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I find it odd that the whole article makes no mention of the significance of Penrose tiles as a quasi-counterexample of the crystallographic restriction, since they exhibit five-fold symmetry. Why is that? Was it deliberately decided to omit what looks like one of the most important facts about Penrose tiles, or am I confused about their nature? Swap ( talk) 04:30, 7 March 2008 (UTC)
What, then, is a "tiling"? Is there such a thing as "aperiodic tiling"? For that matter what is "the penrose tiling"?
Unfortunately, the terms, as used in this article, are thoroughly entrenched, even though they are not mathematically defined. (I've made several changes to this article, and the article on Aperiodic tiling to reflect this. Much more work is needed.)
A tiling is a covering of the plane by a set of tiles with non-overlapping interiors. A given set of tiles "admits" a tiling iff there is a tiling by congruent copies of tiles in the set. One can consider the collection of all tilings admitted by a given set of tiles.
(And here's the trouble-- everything in the paragraph above is often collectively referred to as "tiling"; The Penrose Tiling, approximately, seems to refer to a system consisting of the Penrose tiles (in one of their variations), and all of the tilings they produce)
A given "set of tiles" is aperiodic iff it admits only non-periodic tilings. So aperiodicity, formally, is a property of a set of tiles, not a tiling. This is an important distinction--- mere non-periodicity, a property of tilings, is not so special.
I guess one could mathematically define tiling and Tiling, but as tempting as that might be, I don't think there is any precedent for that kind of nonsense. The fact is, we have a well established, but inadequate, informal term, and a less-known, but more precise mathematical term.
In casual conversation, to be frank, I'm as likely as the next guy to refer to The Penrose Tiling; but it is not a completely meaningful term in itself. This is subtle, but essential. — C. Goodman-Strauss
Several images were not Penrose tilings, but were outputs from L-systems. I've moved them to L-system. Now someone needs to explain in that article what those images actually are, and how the L-system generated them.
A free Microsft Windows program to generate and explore rhombic Penrose tiling is available at http://www.stephencollins.net/penrose. The software was written by Stephen Collins of Splendid Software, in collaboration with the Universities of York, UK and Tsuka, Japan.
LSystems can be generated using the free Software http://jlsystem.sourceforge.net. helohe 12:10, 1 October 2005 (UTC)
Yeah, I know that a reference to a random guy posting at Slashdot is exactly what Wikipedia needs. :) But his post mentions tome interesting things about the tiles, including that Clark Richert has figured at least a part of that at the same time as Penrose. Paranoid 15:05, 12 May 2005 (UTC)
Note that the Penrose tiling is a projection of a five dimensional lattice (which has cubic symmetry) down to two dimensions; thus, the readily apparent symmetry in five dimensions is rather hidden and obfuscated when seen in two.
There is a serious-- indeed critical!-- flaw in this article. The rhombs shown CAN tile periodically. (indeed, of course, any quadrilateral can). The essential aspect of the Penrose tiles is that they are marked in such a way that they can ONLY tile non-periodically. With no discussion or illustration of the matching rules that drive the construction, unfortunately, the article is nonsense.
(The illustrations do show the structure the Penrose tiles are forced to assume, but not the actual tiles themselves)
Here is a reference, chosen by Google: www2.spsu.edu/math/tile/aperiodic/penrose/penrose2.htm --—Preceding unsigned comment added by 69.151.118.199 ( talk)
The tiles are put together with one rule: no two tiles can be touching so as to form a single parallelogram. The tiles can be modified with bumps and dents around the perimeter to enforce this rule, but the tiling looks better if they have smooth sides.
Thank you; I did overlook this. But of course I wouldn't have been the only one. And the standard stripes that are often drawn on the rhombs are quite attractive!
The tiles are put together with one rule: no two tiles can be touching so as to form a single parallelogram. The tiles can be modified with bumps and dents around the perimeter to enforce this rule, but the tiling looks better if they have smooth sides.
I made a christmas and new-year card for my friends using penrose tilings. The program that I wrote for this card can also easily generate other figures, and I believe they are usefull for explaining the matching rules. I have split up the section "Drawing the penrose tiling" two subsections: "L-Systems" and "Deflation". Under the section "Deflation" I have put some figures and explanations about how to generate a penrose tiling based on the matching rule, using of the deflation principle. It is certainly instructive to have some extra words about these matching rules in the introduction. —The preceding unsigned comment was added by Tovrstra ( talk • contribs) 14:35, 30 December 2006 (UTC).
How about a section explaining why the tiling is aperiodic, how it works, and why this is interesting? Torokun 22:05, 28 March 2006 (UTC) And also, why insist (as many others do) that ' given a bounded region of the pattern, no matter how large, that region will be repeated an infinite number of times within the tiling' which is a feature of random systems? 85.187.217.182 23:17, 13 November 2006 (UTC)
See The colossal book of mathematics, Gardner M., Penrose tiles. The tiles featured there are more interesting and should be added. Doomed Rasher 18:19, 2 September 2006 (UTC)
Ref: Kepler/Penrose tiling problem...i am an independent artist/designer and about 20 yrs ago i painted a picture depicting a periodic pattern using the Kepler/Penrose tiles (derived from the dissection of a pentagon)...and this image can be perused on my web-site at:
Best regards pete mcclure.-- 81.86.8.62 13:00, 22 November 2006 (UTC)
Most (all?) of the times that the word appears in the text, it's 'kite.' However, the word appears in some images (and the name of the images) as 'kile.' I expect that 'kite' is correct, though I have no idea. It'd be a good thing for someone knowledgeable in Penrose tiling to make consistent.
—The preceding unsigned comment was added by Stomv ( talk • contribs) 12:45, 23 February 2007 (UTC).
Some more external links which I thought was quite good concerning Penrose Tiles (see below). I'm adding this here so that someone more familiar with the topic can review it and see if it might be useful re the article.
American Mathematical Society article: [2]
Clay Mathematics Institute article: [3]
Scribblesinmindscapes 16:40, 10 January 2007 (UTC)
Rewrote the passage about the Steinhardt & Lu paper and trimmed irrelevant links. The idea is not really new, but the name of Steinhardt, an acknowledged expert, gives it now more weight. Artisanal practices suggest examples, but do not produce mathematical objects; an ellipse does not prove that its daughtsman had a theory of conic sections. 195.96.229.83 13:28, 27 February 2007 (UTC)
A place for links to art could be the end section ('Triva') which we should perhaps rename. I agree that Jos Leys' site mentioned somewhere above should not be missed. 195.96.229.83
Removed ref to Islamic art from lead as it is out of place and incorrect: see above and also reactions to the Steinhardt-Lou paper. One can say that something equivalent to a Penrose tiling might have been obtained as it is done later in the article. 195.96.229.104 ( talk) 09:35, 13 December 2007 (UTC)
The word "Substitution Matrix" is used without reference and introduction. Is it related to the substitutions used to build a penrose tiling and if so, how is it defined? —The preceding unsigned comment was added by 80.202.238.117 ( talk) 20:24, 6 March 2007 (UTC).
Substitutions are transforms which for simple 'linear' cases are represented a matrix, hence the name. The usual notation is New=Matrix.Old, e.g for the penrose tiling:
If the eigenvalues of the substitution matrix are pisot numbers the substitution generates a quasicrystal and the physicists say that it produces Bragg diffraction. 91.92.179.156 23:36, 7 March 2007 (UTC)
the article says "there are many ways (infact, uncountably many).." This is kinda vague and might lead someone to think uncountable is a synonym of infinite. In fact, countability/uncountability really has nothing to do with the size of a set. It should definitely be mentioned that the ways to form a penrose tiling is uncountable, but it should not be confused with an implication about the size of the set. 164.76.162.135 16:56, 5 December 2006 (UTC)
i just decided to go for it and made this change myself 164.76.162.135 17:05, 5 December 2006 (UTC)
Uncountable/countable are characteristics of the two basic infinite sets and the first is more 'powerful' than the second. This funny talk tries to avoid the paradoxes and confusions when dealing with the infinite. The assertion that there are uncountable ways to arrange a Penrose tiling sounds plausible as different tilings within the same perimeter are possible.
There are certainly finitely many connected tilings given any finite number N of tiles, but there are uncountably many tilings of the plane, using the deflation argument. However, it is important to note that only two of the tilings possess five-fold rotational symmetry. This renders most of the statements about five-fold symmetry false. It should be mentioned that these two, and uncountably many others, also possess mirror symmetry; only the two rotationally-symmetric ones possess mirror symmetry through more than one line. The distinction between finite and infinite tilings is crucial here, since a finite subtiling cannot be used to determine which infinite tiling you are in, nor even where you are in that infinite tiling.
Statements about a "rule" that no two rhombs can form a parallelogram are also incorrect, as noted above. The true rule can be seen in the diagram; color the edges of the rhombs as in that diagram, and only allow matching-colored edges to be adjacent.
There doesn't seem to be a rating for an article that has quite a bit of good stuff and some glaring falsehoods. –Dan Hoey 02:28, 24 April 2007 (UTC)
For completeness the 23 combinations are: 1111111111, 111111112, 11111113, 11111122, 1111114, 1111123, 1111222, 111124, 111133, 111223, 11134, 112222, 11224, 11233, 1144, 12223, 1234, 1333, 22222, 2224, 2233, 244, 334. Chris 10:53, 21 June 2007 (UTC)
If counting is OR and unacceptable, I guess that here at least two citations are needed. al 21:36, 28 July 2007 (UTC)
Perhaps something should be said about the earlier steps, when Penrose proposed a 'Keplerian' tilings built with more than two tiles. Apparently the original Penrose tiling was built with pentagons, rhombuses, stars and boats (3/5 of a star). The derivation can be seen on Savard's page [5]. Here is a good introduction [6] which includes all of this and could be linked somewhere. In professional jargon the Penrose tiling(s) are just P1, P2 and P3, which correspond to the pentagonal, kite and dart and rhombus variants. al 22:22, 30 June 2007 (UTC)
Just added a section about the Decagonal covering which seems important in physics. Removed the high-rating tag on this page as I believe that the Penrose tiling is mathematically trivial and pertains more to recreational maths. If people believe that tenfold symmetry is somehow important, here is perhaps an important link [7]. al 17:49, 14 March 2007 (UTC)
Let's try a different tack. My feeling of the importance rating is, again, not so much importance within mathematics, but importance as a contribution to an encyclopedia. How embarrassed should we be if this topic were missing? Not embarrassed at all, it's so trivial as to be unimportant, and there is room for legitimate debate about whether it even meets WP's standards of notability: low importance. The article makes a solid positive contribution to the encyclopedia's overall depth, is on a clearly notable topic, has some applications and connections to other topics, but could be removed without causing us significant embarrassment: mid importance. The topic is notable enough that any encyclopedia worthy of the name should carry it, and it would be a clear embarrassment to us not to carry it: high importance. The topic is central to human knowledge and any educated person should be embarrassed not to know a little about what it is: top importance. That's my own calibration, anyway, and I think it's more conservative than the calibration described at Wikipedia:WikiProject_Mathematics/Wikipedia_1.0. Now, where does Penrose tiling fit on this scale? I think mid or high are both defensible choices, but I'd pick high because I'd be embarrassed to be working on an encyclopedia that doesn't carry an article on a mathematical topic that is so well known in popular culture. — David Eppstein 18:33, 17 March 2007 (UTC)
OK, I see I got it wrong, but I was mislead by the Physics project which explicitely says to rate importance 'within physics'. The Maths project, being tied with the W_1.0, suggests the oposite. Sorry for the trouble. al 18:00, 25 March 2007 (UTC)
Alain Connes considers the space of Penrose tilings a "very interesting `noncommutative' space" and makes it a main example in his book, Noncommutative geometry. So I think calling the topic "mathematically trivial" is a rather narrow viewpoint and one that would presumably change with more experience and knowledge. -- C S (Talk) 22:39, 11 June 2007 (UTC)
I think that the content of 'Early history' paragraph did not match its title and I have modified it. Penrose's name is still buried among technical details but I have tried to connect it with a larger context (not just mathematics). Perhaps Dürer and girih tiles should also be moved here. The discarded ending could be moved to Wang tiles. 91.92.179.156 09:33, 21 September 2007 (UTC)
The opening section states "* any finite region in a tiling appears infinitely many times in that tiling and, in fact, in any other tiling. This property would be expected if the tilings had translational symmetry so it is not a surprising fact given their lack of translational symmetry." Why does the fact that it is expected given translational symmetry (isn't that a tautology anyway?) mean that it isn't surprising given no translational symmetry. I believe the statement due to the largeness of infinite tilings, but I don't see the link to the statement made. -- SGBailey ( talk) 22:48, 22 June 2008 (UTC)
As can be seen by this SVG file with ECMAScript animation of a Penrose tiling using an L-System (essentially following the given axiom/rules in this article), this particular rule set covers many rhombus edges MUCH more then twice.
This leads to MANY edges overlapping each other, especially near the center.
Also some of the pathes are closed (and then can be filled with a color), but others are open and can't be filled.
Does anybody know of a better rule set, only producing closed pathes for the neccessary rhombuses?
Every single rule in the given set forms essentially a narrow or a wide rhombus (2 of them each, the narrow ones have an additional entry/exit line, which can't be removed without breaking the over all tiling. I can provide an SVG file demonstrating the single rules on request).
Would a better rule set even be possible?
Deerwood ( talk) 06:20, 7 July 2008 (UTC)
The introduction here says that Penrose tilings are not correctly described as aperiodic tilings. Yet the introduction of Aperiodic tiling article says "the various Penrose tiles are best known examples of an aperiodic set of tiles" and "the Penrose tiles are an aperiodic set of tiles." Unless I'm missing some subtle semantics, this is confusing to a reader. -- Ds13 ( talk) 07:48, 11 July 2008 (UTC)
I removed the following from the article, as it has multiple errors of sourcing and fact. Since someone reverted the removal with claim that it is "notable" I am taking the time to explain in more detail. The content was so screwed up it was dangerously misleading and could not stay. Perhaps with some work it can be reworded to be accurate and adequately sourced.
This sentence is severely flawed. First up, ownership of intellectual property is always contentious issue, and Wikipedia itself should not take sides on claims of ownership. If Pentaplex asserts ownership to some sort of legally enforceable intellectual property, then we need to be specific in what they claim to own and in what jurisdictions. The ref that was provided here says they have a US patent -- but it also says it's expired. Thus the only reliable source provided shows no ownership rights at all, and only former potential rights (patents can be disputed, ownership comes after it's been tested and prevailed in court), and only in the United States... which is a problem, as Pentaplex is in the UK. If they assert any ownership then it must be something other than this patent. We need a reliable source about their claims if we are going to list these claims... and then they are only claims, not findings of law.
Is there a reliable source about the lawsuit and what exactly the grounds were? It sounds dubious to me, as copyrights are only for fixed forms, not theoretical constructs like tilings.
This is probably the least bothersome sentence. It could use a real source, but it's not exactly confusing or misleading.
In which case no ownership rights were legally established, just that a settlement of some sort was made. And a source would be needed here
OK... the ref cited here doesn't seem to meet Wikipedia standards either. It comes from a law dept. at a genuine school, but it's not a publshed source and appears to be just a handout to students or something. On top of that, claims of a single law professor in an informal way is not a reliable, primary source on a law case. There must be real sources, and ones that can better sort out what really happened. Until that happens the section is extremely misleading (so an expired US patent gives a UK copyright on something most courts say can't be copyrighted at all??) and can't be in the article. DreamGuy ( talk) 14:57, 20 November 2008 (UTC)
I'm unsure of the protocol, so rather than make the (trivial) edit I'm putting this on the Talk page. My concern relates to the usage of "citation needed" tags, and it's bothered me in many articles... perhaps someone can set me straight on this one, so I'll know how to deal with it elsewhere? In this article I'm unsure about the state of the following sentence:
Given that Ammann's name is a link, and that following that link leads you to a Wikipedia article that addresses the statement in some detail and cites some sources of its own (which, unfortunately, don't include links to web-accessible information), is the "citation needed" tag truly needed here? It seems to me there are a few possibilities:
I lean toward the final option, but I don't know if my inclinations are at odds with the customs here. Would somebodt please help a brother out and chime in on this issue? Thanks. 76.105.238.158 ( talk) 15:28, 6 January 2009 (UTC)
Refs
Hi, I was trying to make some images for the Gummelt's decagon section, and I've run into what appears to be an error in the article. It states that the only two allowed overlaps are those shown in this image. However, the following sources seem to say that so long as the red areas only overlap other red areas, anything goes.
Now, as I was drawing this image, I found that I could not get the current "only-two" rule to hold as I built the tiling, but if I used the "so-long-as-they-match" rule, it would work. Is the article wrong, or have I got it mixed up? Cheers, - Inductiveload ( talk) 04:37, 1 February 2009 (UTC)
Reviewer: Wizardman 15:49, 13 November 2009 (UTC) After reading through the article, I found the following issues:
I'll put the article on hold now, as I believe we're getting close to a GA, just some more fine-tuning is needed. Wizardman 16:58, 17 November 2009 (UTC)
One more thing I'd like you to do as we wrap up: re-read the article again, and if there's anything that sounds like it should be reffed, then use something from mathworld and cite it. Lack of cites seems to be the only problem left, so I'll take a look through myself as well. Wizardman 17:25, 18 November 2009 (UTC)
I've made quite a few edits to the article without significantly changing the basic content. However, based upon the sources, I am intending to expand the article to include more discussion e.g. of de Bruijn's work (as described by Senechal). There are also topics such as Ammann bars which need to be discussed, and many things that need to be explained more clearly and in more detail, including various representations of matching rules, and their relation to aperiodic sets. Additionally, there are coherence and ordering issues that need to be fixed (e.g., introducing Robinson triangles before they are used). Help would be appreciated!
I even hope to reinclude a segment on the commercial aspect (Pentaplex and the Kleenex suit) but I don't think it is appropriate to do so until the rest of the article is sufficiently developed and robust. Geometry guy 21:43, 10 December 2009 (UTC)
PS. I found it interesting to discover that the notation P1-P3 comes from Grunbaum and Shephard. This was not in the sources a month ago. The reason for this is that the article owes much of its pre-GA structure to a single remarkable edit by Ael, which improved the article substantially, but did not provide a source for this material (other than adding Luck and Penrose). Now there is a likely candidate for the source Ael was using.
Congratulations for the GA status. Ael 2 ( talk) 10:10, 2 July 2010 (UTC)
There is a saying "Better is the enemy of Good", so I have added the Tie and navette tiling; if it is to stay perhaps somebody would produce a better image. 91.92.179.172 ( talk) 09:57, 6 November 2010 (UTC)
"there are matching rules that specify how tiles may meet each other"
but what are they? brain ( talk) 23:39, 19 April 2010 (UTC)
I'll be going through the article slowly, and will leave suggestions here from time to time. Please be patient, and also with the comments themselves! :) Willow ( talk) 14:37, 26 June 2010 (UTC)
I have a concern with the understandability of the following statements:
A Penrose tiling has many remarkable properties, most notably:
I understand what the writer intended to say, but shouldn't it be better formulated saying that a Penrose tiling cannot be generated by a Primitive cell? I mean if we have an infinite tiling of the plane, even a periodic one, how could we shift it to 'match the original'? If it is already infinite, it makes no sense to shift it in any direction lying in the plane it is embedded in. The correct definition would be saying it cannot be generated by translating a finite primitive cell.
I don't think it is clear to most readers what 'any other tilings' are meant here. I guess it talks about the inflated and deflated tilings in the inflation hierarchy that are generated by the composition / decomposition of the tiles. But this formulation is not very understandable.
I really have trouble to understand what is meant here. How is the geometric structure commonly called a Penrose tiling a quasicrystal? A Penrose tiling is simply an abstract geometric structure and the atomic arrangement in quasicrystals somehow resembles this structure. I don't understand the sense of treating this relationship the other way round and what is got by this.
All these statements should be completely rewritten or ultimately deleted. In the current condition they are simply confusing and have no value for the article.
Thoughts and commons on this are welcome.
Toshio Yamaguchi ( talk) 11:31, 12 October 2010 (UTC)
As it is currently explained, the remarkable feature appears to be its self-similarity and this could be stated more clearly. 195.96.229.83 ( talk) 09:34, 18 November 2010 (UTC)
In the section Background and history - Periodic and aperiodic tilings it says
How is the mere repetition of a part of a tiling descriptive of a periodic tiling? A Penrose tiling, for example, also repeats itself; in fact any finite region in a Penrose tiling (no matter how large) is repeated an infinite number of times in that Penrose tiling. The difference between a periodic tiling and a Penrose tiling is, that in a periodic tiling a finite portion of that tiling is repeated in constant intervals. In a Penrose tiling, every finite region is repeated an infinite number of times in that tiling, but is not repeated in constant intervals. Toshio Yamaguchi ( talk) 11:39, 10 November 2010 (UTC)
I still have to question the rationale for having the third 'remarkable property' in the lead section. I fail to see how this statement, which would be more appropriate in the article on Quasicrystals is helpful for giving the reader an accessible overview of the article's key points. Talking about x-ray diffraction without even briefly describing it or mentioning why it is important in the study of Penrose tilings and without any further description of x-ray diffractograms in the rest of the article doesnt help in making the article accessible to areader. Toshio Yamaguchi ( talk) 18:15, 15 January 2011 (UTC)
There is a deep and illuminating connection between the two, at least imho, but the attempt to mention this was rejected offhand as "too vague to be useful" which is not really an argument. For the general reader the article is probably much too technical and elaborating this would just make matters worse. The interested reader could perhaps be directed to the pinwheel tiling. This is how I see it:... The substitution rules decompose each tile into smaller tiles of the same shape as those used in the tiling (and thus allow larger tiles to be "composed" from smaller ones). This approach makes rather obvious a close link between aperiodic structures and fractals. Ael 2 ( talk) 19:27, 4 November 2010 (UTC)
"The tilings can be generated from one another by the methods of inflation or deflation. For example, in deflation a cluster of tiles is subdivided into smaller pieces following specific procedures. Performing such operations iteratively, one can generate an aperiodic tiling with a much larger number of smaller tiles. These procedures endow the tiling with the property of self-similarity. These properties have suggested, right from the time of the discovery of Penrose tilings, that the tiling is fractal in nature /ref to Gardner chap.1/."
So I have restored the previous text an added this ref available online. Ael 2 ( talk) 10:04, 5 November 2010 (UTC)
I've produced an overlay illustration of a tiling and its inflation overlaid. The article is a bit messy with illustrations (and the text) as it is, especially mixing the various types of tilings. Anyway, this image or something like it could be useful to add -- Sverdrup ( talk) 19:56, 27 September 2011 (UTC)
The article doesn't actually say what a Robinson triangle actually is. The nearest I got to an explanation/definition was Golden triangle (mathematics)#Golden gnomon. The first use of the term in the current article is in the Kite and dart tiling (P2) section:
Both the kite and dart are composed of two triangles, called Robinson triangles, after 1975 notes by Robinson.
How about linking it thus: Robinson triangles ? > MinorProphet ( talk) 16:40, 5 February 2012 (UTC)
The substitution method for both P2 and P3 tilings can be described using Robinson triangles of different sizes. The Robinson triangles arising in P2 tilings (by bisecting kites and darts) are called A-tiles...
Tie-and-Navette tiling showing many properties of the penrose tiling.
Ad Huikeshoven (
talk)
09:48, 2 January 2015 (UTC)
The substitution rules for the P2 tiling as given in the article appear to be redundant: the "sun" and "star" rules just consist of the application of the already-given half-dart and half-kite rules to those particular structures. Is there any reason the sun and star need to be included as separate cases? 130.226.142.243 ( talk) 09:00, 15 January 2015 (UTC)
The article lead currently states that, "The aperiodicity of the Penrose prototiles implies that a shifted copy of a Penrose tiling will never match the original." This is not correct. It is an extraordinary property of the Penrose tiling that any given finite pattern will be repeated within a certain (and remarkably short) distance from it. I don't have a good source handy: does anybody know of one? — Cheers, Steelpillow ( Talk) 05:58, 19 August 2015 (UTC)
The comment(s) below were originally left at Talk:Penrose tiling/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.
This article contains serious inaccuracies, notably in (1) the matching rule and (2) the statements about symmetry. –Dan Hoey 02:33, 24 April 2007 (UTC) |
Last edited at 10:51, 12 February 2008 (UTC). Substituted at 02:27, 5 May 2016 (UTC)
The article is clear that kites and darts, with the standard matching rule, cannot tile the plane periodically. Yet the article contains an image, as at the right, which appears to be of a repeating unit of a periodic tiling. What's going on here? It looks to me like there's a mistake somewhere. Even if there isn't, there's an apparent contradiction that needs explaining. Maproom ( talk) 17:08, 11 July 2018 (UTC)
I've read a couple of books by Penrose and, though he makes my head spin, I managed to keep up — something I cannot yet say for this article. I've reworked the opening of Background and history and may move this up to the lede; I'm not totally happy to set down a jargon-heavy statement and then backfill an explanation of the terminology but haven't yet figured out how better to approach. (As well, the thought occurs to me that a mosaic is essentially a tiling with a potentially infinite set of nonrepeating tiles. Though at risk of coatracking, this may be a valuable illustration for the non-mathematician.)
In addition, I hope to clarify that Penrose tilings predate Penrose, as illustrated at the very end of the article, else the impression is given that he is somehow the inventor.
Weeb Dingle (
talk)
05:29, 13 October 2019 (UTC)
One more thought: aside from the title and references, the term "Penrose tiling" appears 49 times in the article. This feels a little like it may be browbeating the reader. Could this be productively reduced?
Weeb Dingle (
talk)
05:36, 13 October 2019 (UTC)
Our article currently exhibits a photo of a floor tiling at the Pilgrimage Church of Saint John of Nepomuk at Zelena Hora, Czech republic; see right. It does not mention, although it is true, that the two tiles of this tiling (pentagons and thin rhombs), meeting in the vertices of the two types visible in the photo, do not and cannot form a Penrose tiling. I analyzed the same set of tiles in a different context in a blog post [10]. The tiling from the blog post (in a downtown shopping street of Copenhagen) is I think more or less the same as the one in this photo, with the center of symmetry of the tiling at the top center of the photo. It is not a Penrose tiling (it has periodic patches that extend in wedges from the center). But more strongly, as the post explains, this set of prototiles, meeting in this way, cannot be made to form Penrose tilings. Because it is my own blog post, I am not going to add it to this article, but I think it would be appropriate to mention in the caption that it is not a Penrose tiling. — David Eppstein ( talk) 21:05, 20 January 2020 (UTC)
{{u|
Mark viking}} {
Talk}
04:13, 21 January 2020 (UTC)
Is this new discovery better here or in the Aperiodic tiling page? Thanks
I changed the definition of an aperiodic tiling because under the previous definition, the Penrose tiling is not aperiodic (since the tiles can be arranged periodically as in Figure 1). David9550 ( talk) 20:06, 30 March 2023 (UTC)
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I find it odd that the whole article makes no mention of the significance of Penrose tiles as a quasi-counterexample of the crystallographic restriction, since they exhibit five-fold symmetry. Why is that? Was it deliberately decided to omit what looks like one of the most important facts about Penrose tiles, or am I confused about their nature? Swap ( talk) 04:30, 7 March 2008 (UTC)
What, then, is a "tiling"? Is there such a thing as "aperiodic tiling"? For that matter what is "the penrose tiling"?
Unfortunately, the terms, as used in this article, are thoroughly entrenched, even though they are not mathematically defined. (I've made several changes to this article, and the article on Aperiodic tiling to reflect this. Much more work is needed.)
A tiling is a covering of the plane by a set of tiles with non-overlapping interiors. A given set of tiles "admits" a tiling iff there is a tiling by congruent copies of tiles in the set. One can consider the collection of all tilings admitted by a given set of tiles.
(And here's the trouble-- everything in the paragraph above is often collectively referred to as "tiling"; The Penrose Tiling, approximately, seems to refer to a system consisting of the Penrose tiles (in one of their variations), and all of the tilings they produce)
A given "set of tiles" is aperiodic iff it admits only non-periodic tilings. So aperiodicity, formally, is a property of a set of tiles, not a tiling. This is an important distinction--- mere non-periodicity, a property of tilings, is not so special.
I guess one could mathematically define tiling and Tiling, but as tempting as that might be, I don't think there is any precedent for that kind of nonsense. The fact is, we have a well established, but inadequate, informal term, and a less-known, but more precise mathematical term.
In casual conversation, to be frank, I'm as likely as the next guy to refer to The Penrose Tiling; but it is not a completely meaningful term in itself. This is subtle, but essential. — C. Goodman-Strauss
Several images were not Penrose tilings, but were outputs from L-systems. I've moved them to L-system. Now someone needs to explain in that article what those images actually are, and how the L-system generated them.
A free Microsft Windows program to generate and explore rhombic Penrose tiling is available at http://www.stephencollins.net/penrose. The software was written by Stephen Collins of Splendid Software, in collaboration with the Universities of York, UK and Tsuka, Japan.
LSystems can be generated using the free Software http://jlsystem.sourceforge.net. helohe 12:10, 1 October 2005 (UTC)
Yeah, I know that a reference to a random guy posting at Slashdot is exactly what Wikipedia needs. :) But his post mentions tome interesting things about the tiles, including that Clark Richert has figured at least a part of that at the same time as Penrose. Paranoid 15:05, 12 May 2005 (UTC)
Note that the Penrose tiling is a projection of a five dimensional lattice (which has cubic symmetry) down to two dimensions; thus, the readily apparent symmetry in five dimensions is rather hidden and obfuscated when seen in two.
There is a serious-- indeed critical!-- flaw in this article. The rhombs shown CAN tile periodically. (indeed, of course, any quadrilateral can). The essential aspect of the Penrose tiles is that they are marked in such a way that they can ONLY tile non-periodically. With no discussion or illustration of the matching rules that drive the construction, unfortunately, the article is nonsense.
(The illustrations do show the structure the Penrose tiles are forced to assume, but not the actual tiles themselves)
Here is a reference, chosen by Google: www2.spsu.edu/math/tile/aperiodic/penrose/penrose2.htm --—Preceding unsigned comment added by 69.151.118.199 ( talk)
The tiles are put together with one rule: no two tiles can be touching so as to form a single parallelogram. The tiles can be modified with bumps and dents around the perimeter to enforce this rule, but the tiling looks better if they have smooth sides.
Thank you; I did overlook this. But of course I wouldn't have been the only one. And the standard stripes that are often drawn on the rhombs are quite attractive!
The tiles are put together with one rule: no two tiles can be touching so as to form a single parallelogram. The tiles can be modified with bumps and dents around the perimeter to enforce this rule, but the tiling looks better if they have smooth sides.
I made a christmas and new-year card for my friends using penrose tilings. The program that I wrote for this card can also easily generate other figures, and I believe they are usefull for explaining the matching rules. I have split up the section "Drawing the penrose tiling" two subsections: "L-Systems" and "Deflation". Under the section "Deflation" I have put some figures and explanations about how to generate a penrose tiling based on the matching rule, using of the deflation principle. It is certainly instructive to have some extra words about these matching rules in the introduction. —The preceding unsigned comment was added by Tovrstra ( talk • contribs) 14:35, 30 December 2006 (UTC).
How about a section explaining why the tiling is aperiodic, how it works, and why this is interesting? Torokun 22:05, 28 March 2006 (UTC) And also, why insist (as many others do) that ' given a bounded region of the pattern, no matter how large, that region will be repeated an infinite number of times within the tiling' which is a feature of random systems? 85.187.217.182 23:17, 13 November 2006 (UTC)
See The colossal book of mathematics, Gardner M., Penrose tiles. The tiles featured there are more interesting and should be added. Doomed Rasher 18:19, 2 September 2006 (UTC)
Ref: Kepler/Penrose tiling problem...i am an independent artist/designer and about 20 yrs ago i painted a picture depicting a periodic pattern using the Kepler/Penrose tiles (derived from the dissection of a pentagon)...and this image can be perused on my web-site at:
Best regards pete mcclure.-- 81.86.8.62 13:00, 22 November 2006 (UTC)
Most (all?) of the times that the word appears in the text, it's 'kite.' However, the word appears in some images (and the name of the images) as 'kile.' I expect that 'kite' is correct, though I have no idea. It'd be a good thing for someone knowledgeable in Penrose tiling to make consistent.
—The preceding unsigned comment was added by Stomv ( talk • contribs) 12:45, 23 February 2007 (UTC).
Some more external links which I thought was quite good concerning Penrose Tiles (see below). I'm adding this here so that someone more familiar with the topic can review it and see if it might be useful re the article.
American Mathematical Society article: [2]
Clay Mathematics Institute article: [3]
Scribblesinmindscapes 16:40, 10 January 2007 (UTC)
Rewrote the passage about the Steinhardt & Lu paper and trimmed irrelevant links. The idea is not really new, but the name of Steinhardt, an acknowledged expert, gives it now more weight. Artisanal practices suggest examples, but do not produce mathematical objects; an ellipse does not prove that its daughtsman had a theory of conic sections. 195.96.229.83 13:28, 27 February 2007 (UTC)
A place for links to art could be the end section ('Triva') which we should perhaps rename. I agree that Jos Leys' site mentioned somewhere above should not be missed. 195.96.229.83
Removed ref to Islamic art from lead as it is out of place and incorrect: see above and also reactions to the Steinhardt-Lou paper. One can say that something equivalent to a Penrose tiling might have been obtained as it is done later in the article. 195.96.229.104 ( talk) 09:35, 13 December 2007 (UTC)
The word "Substitution Matrix" is used without reference and introduction. Is it related to the substitutions used to build a penrose tiling and if so, how is it defined? —The preceding unsigned comment was added by 80.202.238.117 ( talk) 20:24, 6 March 2007 (UTC).
Substitutions are transforms which for simple 'linear' cases are represented a matrix, hence the name. The usual notation is New=Matrix.Old, e.g for the penrose tiling:
If the eigenvalues of the substitution matrix are pisot numbers the substitution generates a quasicrystal and the physicists say that it produces Bragg diffraction. 91.92.179.156 23:36, 7 March 2007 (UTC)
the article says "there are many ways (infact, uncountably many).." This is kinda vague and might lead someone to think uncountable is a synonym of infinite. In fact, countability/uncountability really has nothing to do with the size of a set. It should definitely be mentioned that the ways to form a penrose tiling is uncountable, but it should not be confused with an implication about the size of the set. 164.76.162.135 16:56, 5 December 2006 (UTC)
i just decided to go for it and made this change myself 164.76.162.135 17:05, 5 December 2006 (UTC)
Uncountable/countable are characteristics of the two basic infinite sets and the first is more 'powerful' than the second. This funny talk tries to avoid the paradoxes and confusions when dealing with the infinite. The assertion that there are uncountable ways to arrange a Penrose tiling sounds plausible as different tilings within the same perimeter are possible.
There are certainly finitely many connected tilings given any finite number N of tiles, but there are uncountably many tilings of the plane, using the deflation argument. However, it is important to note that only two of the tilings possess five-fold rotational symmetry. This renders most of the statements about five-fold symmetry false. It should be mentioned that these two, and uncountably many others, also possess mirror symmetry; only the two rotationally-symmetric ones possess mirror symmetry through more than one line. The distinction between finite and infinite tilings is crucial here, since a finite subtiling cannot be used to determine which infinite tiling you are in, nor even where you are in that infinite tiling.
Statements about a "rule" that no two rhombs can form a parallelogram are also incorrect, as noted above. The true rule can be seen in the diagram; color the edges of the rhombs as in that diagram, and only allow matching-colored edges to be adjacent.
There doesn't seem to be a rating for an article that has quite a bit of good stuff and some glaring falsehoods. –Dan Hoey 02:28, 24 April 2007 (UTC)
For completeness the 23 combinations are: 1111111111, 111111112, 11111113, 11111122, 1111114, 1111123, 1111222, 111124, 111133, 111223, 11134, 112222, 11224, 11233, 1144, 12223, 1234, 1333, 22222, 2224, 2233, 244, 334. Chris 10:53, 21 June 2007 (UTC)
If counting is OR and unacceptable, I guess that here at least two citations are needed. al 21:36, 28 July 2007 (UTC)
Perhaps something should be said about the earlier steps, when Penrose proposed a 'Keplerian' tilings built with more than two tiles. Apparently the original Penrose tiling was built with pentagons, rhombuses, stars and boats (3/5 of a star). The derivation can be seen on Savard's page [5]. Here is a good introduction [6] which includes all of this and could be linked somewhere. In professional jargon the Penrose tiling(s) are just P1, P2 and P3, which correspond to the pentagonal, kite and dart and rhombus variants. al 22:22, 30 June 2007 (UTC)
Just added a section about the Decagonal covering which seems important in physics. Removed the high-rating tag on this page as I believe that the Penrose tiling is mathematically trivial and pertains more to recreational maths. If people believe that tenfold symmetry is somehow important, here is perhaps an important link [7]. al 17:49, 14 March 2007 (UTC)
Let's try a different tack. My feeling of the importance rating is, again, not so much importance within mathematics, but importance as a contribution to an encyclopedia. How embarrassed should we be if this topic were missing? Not embarrassed at all, it's so trivial as to be unimportant, and there is room for legitimate debate about whether it even meets WP's standards of notability: low importance. The article makes a solid positive contribution to the encyclopedia's overall depth, is on a clearly notable topic, has some applications and connections to other topics, but could be removed without causing us significant embarrassment: mid importance. The topic is notable enough that any encyclopedia worthy of the name should carry it, and it would be a clear embarrassment to us not to carry it: high importance. The topic is central to human knowledge and any educated person should be embarrassed not to know a little about what it is: top importance. That's my own calibration, anyway, and I think it's more conservative than the calibration described at Wikipedia:WikiProject_Mathematics/Wikipedia_1.0. Now, where does Penrose tiling fit on this scale? I think mid or high are both defensible choices, but I'd pick high because I'd be embarrassed to be working on an encyclopedia that doesn't carry an article on a mathematical topic that is so well known in popular culture. — David Eppstein 18:33, 17 March 2007 (UTC)
OK, I see I got it wrong, but I was mislead by the Physics project which explicitely says to rate importance 'within physics'. The Maths project, being tied with the W_1.0, suggests the oposite. Sorry for the trouble. al 18:00, 25 March 2007 (UTC)
Alain Connes considers the space of Penrose tilings a "very interesting `noncommutative' space" and makes it a main example in his book, Noncommutative geometry. So I think calling the topic "mathematically trivial" is a rather narrow viewpoint and one that would presumably change with more experience and knowledge. -- C S (Talk) 22:39, 11 June 2007 (UTC)
I think that the content of 'Early history' paragraph did not match its title and I have modified it. Penrose's name is still buried among technical details but I have tried to connect it with a larger context (not just mathematics). Perhaps Dürer and girih tiles should also be moved here. The discarded ending could be moved to Wang tiles. 91.92.179.156 09:33, 21 September 2007 (UTC)
The opening section states "* any finite region in a tiling appears infinitely many times in that tiling and, in fact, in any other tiling. This property would be expected if the tilings had translational symmetry so it is not a surprising fact given their lack of translational symmetry." Why does the fact that it is expected given translational symmetry (isn't that a tautology anyway?) mean that it isn't surprising given no translational symmetry. I believe the statement due to the largeness of infinite tilings, but I don't see the link to the statement made. -- SGBailey ( talk) 22:48, 22 June 2008 (UTC)
As can be seen by this SVG file with ECMAScript animation of a Penrose tiling using an L-System (essentially following the given axiom/rules in this article), this particular rule set covers many rhombus edges MUCH more then twice.
This leads to MANY edges overlapping each other, especially near the center.
Also some of the pathes are closed (and then can be filled with a color), but others are open and can't be filled.
Does anybody know of a better rule set, only producing closed pathes for the neccessary rhombuses?
Every single rule in the given set forms essentially a narrow or a wide rhombus (2 of them each, the narrow ones have an additional entry/exit line, which can't be removed without breaking the over all tiling. I can provide an SVG file demonstrating the single rules on request).
Would a better rule set even be possible?
Deerwood ( talk) 06:20, 7 July 2008 (UTC)
The introduction here says that Penrose tilings are not correctly described as aperiodic tilings. Yet the introduction of Aperiodic tiling article says "the various Penrose tiles are best known examples of an aperiodic set of tiles" and "the Penrose tiles are an aperiodic set of tiles." Unless I'm missing some subtle semantics, this is confusing to a reader. -- Ds13 ( talk) 07:48, 11 July 2008 (UTC)
I removed the following from the article, as it has multiple errors of sourcing and fact. Since someone reverted the removal with claim that it is "notable" I am taking the time to explain in more detail. The content was so screwed up it was dangerously misleading and could not stay. Perhaps with some work it can be reworded to be accurate and adequately sourced.
This sentence is severely flawed. First up, ownership of intellectual property is always contentious issue, and Wikipedia itself should not take sides on claims of ownership. If Pentaplex asserts ownership to some sort of legally enforceable intellectual property, then we need to be specific in what they claim to own and in what jurisdictions. The ref that was provided here says they have a US patent -- but it also says it's expired. Thus the only reliable source provided shows no ownership rights at all, and only former potential rights (patents can be disputed, ownership comes after it's been tested and prevailed in court), and only in the United States... which is a problem, as Pentaplex is in the UK. If they assert any ownership then it must be something other than this patent. We need a reliable source about their claims if we are going to list these claims... and then they are only claims, not findings of law.
Is there a reliable source about the lawsuit and what exactly the grounds were? It sounds dubious to me, as copyrights are only for fixed forms, not theoretical constructs like tilings.
This is probably the least bothersome sentence. It could use a real source, but it's not exactly confusing or misleading.
In which case no ownership rights were legally established, just that a settlement of some sort was made. And a source would be needed here
OK... the ref cited here doesn't seem to meet Wikipedia standards either. It comes from a law dept. at a genuine school, but it's not a publshed source and appears to be just a handout to students or something. On top of that, claims of a single law professor in an informal way is not a reliable, primary source on a law case. There must be real sources, and ones that can better sort out what really happened. Until that happens the section is extremely misleading (so an expired US patent gives a UK copyright on something most courts say can't be copyrighted at all??) and can't be in the article. DreamGuy ( talk) 14:57, 20 November 2008 (UTC)
I'm unsure of the protocol, so rather than make the (trivial) edit I'm putting this on the Talk page. My concern relates to the usage of "citation needed" tags, and it's bothered me in many articles... perhaps someone can set me straight on this one, so I'll know how to deal with it elsewhere? In this article I'm unsure about the state of the following sentence:
Given that Ammann's name is a link, and that following that link leads you to a Wikipedia article that addresses the statement in some detail and cites some sources of its own (which, unfortunately, don't include links to web-accessible information), is the "citation needed" tag truly needed here? It seems to me there are a few possibilities:
I lean toward the final option, but I don't know if my inclinations are at odds with the customs here. Would somebodt please help a brother out and chime in on this issue? Thanks. 76.105.238.158 ( talk) 15:28, 6 January 2009 (UTC)
Refs
Hi, I was trying to make some images for the Gummelt's decagon section, and I've run into what appears to be an error in the article. It states that the only two allowed overlaps are those shown in this image. However, the following sources seem to say that so long as the red areas only overlap other red areas, anything goes.
Now, as I was drawing this image, I found that I could not get the current "only-two" rule to hold as I built the tiling, but if I used the "so-long-as-they-match" rule, it would work. Is the article wrong, or have I got it mixed up? Cheers, - Inductiveload ( talk) 04:37, 1 February 2009 (UTC)
Reviewer: Wizardman 15:49, 13 November 2009 (UTC) After reading through the article, I found the following issues:
I'll put the article on hold now, as I believe we're getting close to a GA, just some more fine-tuning is needed. Wizardman 16:58, 17 November 2009 (UTC)
One more thing I'd like you to do as we wrap up: re-read the article again, and if there's anything that sounds like it should be reffed, then use something from mathworld and cite it. Lack of cites seems to be the only problem left, so I'll take a look through myself as well. Wizardman 17:25, 18 November 2009 (UTC)
I've made quite a few edits to the article without significantly changing the basic content. However, based upon the sources, I am intending to expand the article to include more discussion e.g. of de Bruijn's work (as described by Senechal). There are also topics such as Ammann bars which need to be discussed, and many things that need to be explained more clearly and in more detail, including various representations of matching rules, and their relation to aperiodic sets. Additionally, there are coherence and ordering issues that need to be fixed (e.g., introducing Robinson triangles before they are used). Help would be appreciated!
I even hope to reinclude a segment on the commercial aspect (Pentaplex and the Kleenex suit) but I don't think it is appropriate to do so until the rest of the article is sufficiently developed and robust. Geometry guy 21:43, 10 December 2009 (UTC)
PS. I found it interesting to discover that the notation P1-P3 comes from Grunbaum and Shephard. This was not in the sources a month ago. The reason for this is that the article owes much of its pre-GA structure to a single remarkable edit by Ael, which improved the article substantially, but did not provide a source for this material (other than adding Luck and Penrose). Now there is a likely candidate for the source Ael was using.
Congratulations for the GA status. Ael 2 ( talk) 10:10, 2 July 2010 (UTC)
There is a saying "Better is the enemy of Good", so I have added the Tie and navette tiling; if it is to stay perhaps somebody would produce a better image. 91.92.179.172 ( talk) 09:57, 6 November 2010 (UTC)
"there are matching rules that specify how tiles may meet each other"
but what are they? brain ( talk) 23:39, 19 April 2010 (UTC)
I'll be going through the article slowly, and will leave suggestions here from time to time. Please be patient, and also with the comments themselves! :) Willow ( talk) 14:37, 26 June 2010 (UTC)
I have a concern with the understandability of the following statements:
A Penrose tiling has many remarkable properties, most notably:
I understand what the writer intended to say, but shouldn't it be better formulated saying that a Penrose tiling cannot be generated by a Primitive cell? I mean if we have an infinite tiling of the plane, even a periodic one, how could we shift it to 'match the original'? If it is already infinite, it makes no sense to shift it in any direction lying in the plane it is embedded in. The correct definition would be saying it cannot be generated by translating a finite primitive cell.
I don't think it is clear to most readers what 'any other tilings' are meant here. I guess it talks about the inflated and deflated tilings in the inflation hierarchy that are generated by the composition / decomposition of the tiles. But this formulation is not very understandable.
I really have trouble to understand what is meant here. How is the geometric structure commonly called a Penrose tiling a quasicrystal? A Penrose tiling is simply an abstract geometric structure and the atomic arrangement in quasicrystals somehow resembles this structure. I don't understand the sense of treating this relationship the other way round and what is got by this.
All these statements should be completely rewritten or ultimately deleted. In the current condition they are simply confusing and have no value for the article.
Thoughts and commons on this are welcome.
Toshio Yamaguchi ( talk) 11:31, 12 October 2010 (UTC)
As it is currently explained, the remarkable feature appears to be its self-similarity and this could be stated more clearly. 195.96.229.83 ( talk) 09:34, 18 November 2010 (UTC)
In the section Background and history - Periodic and aperiodic tilings it says
How is the mere repetition of a part of a tiling descriptive of a periodic tiling? A Penrose tiling, for example, also repeats itself; in fact any finite region in a Penrose tiling (no matter how large) is repeated an infinite number of times in that Penrose tiling. The difference between a periodic tiling and a Penrose tiling is, that in a periodic tiling a finite portion of that tiling is repeated in constant intervals. In a Penrose tiling, every finite region is repeated an infinite number of times in that tiling, but is not repeated in constant intervals. Toshio Yamaguchi ( talk) 11:39, 10 November 2010 (UTC)
I still have to question the rationale for having the third 'remarkable property' in the lead section. I fail to see how this statement, which would be more appropriate in the article on Quasicrystals is helpful for giving the reader an accessible overview of the article's key points. Talking about x-ray diffraction without even briefly describing it or mentioning why it is important in the study of Penrose tilings and without any further description of x-ray diffractograms in the rest of the article doesnt help in making the article accessible to areader. Toshio Yamaguchi ( talk) 18:15, 15 January 2011 (UTC)
There is a deep and illuminating connection between the two, at least imho, but the attempt to mention this was rejected offhand as "too vague to be useful" which is not really an argument. For the general reader the article is probably much too technical and elaborating this would just make matters worse. The interested reader could perhaps be directed to the pinwheel tiling. This is how I see it:... The substitution rules decompose each tile into smaller tiles of the same shape as those used in the tiling (and thus allow larger tiles to be "composed" from smaller ones). This approach makes rather obvious a close link between aperiodic structures and fractals. Ael 2 ( talk) 19:27, 4 November 2010 (UTC)
"The tilings can be generated from one another by the methods of inflation or deflation. For example, in deflation a cluster of tiles is subdivided into smaller pieces following specific procedures. Performing such operations iteratively, one can generate an aperiodic tiling with a much larger number of smaller tiles. These procedures endow the tiling with the property of self-similarity. These properties have suggested, right from the time of the discovery of Penrose tilings, that the tiling is fractal in nature /ref to Gardner chap.1/."
So I have restored the previous text an added this ref available online. Ael 2 ( talk) 10:04, 5 November 2010 (UTC)
I've produced an overlay illustration of a tiling and its inflation overlaid. The article is a bit messy with illustrations (and the text) as it is, especially mixing the various types of tilings. Anyway, this image or something like it could be useful to add -- Sverdrup ( talk) 19:56, 27 September 2011 (UTC)
The article doesn't actually say what a Robinson triangle actually is. The nearest I got to an explanation/definition was Golden triangle (mathematics)#Golden gnomon. The first use of the term in the current article is in the Kite and dart tiling (P2) section:
Both the kite and dart are composed of two triangles, called Robinson triangles, after 1975 notes by Robinson.
How about linking it thus: Robinson triangles ? > MinorProphet ( talk) 16:40, 5 February 2012 (UTC)
The substitution method for both P2 and P3 tilings can be described using Robinson triangles of different sizes. The Robinson triangles arising in P2 tilings (by bisecting kites and darts) are called A-tiles...
Tie-and-Navette tiling showing many properties of the penrose tiling.
Ad Huikeshoven (
talk)
09:48, 2 January 2015 (UTC)
The substitution rules for the P2 tiling as given in the article appear to be redundant: the "sun" and "star" rules just consist of the application of the already-given half-dart and half-kite rules to those particular structures. Is there any reason the sun and star need to be included as separate cases? 130.226.142.243 ( talk) 09:00, 15 January 2015 (UTC)
The article lead currently states that, "The aperiodicity of the Penrose prototiles implies that a shifted copy of a Penrose tiling will never match the original." This is not correct. It is an extraordinary property of the Penrose tiling that any given finite pattern will be repeated within a certain (and remarkably short) distance from it. I don't have a good source handy: does anybody know of one? — Cheers, Steelpillow ( Talk) 05:58, 19 August 2015 (UTC)
The comment(s) below were originally left at Talk:Penrose tiling/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.
This article contains serious inaccuracies, notably in (1) the matching rule and (2) the statements about symmetry. –Dan Hoey 02:33, 24 April 2007 (UTC) |
Last edited at 10:51, 12 February 2008 (UTC). Substituted at 02:27, 5 May 2016 (UTC)
The article is clear that kites and darts, with the standard matching rule, cannot tile the plane periodically. Yet the article contains an image, as at the right, which appears to be of a repeating unit of a periodic tiling. What's going on here? It looks to me like there's a mistake somewhere. Even if there isn't, there's an apparent contradiction that needs explaining. Maproom ( talk) 17:08, 11 July 2018 (UTC)
I've read a couple of books by Penrose and, though he makes my head spin, I managed to keep up — something I cannot yet say for this article. I've reworked the opening of Background and history and may move this up to the lede; I'm not totally happy to set down a jargon-heavy statement and then backfill an explanation of the terminology but haven't yet figured out how better to approach. (As well, the thought occurs to me that a mosaic is essentially a tiling with a potentially infinite set of nonrepeating tiles. Though at risk of coatracking, this may be a valuable illustration for the non-mathematician.)
In addition, I hope to clarify that Penrose tilings predate Penrose, as illustrated at the very end of the article, else the impression is given that he is somehow the inventor.
Weeb Dingle (
talk)
05:29, 13 October 2019 (UTC)
One more thought: aside from the title and references, the term "Penrose tiling" appears 49 times in the article. This feels a little like it may be browbeating the reader. Could this be productively reduced?
Weeb Dingle (
talk)
05:36, 13 October 2019 (UTC)
Our article currently exhibits a photo of a floor tiling at the Pilgrimage Church of Saint John of Nepomuk at Zelena Hora, Czech republic; see right. It does not mention, although it is true, that the two tiles of this tiling (pentagons and thin rhombs), meeting in the vertices of the two types visible in the photo, do not and cannot form a Penrose tiling. I analyzed the same set of tiles in a different context in a blog post [10]. The tiling from the blog post (in a downtown shopping street of Copenhagen) is I think more or less the same as the one in this photo, with the center of symmetry of the tiling at the top center of the photo. It is not a Penrose tiling (it has periodic patches that extend in wedges from the center). But more strongly, as the post explains, this set of prototiles, meeting in this way, cannot be made to form Penrose tilings. Because it is my own blog post, I am not going to add it to this article, but I think it would be appropriate to mention in the caption that it is not a Penrose tiling. — David Eppstein ( talk) 21:05, 20 January 2020 (UTC)
{{u|
Mark viking}} {
Talk}
04:13, 21 January 2020 (UTC)
Is this new discovery better here or in the Aperiodic tiling page? Thanks
I changed the definition of an aperiodic tiling because under the previous definition, the Penrose tiling is not aperiodic (since the tiles can be arranged periodically as in Figure 1). David9550 ( talk) 20:06, 30 March 2023 (UTC)