Talk:Penrose tiling

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Mathematics grading:	B Class	High Importance	Field: Geometry and topology

Contents 1 L-systems 2 Clark Richert 3 Projection 4 No Matching Rules Discussed 4.1 Proposal for matching rules 5 Explanation? 6 More penrose tiles needed? 7 Link to a Penrose-base artistic image 8 Kite or Kile? 9 More external links 9.1 Media Hype 9.2 Substitution Matrix 10 use of the word "uncountable" 10.1 Importance

[edit] L-systems

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)

A free Microsft Windows program to generate and explore rhombic Penrose tiling is available at http://www.jkssoftware.com/penrose. The software was written by Stephen Collins of JKS 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)

[edit] Clark Richert

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)

[edit] Projection

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)

[edit] No Matching Rules Discussed

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 --—The preceding unsigned comment was added by 69.151.118.199 (talk • contribs).

The article currently reads (after describing the shapes):

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.

Perhaps this should be more emphasized to avoid confusion by other future readers. --Chan-Ho (Talk) 03:26, 11 March 2006 (UTC)

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!

True. This needs to be covered in the article, preferably with a picture. Reyk _YO! 22:08, 20 May 2006 (UTC)

Is there any reference or better description for that strange rule

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.

? While the second (well-known) method implies the first, the converse does not hold, i.e. the "one rule" is not sufficient to guarantee non-periodicity: One big rhomb and two small ones can be put together to form a hexagon and these tile the plane periodically and obeying that rule (smeone produced some piece ASCII art of this on the talk page of the german article).--Hagman-de 08:58, 10 March 2007 (UTC)

[edit] Proposal for matching rules

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).

[edit] Explanation?

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)

[edit] More penrose tiles needed?

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)

[edit] Link to a Penrose-base artistic image

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:

http://www.peterhugomcclure.com/colour%20images/36.htm

Best regards pete mcclure.--81.86.8.62 13:00, 22 November 2006 (UTC)

Hi Pete. Are you willing to allow use of the image directly on the wikipedia? There's a choice of copyright tags to go with an image here; see WP:TAG.--Niels Ø 09:08, 23 November 2006 (UTC)

Jos Leys has also made some imagery using Penrose Tiles. Jos Leys's website is here, while the Penrose specific images can be perused here. Still, I think these images should only considered if they can add something to the article. Or maybe in a section on the use of Penrose Tiles by artists. Scribblesinmindscapes 17:04, 23 January 2007 (UTC)

I think you misunderstand the significance of Pete's work. He has created a periodic Penrose tiling having translational symmetry in both dimensions (wich I verified in a paint program). This contradicts a statement in the article that the "parallelogram rule" results in an aperiodic tiling. =Axlq 19:21, 27 January 2007 (UTC)

Aaah, okay, I initially just took Pete's comment as artistic, as being indicative of an artistic work inspired by Penrose Tiling. The tiling in his picture is not the Penrose tiling. It has translational symmetry and it uses the constituents of a Penrose tiling to form a tiling with translational symmetry - but the real Penrose tiling must use the constituent tiles in a certain manner as discussed in the wikipedia article. Penrose's original paper also starts by saying: "'kites' and 'darts', which, when matched according to certain simple rules, could tile the entire plane, but only in a non-periodic way". Penrose's article can be found here (and should perhaps be included as an external link): [1] Pete's work does not agree with the pictures in the wikipedia article and I can therefore only assume that it doesn't match the tiles correctly. I think the usage of the words Penrose Tiles for the constituents are a bit misleading as indicated they need to be tiled in a certain way to be aperiodic. I'm not a researcher in the field of aperiodic tilings or quasicrystals so disagreement welcomed. Scribblesinmindscapes 09:24, 28 January 2007 (UTC)

Pete's work falls under the section of this article titled "Rhombus tiling" – not kites and darts – and follows the rhombus tiling rule exactly.

I just confirmed that the picture now shown in that section also has translational symmetry in both dimensions — the region outline itself is tile-able. There is a disconnect between the text in that section and the picture; now that I know the picture has translational symmetry, the text now seems flawed or unclear. =Axlq 19:23, 28 January 2007 (UTC)

[edit] Kite or Kile?

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).

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

[edit] More external links

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)

Another Penrose Tiles article at AMS: [4]

Scribblesinmindscapes 18:48, 27 January 2007 (UTC)

[edit] Media Hype

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)

I think that this paper is important, not just for the link to Islamic architecture, but because it gives tiles which allow you to achieve 10-fold rotational symmetry rather than the mere 5-fold symmetry achievable with the Penrose's own tilings. In both cases there is also an additional reflectional symmetry. So the pattern can be generated by reflections of an 18 degree (for Lu's scheme) or 36 degree (for Penrose's scheme) section between two mirrors. JRSpriggs 07:19, 28 February 2007 (UTC)

If you believe that tenfold symmetry is somehow superior to fivefold perhaps you should appreciate the pinwheel tiling which has symmetry of infinite order. The Radin paper [1] explains that the Penrose tiling has indeed a tenfold statistical symmetry but the local is just fivefold (added this just the other day and meant to clean up references and notes).

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

I am not talking about some kind of average or statistical symmetry. I am talking about a perfect rotational symmetry around a single point. JRSpriggs 10:39, 28 February 2007 (UTC)

A nice symmetry around a central point can be easily achieved by arranging mirror pairs of identical pie-slices from an aperiodic tiling. Their angle should be pi/n. A more elegant solution would be to find an aperiodic decomposition of a triangle and to arrange (pairs of) triangles to form a vertex. Tubingen triangles, Robinson triangles and the pinwheel substitution illustrate this idea.

The main interst in an aperiodic tiling is a lack of symmetry: aperiodic means lacking translational symmetry. The surprise is that only with an aperiodic tiling you can achieve otherwise 'forbidden' symmetries.

Roger Penrose was searching to decrease the number of tiles needed for an aperidic tiling and he arrived at two. The medieval islamic decorators needed some more.

And their interest was to obtain a complicated and elegant motif. In the Mediterranean world interlaced motifs are believed to be a charm against the evil eye, which is apparently one of the sources for interest in them. One might speculate that aperiodic tilings have a particular effect on the perceptive apparatus which could explain their psychological appeal.195.96.229.83

Peter J. Lu has combined inflation rules for two tiles defined by the Islamic architects with one inflation rule of his own devising and gotten a set of rules for three tiles (all used by the architects along with two other tiles) which can tile the plane with perfect ten-fold rotational symmetry. In my opinion, this is an improvement on the five-fold symmetry of the Penrose tiling. I hope to get a link to a file with all three of Lu's tiles. JRSpriggs 09:26, 7 March 2007 (UTC)

[edit] Substitution Matrix

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)

[edit] use of the word "uncountable"

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 is however a serious problem here: if the Penrose tiling is in fact an infinite set of variants, what is the meaning of the definite article? 'The' Penrose tiling is the one that Roger Penrose first proposed, but how to define the rest? In the recent hype an attempt to escape from this ambiguity has been made by speaking about the quasicrystalline Penrose tiling. 'Perfect' would have been a better choice of adjective as the variants are seen to be produced by local 'defects' and thus considered to be 'imperfect'. I suspect that the second illustration in the article has been the cause for the unfortunate word choice but it replaced an older picture of something that was not a Penrose tiling.91.92.179.156 21:08, 7 March 2007 (UTC)

The word uncountable has a special meaning in Mathematics; have a look at the article about the countable set. In short, something can be finite or it can be infinite. The latter can be distinguished into countably infinite and uncountably infinite. In my opinion, you can find a bijection to the set of the natural numbers, so the set of Penrose tilings is countably infinite. This bijection is an algorithm (not really, because it does not terminate, but otherwise it is) creating all possible penrose tilings in an infinite number of steps. It starts with a single tile. Then it enters an endless loop (could be implemented using a width-search) in which it branches a finite number of times to place a neighbouring tile to all possible positions on the current tiling. This way each given possible tiling is generated in a specific amount of steps. This is a bijection to the set of the natural numbers (the numbers of steps are members of this set), so we have the necessary bijection. qed. So it should be countably infinite. Alfe 07:38, 10 March 2007 (UTC)

The article says "It is easy to check that some of the compact patches consisting of three tiles admit two different arrangements within the same perimeter and thus variations are possible.". If that is true and there are a countably infinite number of such patches (as there must be), then the number of variations which can be achieved this way is uncountable in the technical mathematical sense used in set theory. JRSpriggs 11:10, 10 March 2007 (UTC)

That's not a proof, though. The proof of uncountability that I know of is by deflation. It's easier to prove that there are uncountable choices of a single tile in a tiling; since each tiling only has countably many tiles, it amounts to the same thing. To choose a single tile in a tiling, first choose whether it is a kite or a dart, second choose what kind of patch this tile is part of at the second level of deflation, etc. Thus one can specify the whole tiling by an infinite sequence of finite choices of this type. The number of sequences of choices that can be made is the number of the continuum. —David Eppstein 16:05, 10 March 2007 (UTC)

Yes, David. My argument was for the rhombus tilings. Yours works for the kite and dart tilings (and would work also for the rhombus tiles with a different choice of words). What would you like me to clarify or justify to make my proof complete in your eyes? JRSpriggs 09:54, 11 March 2007 (UTC)

The algorithm I presented above produces a countably infinite number of finite tilings. So it proves that the number of finite tilings is countably infinite. It seems to be of more importance, though, what the number of tilings is which fill the whole infinite plane. That number may likely be uncountably infinite. The text could point out that difference more clearly. Alfe 01:47, 15 March 2007 (UTC)

[edit] Importance

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 [5].al 17:49, 14 March 2007 (UTC)

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)

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)

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)

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)

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)

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)

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)