Talk:Penrose tiling

message: 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)
thread_title: No [[crystallographic restriction]]? 

message: 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.
:Now I too wonder how the L-system generates the tilings. I have one guess: it might use deflation. Is this correct? —Sverdrup(talk) 15:31, 27 Dec 2003 (UTC)
thread_title: L-systems

message: 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)
replies: text: A very important distinction is that Richert does not claim he had invented the matching rules; as I understand it, he only is claiming to have discovered the pattern they force. This is the crux of the matter: the Penrose tiles are interesting because they can only form this particular non-periodic structure. Of course, Kimberly-Clark was only printing the pattern on it's tissues, not making any infringement, I think, on the matching rules and tiles themselves. --69.152.216.28 18:53, 17 September 2007 (UTC)
thread_title:  Clark Richert 

message: 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.
:I heard that it is not a projection of the entire five dimensional lattice, but just of the part of the lattice within a slab between two parallel four-dimensional flats. Is that right? JRSpriggs 07:19, 28 February 2007 (UTC)
thread_title: Projection

message: 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)
thread_title:  Proposal for matching rules 

message: 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)
thread_title:  Explanation? 

message: 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)
thread_title:  More penrose tiles needed? 

message: 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)
replies: text: Probably some typing error; never heard about 'kile', definitely.al 20:39, 26 February 2007 (UTC)

text: "Kile" in norwegian means a wedge (or to tickle) Cuddlyable3 07:57, 15 May 2007 (UTC)
thread_title:  Kite or Kile? 

message: 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)
thread_title: Substitution Matrix

message: 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 .
Here is a good introduction  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)
thread_title: The Pentagonal Penrose Tiling

message: 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 .al 17:49, 14 March 2007 (UTC)
replies: text: Please don't remove the rating template. If you feel that "High" is the wrong importance, that can be changed, but changing it is very different from ripping out the whole rating block. As for mathematical triviality, I disagree but more importantly I think that mathematical depth is far from the only thing determining importance. —David Eppstein 20:57, 14 March 2007 (UTC)

text: Sorry for the tag; please explain why the importance is high.Imho it is very low, the Penrose tiling being just an example.I believe that the quality is also low and a lot of rewriting has to be done (Be bold).al 18:32, 16 March 2007 (UTC)
replies: text: My feeling is that in terms of mathematical depth the importance of Penrose tilings should be mid, not low. The concept of a periodic tiling itself is nontrivial, as is the construction of Penrose tilings by inflation and deflation, as is the relation to duals of pentagrids, as is the connection to sections of higher dimensional lattices. However, since Penrose tilings are so well known popularly, and have some relation to physical quasicrystals as well, I think it's enough to boost the importance to high. As a similar example, I've been editing regular number recently; in terms of mathematical content it is much more trivial (merely the numbers which have only 2, 3, or 5 in their prime factorization) so in mathematical depth I would rate it as low, but again it is a concept with several important applications outside of mathematics, due to which I boosted the rating to mid. —David Eppstein 18:51, 16 March 2007 (UTC)

text: There is no doubt that the  PT is the paradigmatic case, but it is just an instance of an aperiodic tiling. Everything that is known about  the PT, has been found to be valid mutatis mutandis for the Ammann-Beenker tiling. As I see it, aperiodic to periodic  is just as rational to irrational. The binary Fibonacci word can be inflated, deflated and projected and it offers a more obvious access to understanding the properties of the PT, which is more or less its generalization. Socolar's multigrid approach is perhaps 'deep', but it is also a much more general topic than PT. Briefly: popular is not the same as important. Shall we lower the rating with one or two notches? al 17:03, 17 March 2007 (UTC)

text: I would go for mid importance for this article, but I'd accept high. Low is really for the very obscure stuff, see :Category:Low-importance mathematics articles (although there are a few there which I think deserve mid). I do think this is mathematically important as it opened up a new area of investigation, it what was beleived to be a closed book. It further prompted investigation into real world crystals which had five-fold symmetry where none were thought to exist. Hence is has implication outside of mathematics, which is one reason for for an increased rating. --Salix alba (talk) 22:00, 17 March 2007 (UTC)
thread_title: Importance

message: 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)
thread_title:  Early history 

message: 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)
thread_title:  Surprising 

message: 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)
thread_title:  Is there a better L-System rule set? 

message: 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)
replies: text: I noticed that contradiction as well, and you seem to be correct. Under the citation I just added (), which distinguishes between aperiodic and nonperiodic tiling, the Penrose tiling is indeed aperiodic, since it is not also possible to construct a periodic tiling from Penrose tiles. I've updated the article accordingly. --ArthurDenture (talk) 05:05, 26 August 2008 (UTC)
thread_title:  Contradiction between Penrose and Aperiodic articles? 

message: 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:
:"Robert Ammann independently discovered the tiling at approximately the same time as Penrose.[citation needed]"
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:
*Cite the same sources as Ammann's Wikipedia page.  This seems a bit pedantic.
*Cite Ammann's Wikipedia page itself.  This seems redundant, since that link is already in the sentence.
*Find newer and more useful citation sources, and update the article, Ammann's article, or both.  While this would solidify things a bit, it also just adds a layer of abstraction to the question: the placement and propagation of the citations would still need to be determined.
*Remove the "citation needed" tag.  Assume that the link at the beginning of the sentence is a sufficient guide for interested readers.
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)
replies: text: Removing the tag seems OK to me. The obvious problem is that the person who originally put it there may, without reading this discussion, put it back. Nevertheless I should try removing the tag, and, if it comes back, copying the references from the other article to here. JamesBWatson (talk) 15:41, 22 January 2009 (UTC)
replies: text: Thanks for your input.  I just removed the one instance mentioned above, and will observe how things play out in the future.  76.105.238.158 (talk) 02:00, 28 January 2009 (UTC)
thread_title: "Citation needed"? 

message: 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.
*Source 1 (p. 2)
*Source 2 (p. 2)
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)
replies: text: That's a nice image you made. Is it public domain? Thanks for helping to clarify a confusion. My understanding from the sources is that Gummelt's matching rules permit only two kinds of overlapping region (by one or two red kites), but such an overlap can be achieved in multiple ways. I have clarified the article and image based upon the article by Lord et al. Geometry guy 22:16, 13 December 2009 (UTC)
thread_title:  Gummelt's Decagons 

message: 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)
replies: text: There is an interesting approach in the French version: it starts with P0 which is the decomposition into Robinson triangles. Later it is noted that decorating the rhombuses allows to transform P3 into P1. Next by labeling the pentagons' corners  1,3,5,2,4 the P0 is recovered and a by-product, the (bow)tie-and-navette tiling is also mentioned, see there ref.3 Lück R., Penrose Sublattices, J. of Non Crystaline Solids 117-8 (90)832-5. Perhaps this could be sneaked in somewhere. Fig4 of the article presents a "genealogy" of related tilings;  Grunbaum and Shephard are cited, while the P1-P3 naming  is used as a matter of fact (and probably I took it from here, but my memory is blank about the 'P0'). 
Congratulations for the GA status.Ael 2 (talk) 10:10, 2 July 2010 (UTC)
thread_title:  Plans 

message: "there are matching rules that specify how tiles may meet each other"
but what are they?   brain (talk) 23:39, 19 April 2010 (UTC)
thread_title:  P1 matching rules 

message: 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)
replies: text: Hey Willow, thanks for commenting here! There is certainly no rush, as there is still quite a bit to do content-wise to go from broad to comprehensive. Note also that the lead is in very poor shape (don't tell anyone at GA :) ), but I would rather hold off rewriting it until the body of the article is more complete. Geometry guy 20:13, 29 June 2010 (UTC)
thread_title:  Kindly meant suggestions 

message: 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)
thread_title: Third remarkable property

message: 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)
replies: text: The thing to do is find a reliable source that relates Penrose tilings to fractals. Then the material in that source will guide you as to how to present the idea. Geometry guy 23:42, 4 November 2010 (UTC)
replies: text: Here is the beginning of the article On the fractal nature of Penrose tiling by Ramachandrarao P., in Current Science 79(2000)p364:
"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)

text: "A close link between aperiodic structures and fractals" really was extremely vague, and what it appeared to vaguely hint at was simply wrong - most aperiodic structures are not fractals at all. What the source you quote actually says is that the inflation/deflation construction shows that the Penrose tiling has a scaling self-similarity, and so can be though of as a fractal. I have amended the text in our article to correctly say this. Gandalf61 (talk) 13:00, 5 November 2010 (UTC)
thread_title: Linking Penrose tilings to fractals?

message: 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)
thread_title:  Illustration of inflated tiling 

message: Tie-and-Navette tiling showing many properties of the penrose tiling. Ad Huikeshoven (talk) 09:48, 2 January 2015 (UTC)
thread_title:  Tie and navette tiling 

message: 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)
thread_title:  Dart and kite substitution rules 

message: 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)
thread_title:  Periodicity and matching 

message: Substituted at 02:27, 5 May 2016 (UTC)
thread_title: Assessment comment

message: 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)
replies: text: The image isn't a repeating unit of a periodic tiling. I looks symmetric Up/Down but not Left/Right, and is also cropped in a way that the edges don't match up. Jonpatterns (talk) 17:17, 11 July 2018 (UTC)
replies: text: In that case, captioning it "Wallpaper" is misleading, it suggests the word is being used as in Wallpaper group. Maproom (talk) 18:54, 11 July 2018 (UTC)
replies: text: I can see the confusion, as wallpaper was used in the loose way meaning Wallpaper (computing). As this is a maths article I've updated the describe to Part of the plane covered by Penrose tiling of type P2 (kite and dart). Created by applying several deflations, see section below.. Jonpatterns (talk) 11:30, 13 July 2018 (UTC)
thread_title:  A contradiction? 

message: 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 . 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)
replies: text: I moved it up a section to the "related tilings" discussion. It's not that great a picture (poor angle, not much color contrast); maybe we should just leave it out altogether? XOR'easter (talk) 21:08, 20 January 2020 (UTC)
replies: text: I do think it's interesting that people other than Kepler were using tilings of this type, long before Penrose, so my (weak) preference would be to keep it in. —David Eppstein (talk) 21:20, 20 January 2020 (UTC)
replies: text: That's a fair point. XOR'easter (talk) 02:40, 21 January 2020 (UTC)

text:  I did a little image processing on the picture to correct exposure and bring out the tiling a bit more strongly. See if you think it is an improvement. -- {Talk} 04:13, 21 January 2020 (UTC)
replies: text:  Yes, that does look better. Thank you. XOR'easter (talk) 15:42, 21 January 2020 (UTC)

text: I agree it has periodic patches in wedges, but the pattern as a whole never repeats. Thus it lacks translational symmetry yet has reflection symmetry and rotational symmetry. Isn't that the definition of a Penrose Tiling as given in this article? --Skintigh (talk) 21:21, 10 December 2021 (UTC)
replies: text: No. That is neither the definition of a Penrose tiling (which is a specific aperiodic tiling, different from this tiling and from other aperiodic tilings) nor the definition of an aperiodic tiling (which requires that it does not contain arbitrarily-large periodic patches, or sometimes more strongly that the local rules used to piece together the tiles cannot produce arbitrarily-large periodic patches). Being aperiodic is a much stronger condition than merely not being periodic. —David Eppstein (talk) 22:54, 10 December 2021 (UTC)
replies: text:  I didn't realize the term aperiodic tiling implied more than nonperiodic. Like Aperiodic tiles I wonder if set of tiles could be periodic in one direction and "aperiodic" in the other, a frieze pattern symmetry, or maybe it is just an aperiodic tiling on an infinite cylinder rather than infinite plane? Tom Ruen (talk) 08:10, 11 December 2021 (UTC)
replies: text: There's an example like that for the hyperbolic plane: the binary tiling has versions that have infinite one-dimensional symmetry groups, but cannot have a two-dimensional family of symmetries. —David Eppstein (talk) 08:52, 11 December 2021 (UTC)
thread_title:  Saint John of Nepomuk 

message: Is this new discovery better here or in the Aperiodic tiling page?  Thanks
: https://phys.org/news/2023-03-geometric-tiled.html
: Samw (talk) 19:17, 24 March 2023 (UTC)
replies: text: Better in the einstein problem page. Definitely not here. This article is about a specific and different family of aperiodic tilings, not about the general concept of aperiodic tilings. —David Eppstein (talk) 23:27, 24 March 2023 (UTC)
thread_title:  Einstein shape 

message: 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)
thread_title:  Definition of aperiodic tiling