incomprehensible. 

message: What is meant by "the undecidability of the Domino Problem ensures that there must be infinitely many distinct principles of construction"? To me Berger's construction clearly has the "non-periodic hierarchical structure" mentioned in the sentence before. The hierarchical structure just allows a turing machine to be embedded in it but the non-periodicity very much seems to come from the hierarchical structure. What is meant by distinct principles of construction? Dskloet (talk) 20:11, 4 April 2010 (UTC)
thread_title: Principles of construction

message: Shouldn't there be a seperate article on the tiles that can only generate non-periodic tilings and not only the tilings obtained from them? I think most papers on this topic discuss the sets of tiles generating these tilings. Toshio Yamaguchi (talk) 00:13, 12 July 2010 (UTC)
thread_title: Own article about aperiodic tiles

message: I re-added an opening to the article that makes it somewhat comprehensible. The old opening sentence, "An aperiodic tiling is a tiling obtained from an aperiodic set of tiles," approaches self-parody.  — Preceding unsigned comment added by Garry SF (talk • contribs) 06:50, 22 December 2010 (UTC)
thread_title: Opening of article

message: I quote:
"Wang showed that such an algorithm exists if every finite set of prototiles that admits a tiling of the plane also admits a periodic tiling.
[...]
Hence, when in 1966 Robert Berger demonstrated that the tiling problem is in fact not decidable,[7] it followed logically that there must exist an aperiodic set of prototiles."
From the first statement it does not follow that there must exist an aperiodic set of prototiles, so the second statement is wrong. I don't know however, if Wang actually showed an equivalence between (exists algorithm) and (if exists tiling then exists periodic tiling) rather than the mere implication in the first statement in the quote. My reading of "Tilings and patterns" by Grünbaum and Shephard suggests no, but not unambiguously so. 
I think the text should be changed to something like
[In 1966 Robert Berger found an aperiodic tile set] (most likely)
or
[Wang showed equivalence rather than implication and Berger showed undecidability of the tiling problem.]
according to which is the case.
Yuide (talk) 16:04, 23 February 2011 (UTC)
thread_title:  Error in history section 

message: Just to be clear, are these patterns aperiodic in terms of the basic tiles only or are they also aperiodic in terms of larger structures (say a group of 679 tiles)? These patterns are mindboggling: intuition would tell you that there is always some periodicity in an infinite area (you'd think that after a while all possible combinations would be exhausted), but apparently that's not true. 195.169.213.92 (talk) 12:46, 5 October 2011 (UTC)
replies: text: Find a patch of tiles you like.  You will also find that same patch somewhere else.  But if you look double the distance in the same direction, and triple the distance, etc, you will not find that patch at all the places you would expect if it was crystalline/periodic. --99of9 (talk) 12:53, 5 October 2011 (UTC)

text: I am not sure if I really understand what you are trying to ask. Aperiodicity is a property of the tiles (for example the two Penrose rhombs). These tiles are aperiodic if and only if all tilings they can produce are nonperiodic, ie. they can not produce any periodic tilings. I don't think it makes any sense to look at something like your group of 679 tiles as it only makes sense to talk about periodicity and aperiodicity / nonperiodicity when looking at infinite patterns of tiles. For example, how would you determine if your patch of 679 tiles is part of a larger periodically repeating structure or not by only looking at this patch locally? The answer is you can't. And you are perhaps correct that aperiodicity is a quite counterintuitive phenomenon. After all, tilings had already been studied by people such as Albrecht Dürer and Johannes Kepler but the theory of aperiodic / nonperiodic tilings only developed in the 1960s. Toshio Yamaguchi (talk) 13:06, 5 October 2011 (UTC)
replies: text: I was talking about infinite patterns. What I meant was that intuition would tell you that if you use a finite number of different tiles, you will run out of arrangements at some point (say you do find patches of tiles at periodic distances, but they have been rotated by 3 degrees, then after 120 steps you will have some sort of repetition). If it takes 679 tiles to run out of arrangements then the structure of the first to the 679th tile may repeat. It also may not repeat because the next 679 tiles have the same arrangements, but in a different order. However, after a while you will run out of orders as well and you will have a larger structure that might repeat itself, etc... For the whole infinite pattern to be aperiodic one would require every successive step of larger structures to have different orders after cycling tthrough all combinations to keep any one of them from repeating (it must also be true that re-ordering some big structure doesn't incidentally cause smaller structures to become periodic) so that any repeating structure would have to be infinitely large itself. The fact that mathematicians have found patterns that fit these conditions is quit counter intuitive. I can't imagine any mathematics student (except maybe for a true genius) who hears about aperiodic tilings for the first time thinking "yeah sure, it makes sense that one can construct a never repeating pattern that fills space out of a handful of tiles, and then prove it", because the human mind expects a pattern to emerge at some point, even if it's only in terms of structures of trillions of tiles. the fact that it took mathematicians centuries to figure it out underscores my point I think. I feel the article doesn't acknowledge this and leaves the lay reader with too many conceptual questions.87.212.243.106 (talk) 08:53, 7 October 2011 (UTC)
replies: text: I still somehow don't get your point. You say "say you do find patches of tiles at periodic distances, but they have been rotated by 3 degrees, then after 120 steps you will have some sort of repetition". One important point is that mere repetition is not unique to periodic tilings. For example the patterns (that is, any finite patch of tiles no matter how large) in a Penrose tiling also repeat an infinite number of times. This necessarily follows from the hierarchical order in these tiles. For any finite patch (even one containing some 10^{100000000} tiles) there exists a larger tile in the inflation hierarchy that completely contains this patch and thus wherever the inflated tile appears in the inflated tiling (which is an infinite number of times in an infinite tiling), this patch of 10^{100000000} tiles appears in the original tiling. It is important to understand what periodicity actually means. In tiling theory it means, there exists a fundamental domain that can generate the tiling only using translation. However in an aperiodic tiling, such a fundamental domain does not exist. I am all open to make this article clearer for reader, but in this particular topic one has to be very careful not to generate statements that are factually incorrect in the process of this simplification. Do you have any explicit suggestions how the article could be made clearer? Toshio Yamaguchi (talk) 09:41, 7 October 2011 (UTC)
replies: text: In my example only rotation (in the same direction every time) prevents periodicity, therefore, after a while the rotations add up to a multiple of 360 degrees and the whole process starts over again. In this example there is periodicity of groups of tiles because you CAN translate over a certain distance and find the exact same group. I know I used a specific example: if the rotations didn't have to be in the same direction then you can postpone periodicity by arranging the tiles between two tiles with the same direction differently each time, but that process will also run out of combinations at some point after which you get a larger structure that threatens to be periodic, unless you repeat the whole thing treating these structures as tiles and rotating them relative to each other and after a while you have to take it to the next level with an even bigger structure, and so on, and without inadvertently having the rotation of a structure in the chain make a smaller structure down the chain become periodic. This has to go on indefinitely to make the tiling aperiodic across space. It is mind boggling that mathematicians have found such tilings (after centuries of work) while this article treats it like it is trivial. Similar to wikipedia pages on relativity there should be a clear example to guide the reader through the concept and explain why/how it works. The Robinson 1971 example in this article is a start but should be expanded. I'm a physics person who lacks the mathematical knowledge and teaching skills to expand the Robinson example but maybe someone more knowledgeable could do it, maybe you? I'm just a casual editor to wikipedia and I'm not a mathematician so I'd understand if you guys reject a change.195.169.213.92 (talk) 16:55, 8 October 2011 (UTC)
replies: text: I don't think the phenomenon of aperiodicity can be understood well without understanding what an "Aperiodic hierarchical tiling" (as Goodman-Strauss calls it) is, since most aperiodic tilings known so far belong to this type. What about adding a graphic such as this one (see Figure 15, page 53) to the article? I could make a better annotated version of this image I created (showing how the primitive cell is translated to form a tiling) and then showing that such a primitive cell cannot exist in the Robinson tilings because of the hierarchy (which is visible through the squares with the rounded corners) the tiles form when added together. Toshio Yamaguchi (talk) 01:27, 9 October 2011 (UTC)
thread_title:  Aperiodic in terms of what? 

message: I think a lot of the confusion in the discussions of this article probably started through the confusing title of this article. As the first paragraph of the article tries to make clear, there is actually no such thing as aperiodic tilings, but rather aperiodic tiles. However, several comments on the talk page show that many people are not understanding.
The point was made in a 2007 discussion above that the article should be renamed "Aperiodic sets of tiles". (I suggest that "Aperiodic tiles" might be more succinct.) C Goodman-Strauss, an expert in this field, at first resisted this idea, but then accepted that the article should be renamed, noting in a later discussion that there really was no such thing as an "aperiodic tiling". But the change was never made.
Is there any reason not to now rename the article and simplify the opening paragraph? --seberle (talk) 21:32, 11 June 2012 (UTC)
replies: text: The preceding anonymous comment is a good example of what I mean. The sentence is meaningful, but I doubt that anyone can understand it if they do not already understand about aperiodic tiles. Remember, this is an encyclopedia entry and most people who come here want to understand a topic they do not currently understand. We really need a simpler, more direct presentation, beginning with a more accurate, less confusing title. Do others agree? --seberle (talk) 03:13, 3 November 2012 (UTC)
replies: text: Maybe this is a case where Wikipedia should consider starting an article in a style OTHER THAN "an X Y is an X that is Y ...." --2607:FEA8:86DC:B0C0:541:CB0F:2EEA:D251 (talk) 22:33, 10 December 2021 (UTC)
thread_title:  Article title 

message: This is an incredibly convoluted sentence: "An aperiodic tiling is a tiling obtained from an aperiodic set of tiles." It might as well be "A left turn is a turn that goes left." This article needs to better summarize the topic in the introduction, however, that summarization appears to have become worse over time.  — Preceding unsigned comment added by 12.28.104.10 (talk) 14:03, 11 July 2013 (UTC)
replies: text: If you can help make the introduction clearer, please do so! It is an unfortunate fact that "aperiodic tilings" suffers from a confusing definition. It is not tautological, but it is not easy to grasp either. Aperiodic tilings are not really about the tilings at all, but rather about the tiles. This is what the first sentence is stating. Aperiodic tilings are defined in terms of aperiodic tiles, which are then defined in the second sentence. Aperiodic tiles are then defined in terms of nonperiodic tilings, which seems circular unless you step through the different definitions carefully, understanding that aperiodic tilings and nonperiodic tilings are not the same thing. It has been lamented several times both on this Talk Page and in other sources that the definition is confusing. The title of the article should really be "Aperiodic Tiles". But I think the article is correctly following current (confusing) usage in the literature. Suggestions for making this easier for the reader are welcome! --seberle (talk) 15:30, 17 July 2013 (UTC)
thread_title:  Tautological first sentence 

message: Statements like this are unsatisfying (maybe embarassing) for a mathematical field: "There is no mathematical concept of aperiodic tiling per se." In the past, unfortunately such a statement was just true. I think now things have changed to the better. If the term "aperiodic tiling" has no precise meaning, it should get one. Finally it got one. Baake and Grimm, "Aperiodic Order", Cambridge University Press 2013. Def 4.13 defines an aperiodic (symbolic, 1D) sequence: "a bi-infinite sequence is called aperiodic, when its hull contains no periodic sequence." Def 5.12 defines an aperiodic point set (in R^d): "A point set is non-periodic [if it has no period]. An FLC set is called aperiodic, if all elements of its hull are non-periodic." In short this means roughly "aperiodic = non-periodic & repetitive". By the preceding comment in the book (p. 138) this definitions apply to FLC tilings too. This definition covers the demand for ruling out boring non-periodic tilings like ....aaaaaaaabaaaaaaa....
Regarding "aperiodic set of tiles" (a term we want to keep) Def 5.21 says "A finite set of prototiles is called aperiodic [if it tiles R^d and allows only nonperiodic tilings]". Exactly as it was used during the last decades. I'm very happy with this. I suggest editing the page according to these definitions. This solves many (the majority?) of the objections on this talk page. The changes would include roughly: 
1. Change intro
2. Add stuff to History regarding substitution rules, model sets (what else?)
3. Keep "Constructions" and "Physics", delete "Terminology".
Objections? --Frettloe (talk) 16:25, 13 November 2013 (UTC)
replies: text: Objection: this definition of aperiodicity does not have anything to do with aperiodic tile set, that is, with the idea of non-periodicity enforced by local constraints. Actually, I do not see the difference with the definition of quasiperiodicity (non-periodic repetitive). 82.225.164.246 (talk) 10:18, 8 January 2014 (UTC)
replies: text: First point: agreed, it has not much to do with "aperiodic tile set". This is why there is now a new page for "aperiodic set of tiles". Second point: agreed, what you call "quasiperiodic" is pretty much the same as what is described here as "aperiodic". Can you name a source for your definition of "quasiperiodic"? On wikipedia, for instance, quasiperiodic means something different from your definition. Frettloe (talk) 02:35, 9 January 2014 (UTC)
thread_title:  Terminology again 

message: This example pretends to show five and ten fold rotational symmetry among many points. Maybe you can see many rhombs, darts, kites, pentagons. The tiling consists of two basic shapes: a bowtie (concave hexagon) and a "boat" (flat convex hexagon). In the picture the bowties are gray, and the boats are white. Ad Huikeshoven (talk) 09:38, 2 January 2015 (UTC)
thread_title:  another example 

message: I think this article should mention the first einstein tile the Socolar-Taylor tile. This is a single tile which is aperiodic in the Euclidean plane. 121.45.79.203 (talk) 13:25, 25 October 2015 (UTC)
thread_title:  Socolar-Taylor 

message: 11-fold and 17-fold tilings exist.  They are stated to be "quasi-periodic".  Whether or not they are aperiodic might need further investigation.  Sites are:
Quasi-crystals and the Golden Ratio
Symmetry and chaos
What are Quasicrystals?
agb  — Preceding unsigned comment added by 143.43.206.194 (talk) 20:37, 6 February 2017 (UTC)
thread_title:  Possibly missing n-folds 

message: Most of the above discussion was written before someone made an entirely separate Aperiodic set of prototiles article, which I just noticed...
Maybe we should just merge into that article? It looks to me like this article talks almost entirely about aperiodic sets of prototiles. --2607:FEA8:86DC:B0C0:541:CB0F:2EEA:D251 (talk) 02:25, 11 December 2021 (UTC)
replies: text: I fully agree to this suggestion! Many times people said in this discussion that the page's title is mathematically incorrect. This is definitely true, so one of the authors should take this step. (I cannot do it properly since I am not a native speaker.) --Nessaalk (talk) 11:19, 17 April 2023 (UTC)
replies: text: If I understand correctly, the title for this article, "Aperiodic tiling," used to be mathematically off because only prototiles could be properly said to have the property of aperiodicity. An "aperiodic tiling" simply referred to a tiling made with aperiodic tiles. But I think the definition has shifted, so now there is a distinct definition for an aperiodic tiling which does not depend on the definition of aperiodic prototiles. Currently, the article opens, "An aperiodic tiling is a non-periodic tiling with the additional property that it does not contain arbitrarily large periodic regions or patches." I am concerned this is not the definition we find in standard literature. --seberle (talk) 13:56, 30 April 2023 (UTC)
thread_title:  Aperiodic set of prototiles 

message: They said ...aaabaaa... Is aperiodic in 1d, but isn't it symmetric to reflection? And if it ends somewhere it's a finite tiling, which is not by definition infinite...
I'm not sure if to fix it cuz I might be mistaken someone make sense of this please? 2A0D:6FC7:526:BCA:A78:5634:1232:5476 (talk) 16:56, 3 August 2023 (UTC)
thread_title:  I'm not sure bout a thing here