Talk:Pythagorean tiling

I am standing on this right now
it can be split into substrings of the form "01" and "0" (that is, there are no two consecutive ones) and if these two substrings are consistently replaced by the shorter strings "0" and "1" then another string with the same structure results. I tried that on my tile floor that I am standing on now and got it. Turtleguy1134 (talk) 22:00, 23 October 2011 (UTC)

Proof of theorem
Here is a part of the current article: "This tiling is called the Pythagorean tiling because it has been used as the basis of proofs of the Pythagorean theorem by the ninth-century Arabic mathematicians Al-Nayrizi and Thābit ibn Qurra, and by the 19th-century British amateur mathematician Henry Perigal". Why not expose such a proof on "Pythagorean_theorem", where a link was just created to this article? 109.6.129.249 (talk) 16:22, 17 October 2012 (UTC)


 * On one hand, it is difficult to find this link to the article. On the other hand, the proof of the theorem given here is not clear. 109.6.129.249 (talk) 11:48, 18 October 2012 (UTC)


 * The article has been viewed 373 times in September. "Pythagorean_theorem" has been viewed 162696 times in the same month.  The title "Pythagorean tiling" has no meaning for most people.  It is not reasonable to expose within this article  one proof or more of the theorem through a tiling.  109.6.129.249 (talk) 14:51, 31 October 2012 (UTC)

A click to go to "Pythagorean tiling" is scarce, and no source was given to this title since the creation of the article. Moreover, no reason to isolate on an article a partial proof of the Pythagorean theorem, where a given right triangle is not isosceles. Really numerous are proofs of the theorem through a tiling? At least one complete proof through a tiling is logically expected below the title "Pythagorean theorem". 109.6.129.249 (talk) 15:18, 2 November 2012 (UTC)
 * The title is given in footnote [1] of the article, and has been since the creation of the article. And the article is not primarily about a particular proof of the Pythagorean theorem; that's one use of this tiling, and concerns one section of the article, but the tiling has other uses and properties described in the article's other sections. —David Eppstein (talk) 16:34, 2 November 2012 (UTC)
 * It is suggested that  the  section  be merged&hellip;  109.6.129.249 (talk) 09:46, 3 November 2012 (UTC)
 * Many people could enjoy the image if inserted in a section about history  of the Pythagorean theorem,
 * while it is difficult to know immediately what proofs are behind the image. Thus the author of the image
 * could be satisfied when the section will be removed.
 * 194.153.110.5 (talk) 11:52, 7 November 2012 (UTC)
 * Speak for yourself. —David Eppstein (talk) 16:23, 7 November 2012 (UTC)

Speak for yourself, what does that mean? Do you think that this section of the article and its image are revealing a principle of proofs of the theorem? What is the  topic  of this section of the article? 109.6.129.249 (talk) 13:34, 8 November 2012 (UTC)

A tiling  by  squares is  created  from  a  right  triangle. On  the   first   row,  three   images   show a  grid in  dashed  red  which  takes  a  particular  position relative  to  the  tiling.

The Pythagorean theorem does not exclude the particular case where a right triangle is isosceles. The banner is now removed, that suggested  that  the section about the theorem could be transported, not the whole article, into section "Proofs"of "Pythagorean theorem", because all proofs of the theorem through a tiling are still valid in case of  isosceles  right  triangle  ( a = b ), while the article says that a "Pythagorean tiling" is a tiling by squares of  two  different  sizes  ( a &ne; b ). If we want to expose in this article a correct proof of the theorem through a tiling, we have to modify the definition of a "Pythagorean tiling". The new definition would be: a Pythagorean tiling is a tiling of the Euclidean plane by squares of equal  or  different  sizes. 109.6.129.249 (talk) 16:29, 1 December 2012 (UTC)


 * Oppose merger. I see no reason why this article should not mention that Pythagorean tilings have been used to prove Pythagoras' theorem.  In fact, it seems perverse not to, since this is the very reason they are called Pythagorean tilings.  As to whether the main article Pythagorean theorem should contain an independent discussion of this proof (that's not what "merge" means, by the way), that is an issue that should be discussed at Talk:Pythagorean theorem rather than here.  In fact, there already is a section there responding to an earlier addition of content related to this proof to that page, and there seems to be no consensus to include it.   Sławomir Biały  (talk) 18:27, 29 November 2012 (UTC)
 * Oppose merger for basically the same reasons. I also oppose inclusion of symbol-packed images like "A pattern of Pythagorean tiling.svg" because I think all the formulas make it more confusing than helpful. —David Eppstein (talk) 23:30, 29 November 2012 (UTC)
 * Oppose merger. Even if there is a modest amount of overlap with the other page, this is quite common by wiki standards. Tkuvho (talk) 09:18, 30 November 2012 (UTC)
 * Oppose merger as these are distinct and reasonably substantial topics. A modest overlap is no bad thing.  Deltahedron (talk) 20:23, 30 November 2012 (UTC)
 * Oppose merger. Overlap between the topics of Wikipedia articles is inevitable; nevertheless we prefer to separate articles rather than to have one enormous master article on everything.  This is a page on Pythagorean tiling, and a proof of the Pythagorean theorem by tiling fits extremely well here.  Ozob (talk) 13:17, 1 December 2012 (UTC)

Additions to the page
Proposed additions to this page should be discussed here in small installments. Confrontational edits should be avoided. Tkuvho (talk) 17:24, 29 November 2012 (UTC)
 * About the suggestion to merge, see "Discuss"!  109.6.129.249 (talk) 17:41, 29 November 2012 (UTC)
 * But that's a link to the above discussion, which doesn't make any kind of case for a merger. The only pertinent comment there regarding a merger is my own "oppose" vote.  Sławomir Biały  (talk)
 * Please also note a possibly related discussion at Commons:Commons:Administrators' noticeboard/User problems. I am also opposed to any merger; I believe this is adequately notable as a standalone topic and that any inclusion of its content at the Pythagorean theorem article should have zero effect on what is included here. —David Eppstein (talk) 23:29, 29 November 2012 (UTC)

Six piece dissection of two squares into a different two squares
I think the article may already be a bit image heavy for the amount of text it has, so I'm reluctant to add this, but to the anonymous editor who was confused by the sentence in the article about getting a six-piece dissection of two squares into a different two squares by using two overlaid tilings, perhaps this image will help enlighten you. The two green squares can be dissected in six pieces into the two red squares; in each case, the larger of the two squares is split into five pieces (a square in its center surrounded by four congruent irregular quadrilaterals) and the smaller of the two squares is unsplit. —David Eppstein (talk) 23:54, 29 November 2012 (UTC)
 * What connection  with  the  theorem? The section title begins with "Pythagorean theorem"… — Aughost (talk) 02:34, 30 November 2012 (UTC)
 * ...and ends with "dissections". This is a dissection. The connection to the theorem is a different set of dissections, the 5-piece two-squares-to-one dissections. I think this one is relevant enough to mention in the article (at least at the single sentence length it currently is given) but may not deserve its own separate section. —David Eppstein (talk) 03:12, 30 November 2012 (UTC)
 * So you have to define a "dissection", with a source. With planar surfaces, would you explain here what is a "dissection", please? — Aughost (talk) 09:29, 30 November 2012 (UTC)
 * There is a source already given. It is an entire book about dissections. Additionally, the article gives a wikilink to a separate article with a definition. I have no idea what you mean by "with planar surfaces". —David Eppstein (talk) 13:47, 30 November 2012 (UTC)
 * It's also clear from the context (to me at least) what is meant by the term "dissection".  Sławomir Biały  (talk) 13:48, 30 November 2012 (UTC)
 * The current section about the theorem contains a link to "Dissection problem"  in its last paragraph. A link to "Dissection puzzle"  would be more instructive in the first paragraph. — Aughost (talk) 16:08, 30 November 2012 (UTC)
 * There are two kinds of dissection puzzle: the ones where you are given a fixed set of pieces and must rearrange them to resemble drawings of real-world objects (e.g. tangram) and the ones where you are given two sets of shapes with equal total area and must find the pieces yourself, generally aiming for as few pieces as possible. The second type of puzzle is the one that's relevant here, and that's the one given in the existing link. But the meanings are similar enough that it might make sense to merge the two dissection articles. —David Eppstein (talk) 16:16, 30 November 2012 (UTC)

Puzzle
Easy to understand, without any doubt about a word like "dissection", two different assemblages of a same set of puzzle pieces have equal areas. It is a principle of proof of the Pythagorean theorem, badly exposed in the current section about the theorem, where we see only one shape formed by five pieces: a square, the size of which is denoted by c  in the text. Here the image is better, with the two different shapes used to prove the theorem. 194.153.110.5 (talk) 13:27, 30 November 2012 (UTC)
 * It may be "badly expressed" in the current version, but those two versions shown in the current version are not original research, because they both appear in that form in the mathematical literature. I don't think the same thing can be said for your figure. Additionally, your figure is related to Perigal's dissection but does not show the much older one of Al-Nayrizi. —David Eppstein (talk) 13:51, 30 November 2012 (UTC)

No original research, as said above. With these two positions of the grid in dashed red relative to the tiling, we get the same cuttings as on the third image of this section. 194.153.110.5 (talk) 14:29, 30 November 2012 (UTC)

Group
A group of geometric transformations leaves unchanged the tiling and the grid. So the grid must be displayed in the first section of the article. The first image here shows four arrows, for a same transformation that preserves grid and tiling: a translation. However, in introduction of the article is defined a "Pythagorean tiling": a base of proofs of the theorem. And all copies of the original triangle are very visible in another image, where all hypotenuses are in dashed red. With that image without arrows, as first image with something about mathematics, we would respect chronological order of historical discoveries about such grids.

What image to illustrate the group of transformations, what do you think? 109.6.129.249 (talk) 10:26, 1 December 2012 (UTC)

We see no grid in the current article, it will be necessary to show a grid, of course. To avoid too numerous images, a same image will deal with several subjects, indeed. On the image near the lead, I prefer a grid that shows periodic copies  of  the  original  triangle. 194.153.110.5 (talk) 13:37, 1 December 2012 (UTC)

I am happy enough with the undecorated image that apperas first in the article. I think the fact that it is periodic is obvious. It might be possible to draw an image that illustrates both the translational and rotational symmetries of the tiling, but just superimposing another square grid on top of it won't do that. —David Eppstein (talk) 16:02, 1 December 2012 (UTC)

I am not sure to understand. Anyway, on 'Commons' everybody can search an image about our subject. Maybe someone will soon create a marvelous image&hellip; 109.6.129.249 (talk) 16:44, 1 December 2012 (UTC)
 * In fact, a user on Commons whom I strongly suspect to be the same as Aughost/the various IP editors here has made it much more difficult to find these images on commons, by removing most of the images from the Pythagorean tiling category there. —David Eppstein (talk) 16:55, 1 December 2012 (UTC)
 * I agree to insert this image in first section of the article. — Aughost (talk) 17:48, 1 December 2012 (UTC)
 * You agree with yourself? Or with who else? —David Eppstein (talk) 19:06, 1 December 2012 (UTC)
 * I think the proposed image is not very helpful.  Sławomir Biały  (talk) 02:25, 2 December 2012 (UTC)
 * What connection between the article and this question: are you two friends or a same person? More suitable, what are your arguments about this image? — Aughost (talk) 08:25, 2 December 2012 (UTC)
 * There was a request for attention at WT:WPM. The issue with your proposed image is that it contains far too many irrelevant symbols.  The image currently contained in the article illustrating the dissection is quite adequate.  A child with no special knowledge of trigonometry could understand how it works.  What I have not seen is any compelling reason we should add any of the other proposed images.   Sławomir Biały  (talk) 13:03, 2 December 2012 (UTC)
 * How to illustrate the first section of the article, that is the question. Ask children what does mean the section title: "Topology and symmetry", most of their answers will be that they don't know. Children gradually learn by reading more and looking at images.  Please, would you avoid doubtful generalities and come to our subject? — Aughost (talk) 14:34, 2 December 2012 (UTC)
 * The proposed image apparently has nothing to do with that section.  Sławomir Biały  (talk) 16:12, 2 December 2012 (UTC)
 * Absurdity! — Aughost (talk) 06:50, 3 December 2012 (UTC)