# Talk:Aperiodic tiling

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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)

## Rename or what?

According to Experts 'aperiodic tiling' is a misnomer: there are aperiodic sets of tiles and nonperiodic tilings. A few options

• rename the page to 'nonperiodic tiling'
• rename the page to 'aperiodic sets of tiles'
• create two different pages and redirect the search for 'aperiodic tiling' to one of them

I would go for (1).al 09:43, 24 September 2007 (UTC)

'nonperiodic tiling' is too wide a description for the page; most nonperiodic tilings are of no particular interest, the interest lies in the properties of the sets of tiles or of narrower classes of nonperiodic tiling. I'd suggest the page instead describes 'aperiodic tiling' as being the field of study of aperiodic sets of tiles (before going on to note that the name is used loosely to refer to tilings by such sets of tiles). Where the properties of particular types of tilings that may arise both from aperiodic sets of tiles and otherwise are to be described, other articles may be used for details of those classes of nonperiodic tilings; for example, substitution tiling, or quasiperiodic tiling turned from a disambiguation page into an actual article. Joseph Myers 19:04, 25 September 2007 (UTC)
Unfortunately, 'aperiodic tiling' is exactly the way even experts talk about this topic--- even though it is not mathematically well-defined. I have a few guesses about how this came about; partly, one feels one is tiling; the tiling in the term feels like a verb, and is a different word than the noun tiling which is mathematically defined. Second, the topic was popularized as recreational mathematics; the subtle point that the Penrose tiles admit, in fact, uncountably many tilings was (and is) often overlooked. How much easier things would be if the Penrose tiles had been popularized instead of the the Penrose tiling. But here we are. I think this is the correct title for this page and topic, with a careful discussion of the problem. I hereby invoke my Expert status.C G Strauss 20:09, 6 October 2007 (UTC)
Maybe option (2) really is the way to go. I've been resisting this, but maybe it really would be simpler. Myers' disambiguation idea is pretty good as well.--C Goodman-Strauss 22:50, 7 October 2007 (UTC)

This discussion is really hard to follow because no-one is defining terms like "set". Is the set of tiles in an infinite tiling an infinite number of tiles? Or is it the set of distinct prototiles that are used repeatedly to fill the space. If the latter (as in the first para of the article), then it is the "tiling" that is periodic or aperiodic, not the set of prototiles. Two different tiles cannot, in themselves, be periodic or aperiodic. Sets of distinct prototiles may necessarily make periodic or aperiodic tilings, or possibly either depending on their configuration, but the sets themselves are neither periodic or aperiodic. Is a rectangle a periodic tile? No. It is a figure that makes either periodic or aperiodic tilings, depending how its replicates are laid down.

Moreover, the distinction between "aperiodic" and "nonperiodic" is a null set, since "a" is Greek for the Latin "non". A tiling is either periodic to infinity or it is not. "Quasiperiodic" makes sense because it means"approximately periodic".

Christopher Tyler  Cwtyler (talk) 00:25, 3 August 2011 (UTC)

How do we know that the assembly is necessarily non-local? Who showed this? --131.215.220.112 23:14, 3 August 2005 (UTC)

Dworkin and Shieh showed this; their remarkably simple proof (for 2D tiles with finitely many rotations) is sketched in Senechal's Quasicrystals and Geometry, with further references there.
Conversely, Socolar has given probabistic methods for assembling the Penrose tiles: given an ε, and an R, he gives probabilities to certain things sticking, so that a defect free patch of radius R is assembled (after some time) with probability 1-ε. Having fixed the rules, though, the probability of correct assembly falls off exponentially in R; moreover the time required increases exponentially as ε decreases. The critical thing, though, is that the basic idea seems to generalize to (all?) aperiodic hierarchical tilings
(am misusing the word tiling here, but there's no better way to state what I mean--- this is the heart of the problem the title of this page embodies!)--C G Strauss 20:18, 6 October 2007 (UTC)

Little thing: in some loose sense, "tilings" are arguably not the same as "tesselations"; I don't know of any formal distinction, but I have never seen the word "tesselation" except in the context of periodic tilings. There is no article defining "tiling" however.69.152.222.206 (talk) 01:14, 13 June 2008 (UTC)cgstraus

## Help rewrite

I have attempted to rewrite the article, but nevertheless I believe that it still needs the attention of an expert and serious clean up. The old paragraphs have been incorporated in the text, even if I don't think an expert would agree.al 15:54, 17 January 2007 (UTC) Removed an addition which does not fit in 'Mathematical Considerations': a new section at the end suggests that it was a first case of practical use.al 17:55, 3 March 2007 (UTC)

It is getting better, but rewriting is a tricky business and now the introduction is rather awkward. Why the restriction 'in geometry'? A more general approach includes Meyer, Delone, and model sets but their mention has disappeared. (There are entries for Voronoi cell and Dual graphs). Substitution is a key concept but it is algebric.
The definition of aperiodic as non-periodic looks like a tautology.
Hi there, I guess I qualify as an 'expert'; the article is looking much better than it did in the past. If I get some time I will help out some. I have few comments I'll sprinkle through the talk page. CGS
I will change the lead para. One big problem in the subject is that there really is NO SUCH THING as an "aperiodic tiling" per se. Unfortunately, even experts use the term-- I do myself all the time. Really, "aperiodicity" is a property of a set of tiles. A given tiling might be non-periodic or periodic; a given set of tiles might not admit any tilings at all, or admit periodic tilings (among others); an aperiodic set of tiles admits tilings, but only non-periodic ones. I think it is amazing that this is possible-- translational order must be subtly wrecked at all scales.
The real problem is with the word "tiling"-- in English, it is somewhat ambiguous, and is used in more than one sense in discussing this topic (though there is a single well-defined notion: a collection of non-overlapping tiles covering whatever space we're working in, the word "tiling" is also used, informally for the entire system embodied by a given set of tiles, i.e. the set of tilings admitted by the tiles; an example of this is "the Penrose Tiling" meaning "the Penrose tiles and all the tilings they admit") This is sort of a hangover from some bad choices of language early on. CGS

An important point would be to distinguish between two approaches:

• aperiodic tilings as generated or produced by aperiodicity-enforcing tiles
• aperiodic tilings as illustrations of a more general idea

which are not exactly synonymous with 'local'and 'global'.

The definition of aperiodicity by a set of aperiodic-enforcing tiles is standard one, but I do not know how it works in one dimension e.g. for the tiling of the line by segments according to the Fibonacci word.

There can exist no aperiodic set of tiles in the line. It is a nice exercise to prove this. CGS

In a Penrose tiling, generated by its decorated tiles, it is possible to introduce local 'defects' which do not affect its aperiodicity. al 21:49, 27 March 2007 (UTC)

This comment does not make sense; I think Ael means "non-periodicity?" The tiles are aperiodic because they admit only non-periodic tilings; if you change the defn of tiling to allow gaps, then all kinds of bad examples can creep in. CGS
The "in geometry" is simply to set the scene for the context in which the term "aperiodic tiling" is defined, as in WP:LEAD. The lead section should also try to be accessible. Making the distinction between nonperiodic and aperiodic clear in a way accessible to non-specialists is a tricky problem (but an important one to solve here so that readers do understand what the article is about), and Senechal points out (p169) that the terms are frequently misused even by those familiar with the distinction.
Following the standard terminology, you have the aperiodic tilings (by aperiodic protosets), and then the various constructions that may be used to generate or describe them, and related concepts; these constructions may also generate objects of interest but outside the scope of aperiodic tilings, and those and more general discussion of related concepts belong in other articles such as substitution tiling. This article needs to discuss the various ideas relating to aperiodic tilings; but further information about these ideas in more general contexts belongs in those other articles. More detail regarding Penrose tilings with defects belongs in Penrose tiling.
Some of the links to related concepts are present here, others may need to be added. I deliberately left the tag for expert attention, since it's still needed for those sections relating to other concepts. Joseph Myers 00:38, 28 March 2007 (UTC)
Notes on four problems with the article. The latter are true of Penrose tilings, and I suspect true of Amman tilings, and perhaps true of all known aperiodic tilings.
1. The first sentence is manifestly redundant, in that the tiling must be non-periodic if the prototiles do not admit a periodic tiling. If this is to remain for accessibility, it should be mentioned that the statement is redundant, to help people who wonder if there's something they are missing.
2. I advise using the more common terminology that a tiling is periodic if it admits any translational symmetry; the distinction between "fully periodic" (called "periodic" in the article) and "subperiodic" is not useful in this context. Aperiodic tilings have tiles that admit no "subperiodic" tiling, either. This is in conflict with the terminology of section 1, but I believe it is standard.
3. Another problem is that Penrose tilings (and, I believe, Amman tilings) are not unique. The Penrose tilings admit more than one tiling of the plane—uncountably infinitely many tilings, in fact. Only two of these tilings have fivefold rotational symmetry; these two also have mirror symmetry through five lines meeting in the rotational center. (There is a "statistical" symmetry only in that there are five distinguished families of parallel lines). There are other (uncountably many) Penrose tilings that possess mirror symmetry through a single axis.
In fact, this is a feature of ALL aperiodic sets of tiles that work by forcing a hierarchical structure. I believe Danzer and Dolbilin proved it true of all aperiodic sets of tiles-- that is, that all aperiodic sets of tiles necessarily admit uncountably many tilings.
4. The fact that there are multiple Penrose tilings is paradoxical, in that it is impossible to distinguish two tilings by examination of any finite subset of the plane. It would be impossible to verify that a given point or line is the rotational center or mirror axis by examination of a tiling, even if we were given that the tiling is symmetric. It would only be possible to show that a given point or line is not the center or axis, and then by a process of examination that does not have an upper bound. This is true because of the theorem that given any patch of diameter D in a tiling, we can find a congruent patch within a distance of 2D from any point in any tiling.
The third and fourth facts, which are proved in Gruenbaum and Shephard and asserted in the Martin Gardner columns, contradicts several statements about periodicity in the article. But I haven't time to edit it now. I hope these concepts will help whoever does work with the article, and that they look up the extent to which these apply to Amman and other aperiodic tilings. –Dan Hoeytalk 01:27, 24 April 2007 (UTC)

To Dan Hoey (talk · contribs): Regarding your first point. The first sentence says "In geometry, an aperiodic tiling is a tiling which never repeats itself, by a (finite) set of prototiles not admitting any tiling that does repeat itself.". It is not redundant. You are overlooking the distinction between nonperiodic and aperiodic. An aperiodic tiling is not merely nonperiodic, it is a tiling by a set of tiles which cannot tile the plane (or whatever space) periodically. JRSpriggs 10:28, 24 April 2007 (UTC)

I fully understand the difference between nonperiodic and aperiodic. I failed to make it clear that it the first clause of that sentence that is redundant. A non-redundant form of that statement would be "In geometry, an aperiodic tiling is a tiling by a finite set of prototiles that do not admit any periodic tiling.". It is redundant to say that the tiling is nonperiodic, because that is immediate from the statement that its prototiles can only tile nonperiodically. Incidentally, I consider the rewording "… that does not repeat itself…" to be an unhelpful dumbing-down of "nonperiodic".–Dan Hoeytalk 12:26, 24 April 2007 (UTC)
I do agree that the redundancy is rather unwieldy. Just saying that aperiodic tilings are the kind that result from finite sets of prototiles that cannot admit a periodic tiling should be sufficient. Somehow breaking the definition into two parts ( 1) it is a nonperiodic tiling 2) tiles cannot admit a periodic tiling), may seem on first thought to be a gentler approach, but I find it confusing actually. --C S (Talk) 21:43, 24 April 2007 (UTC)
By the way, I retract my comment that the the phrase "… that does not repeat itself…" is merely "unhelpful". The phrase is simply wrong. After all, every Penrose tiling repeats all finite patches of itself, it just doesn't do so periodically. I don't know of better words than "periodic"/"nonperiodic"; if the reader has to scroll down to the Terminology section to find out what the intro means, that's a good thing, because without understanding what periodic means there's no way of understating what aperiodic means. By the way, the terminology section (or an appropriate section of periodic function) should define the usage "fully periodic" for a periodicity whose translation vectors spans the space. I'm pretty sure that this article should specify "In this article, the term periodic refers to a tiling that is either quasiperiodic or fully periodic."–Dan Hoeytalk 18:24, 30 April 2007 (UTC)

## Terminology

The articles periodic function and substitution tiling both use aperiodic as synonymous with nonperiodic. A distinction, disregarded even by those who understand it, is probably inappropriate, pace Senechal.

• (1)any tiling by that set (..) is nonperiodic

('that set'= set of tiles said to be aperiodic)

• (2)A tiling by an aperiodic protoset is said to be an aperiodic tiling.

As it stands, we have to deal with: not periodic, nonperiodic, aperiodic and quasiperiodic. And we note that 'periodic' is originally temporal and hence one-dimensional.al 18:50, 28 March 2007 (UTC)

Well, we can't invent new terminology here, so all we can do is establish (here or at Wikipedia:WikiProject Mathematics/Conventions) which existing terminology to use in Wikipedia for which concepts, and fix pages not following that terminology, while making sure pages note that there is confusion and ambiguity in the terminology in practical use. I might quote Goodman-Strauss on the tilings mailing list today [1]:
This confusion is widespread, and I think it is incumbent on all that are interested in tilings to make a real effort to correct any conflation of the terms "aperiodic" and "non-periodic", at least when discussing tilings.
to illustrate the established understanding that these terms have particular meanings in the field of tiling but are nevertheless sometimes confused. Joseph Myers 19:55, 28 March 2007 (UTC)
I fixed substitution tiling because its conflation of "aperiodic" with "nonperiodic" is not common in tiling literature. Periodic function still uses the terms synonymously; I don't have a good enough handle on the literature to know whether this is legitimate. –Dan Hoeytalk 18:49, 30 April 2007 (UTC)

## removed some nonsense

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)

## removed more nonsense

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)

## An absolutely comprehensive list?

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)

This article should include mention of the ?new? Socolar&Taylor tile: http://maxwelldemon.com/2010/04/01/socolar_taylor_aperiodic_tile/ —Preceding unsigned comment added by 217.155.231.166 (talk) 22:59, 24 May 2010 (UTC)

## aperiodic tilings in medieval Islamic architecture

I would like to discuss removing the mentioned section. It is devoted entirely to the Lu-Steinhardt paper. Admittedly, their paper got a lot of attention, and they made no obvious mistake. But there are experts in the field - i.e. both experts in islamic art and mathematics - who do not share the conclusions of Lu and Steinhardt. So the statement 'that Islamic architects came close to discovering aperiodic tilings some five centuries before they were discovered by Western mathematicians' is arguable. Probably it's just wrong. The preceding statement 'These could be used to construct nonperiodic tilings with fivefold or tenfold rotational symmetry' means nothing. It is also true for rhombs with angles of 72 and 118 degree. If noone gives me a good reason against it, I will remove the section. Please excuse my bad English. Cheers. Frettloe

## Incomprehensible.

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)

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)

It should be done better, with clearly stated definitions (and what about "strong aperiodic"?). and not in the introduction, but in a section "definition" somewhere between history and construction.

## Islamic art

These two info. are misplaced. I've supressed them, but they should be putted in a right place. maybe in the history section ? Note that non-periodic tilings are necessarily infinite. So I would speak about "tilings with non-periodic symmetries" instead of "aperiodic tilings".

In Islamic architecture the geometric space-enclosing system known as muqarnas appears to be aperiodic. Numerous computer rendered examples can be seen here.

Muqarnas, the Arabic/Islamic space-enclosing system, appear to be non-periodic while lending themselves to aperiodic forms. Consider examples at the image gallery: [2]

## Principles of construction

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)

## Own article about aperiodic tiles

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)

## Opening of article

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 (talkcontribs) 06:50, 22 December 2010 (UTC)

• Saying that, like in the newly introduced introduction, the patterns in aperiodic tilings never repeat is factually incorrect. For example in a Penrose tiling, every finite patch is repeated an infinite number of tiles. The reason for this lies in the hierarchical nature of the tilings. Thus, whenever a single tile in the inflated tiling appears, the cooresponding 'cluster' or 'patch' of tiles in the original tiling appears. And since there are infinitely many tiles in an infinite inflated tiling, the corresponding 'clusters' or 'patches' occur an infinite number of times in the original tiling. So every 'bounded region' in a Penrose tiling, no matter how large, appears infinitely often. Thus the lead section is incorrect. Toshio Yamaguchi (talk) 12:47, 14 January 2011 (UTC)
• For a more illustrative example of what I pointed out, please see [3]. Please have a look at the illustrations on page 4. one can see, that the L-tiles are grouped together to form larger L-tiles. Each larger L-tile is actually a 'cluster' or 'patch' of some trilobites and some crosses. Thus, everywhere the larger L-tile appears, the cluster of trilobites and crosses appears. This process of recomposing smaller L-tiles into larger L-tiles continues ad infinitum. The larger the L-tiles grow, the larger will the corresponding cluster of trilobites and crosses be and this corresponding cluster will appear everywhere an L-tile appears. This each finite cluster of trilobites and crosses is repeated infinitely often. Therefore mere repetition of a tiling is not unique to periodic tilings. For this reason I reverted the most recent edits.Toshio Yamaguchi (talk) 13:29, 14 January 2011 (UTC)

## Error in history section

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)

• No, it's right as is: Wang showed "If there is no aperiodic tileset, then the tiling problem is decidable", and Berger showed that the tiling problem is undecidable; it follows that there is an aperiodic tileset. Of course, Berger wanted to give an explicit aperiodic tileset as well. Ben Standeven (talk) 21:08, 3 April 2011 (UTC)

## Aperiodic in terms of what?

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)

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)
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)
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)
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 ${\displaystyle 10^{100000000}}$ 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 ${\displaystyle 10^{100000000}}$ 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)
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)

## Article title

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)

"A set of tile-types (or prototiles) is aperiodic if there are some tilings using only these types, and all such tilings are non-periodic." This is not a meaningful English sentence. Please re-write. — Preceding unsigned comment added by 122.213.146.102 (talk) 02:46, 24 July 2012 (UTC)

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)

## Tautological first sentence

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)

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)

## Terminology again

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)

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)
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)

## another example

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)

## Socolar-Taylor

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)