Talk:Aperiodic tiling

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[edit] Time to clean up this page!

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)

[edit] 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)


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

[edit] 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)

[edit] 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.

As I read it:

  • (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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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