Talk:Tessellation

Non-Euclidian Geometries -- Spherical Geometry?

"It is possible to tessellate in non-Euclidean geometries such as hyperbolic geometry."
What about spherical geometry? Has nothing been done in this area? Is it somehow not possible due to a quirk of mathematics? I'd suggest at least a mention of this would be expected, since you mention the two other main branches of geometries.
--167.230.96.9 (talk) 20:58, 19 August 2015 (UTC)

Agreed. Spherical polyhedron has regular, uniform, and uniform dual tilings of the sphere. These could be mentioned in Tessellation#Tessellations_in_non-Euclidean_geometries somewhere. And the regular 4-polytopes also can projected onto the 3-sphere as tessellations/honeycombs and viewed in stereographic projection. Like 120-cell projected here File:Stereographic_polytope_120cell_faces.png. Tom Ruen (talk) 21:14, 19 August 2015 (UTC)

By loose analogy to the seventeen wallpaper groups, there is a family of tessellations of the sphere for each of the point groups in three dimensions. —Tamfang (talk) 09:53, 16 April 2023 (UTC)

Assessment comment

The comment(s) below were originally left at Talk:Tessellation/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 The comments may be irrelevant or outdated; if so, please feel free to remove this section.

At present this is largely a random collection of a few specific topics. It needs going through with reference to Tilings and Patterns to make sure that all the main topics and terminology are covered in a coherent arrangement; in the process, most of the "See also" links should become links at appropriate points in the body of the article instead. Joseph Myers 02:20, 20 May 2007 (UTC)

Last edited at 02:20, 20 May 2007 (UTC). Substituted at 20:16, 1 May 2016 (UTC)

Introductory video

A nicely done "educational animation" was added by an anonymous user to "explain the basic principles behind tessellating patterns." I have removed it for a number of concerns.

1. It introduces some artistic concepts about creating tessellation art, but does not really explain the basic idea of what a tessellation is. If the video is restored, perhaps it should go later in the article?
2. The video ends with an ad for a certain website. This is not appropriate for the introduction to an article. Perhaps as an external link? I have not checked the website.
3. It claims that only 3 "basic shapes" can tessellate. It is true that only 3 regular polygons tessellate, and many simple artistic tessellations follow these three patterns, but there are many other possible shapes and patterns in tessellations, such as the Arabic tessellation in the next image.
4. The video claims there are 3 "basic principles" for tessellating: translation, rotation, glide reflection. Is reflection being considered a type of glide reflection? In any case, these are symmetries found in many (not all) tessellations, not "basic principles." E.g. Dirichlet tilings generally have no symmetry at all.

--seberle (talk) 06:19, 21 June 2016 (UTC)

I agree with your decision to remove this animation. Any one of your four reasons would probably have been justification enough; the combination of all 4 makes the decision easy. Darrah (talk) 23:12, 22 June 2016 (UTC)

I'd really love to see that video. Too bad it has been removed Nonkuli (talk) 13:11, 13 August 2016 (UTC)

Nonkuli: The video is available on youtube here. There are hundreds of other similar videos that you can find with searches on youtube like "Tessellation Introduction", and if we were going to put one here, many of the other candidates do a better job of showing a wider variety of the basics, IMHO. Darrah (talk) 08:09, 21 August 2016 (UTC)

Expand Manufacturing Section

The "In Manufacturing" section looks tiny. I think we should add more to this section. — Preceding unsigned comment added by Jimli536 (talk • contribs) 23:23, 31 August 2016 (UTC)

Creating a separate Tiling (mathematics) page

There is tons of material on mathematical tiling/tessellation. So much so that adding it all to this page would be tedious, especially for readers coming to this article for the art, manufacturing etc aspects who don't have any interest. I've been discussing on the Talk:Polyomino page the possibility of creating a separate page purely dedicated to all of the finer mathematical points of tiling, which would include Discrete Tilings (e.g. Z^n for starters... there's a lot of literature here) and also the more usual tilings by sets of polygons, and then leaving a link to that more comprehensive article here. Mathguy9109 (talk) 15:37, 21 June 2018 (UTC)

There is no difference between tessellation and tiling, so there is no justification for having two articles on those two different titles. We have articles on subjects, not on their names; see WP:NOTDICT. —David Eppstein (talk) 14:09, 22 June 2018 (UTC)

Do I need to start giving notable examples where what you're saying doesn't apply. What a terse and thoughtless reply. Mathguy9109 (talk) 15:58, 22 June 2018 (UTC)

The point is: if you think there is important content on tessellations that this article does not describe, then it should be added to this article, or as a separate article on a special case of tessellations. We should not have two different articles on the same topic merely because "adding it all to this page would be tedious". We are here to benefit the readers, not to alleviate the tedium of the editors. I note that you have also tried to make the same suggestion at Talk:Polyomino and received more or less the same response. —David Eppstein (talk) 17:37, 22 June 2018 (UTC)

Yes, this is the main topic. If there is a lot of material on one particular kind of tessellation then of course it will make sense to add or extend an article on that subtopic, and if need be to extend the summary here briefly. Chiswick Chap (talk) 18:13, 22 June 2018 (UTC)

If it is indeed the case that discrete tilings are not treated in any Wikipedia article—I was unable to find any mention—then a starting point might be to create an article on that topic.

Two comments:

(1) Now that I've had a closer look at the Tessellations article, I like the structure the way it is. I see it as positive that practical, artistic, and mathematical aspects are combined into a single article. This is what general-interest articles on mathematics topics that have large overlap with other areas should strive to be. If readers come here looking for the artistic or manufacturing aspects, and are exposed to some mathematics along the way, I see that as a good thing. A lot of work by many editors has gone into making the article what it is today, and it has achieved Good Article status.

(2) I'm not sure I have grasped the source of your dissatisfaction with the article. I do see that there is a literature on discrete tilings that seems not to be mentioned on Wikipedia, and it would be good to fill that gap. Couldn't this be done, however, by simply creating an article on the topic and linking to it from the "Introduction to tessellations" subsection of "In mathematics"? Perhaps your point is that it is mathematically more productive to regard discrete tilings as tilings of the space Z² rather than as a restricted class of polygonal tilings of R². (I don't know whether this is the case. I'm just speculating on your motivations.) If that's the issue, perhaps a discussion could be added either under "Tessellations in non-Euclidean geometries", or in a new subsection devoted to discrete tilings. In an article aimed at general readership, I don't think it would be helpful to do a wholesale rewrite describing some very general theory in which tilings of Z² and tilings of R² are treated on an equal footing. Again, I don't know if there is such a theory—I'm just speculating, based on comments you made at Talk:Polyomino. If there is such a general theory, then perhaps a specialized article could be created on that theory, again with a link from the main article. Will Orrick (talk) 20:23, 22 June 2018 (UTC)

I think it would certainly be possible to make a separate article on integer tilings with a brief link to it from this article. (I would prefer the name integer tilings to discrete tilings because in this context the word discrete is too ambiguous.) —David Eppstein (talk) 21:00, 22 June 2018 (UTC)

Shouldn't the extended Girih tilings have its spot somewhere?

When I read the topic of Tessellation I miss something.

• Under the "In Mathematics" topic there are a number of sub topics, starting with the introduction that explains for example regular and semi-regular tessellations.
• A few sub-topics down the section for non-regular tessellations begins, called "Tessellations with polygons". In the end it talks about different monohedral tilings based on the number of corners, and there it links to "Pentagonal tiling".
• A few sub-topics above we have the "Aperiodic tilings" section, which have a picture of a 5-fold Penrose P3 tiling pattern.

Not until we step out of the "In Mathematics" topic and enter the "In Art" topic we get to see a reference and a link to 5-fold patterns - the Girih tiling page. Being in the art section hints that the tiles themselves don't have merits, and it's only the decoration and lines on top of the Girih tiles that is important - which I agree is true for art - but my question is if this isn't interesting from a mathematical point of view too? It should render a mentioning in the "In Mathematics" topic.

It also should have its own separate topic. I know the topic of Girih tiles already exist, but there are a few problems with it:

• The tiles defined as Girih tiles (by Lu and Steinhardt) are only five - and there are far more of them, but a classification and complete list have yet to be published.

  

  Excerpt from a draft on classification of tiles in the extended world of Girih.

  Jay Bonner is the most prominent figure in this matter, and he actually published a paper with a bigger set of these tiles long before Lu & Steinhardt, but it didn't get any acknowledgement. Myself I'm looking into the classification issue, but haven't published anything serious yet. See the draft to the right:
• The name "girih" implies that it is purely a Islamic geometry issue (even though the Islamic Geometric Art community haven't really embraced the tiling method). 5-fold pattern is universal math. We are missing that chapter/page, and from there the five Girih tiles are a sub-set and should be linked to.
• If we take a look at the Topkapı Scroll it contains several non-equilateral prototiles (including the Kite from Penrose P2 tile set).

These non-equilateral prototiles together with the extended set of the Girih tiles, should get their rightful spot on Wikipedia. I just don't know where. Please help me! Rixn99 (talk) 09:42, 16 September 2018 (UTC)

OK, thanks, and two immediate comments. Firstly, it would be preferable to mention and link Girih only once in this article. Secondly, we are allowed only to rely and cite existing published sources. If there's a "page" missing (a Wikipedia article, I take it) then feel free to write and cite it; we would then summarize that article here, as a new paragraph or as a new section. Failing that, if you know of suitable sources on the mathematics of girih, then feel free to write a brief mention of the topic here, citing those sources.Chiswick Chap (talk) 10:45, 16 September 2018 (UTC)

Thanks for your answer. I'll try to address each issue, starting with the "mentioning" issue first:

1) I'm not sure I understand you - I only link to the Girih Tiles page once (and no link to the Girih page)? For the mentioning of the word "girih", I didn't know I couldn't use it several times. That would limit my option on communicate what I intend to say. That can't be the objective. I suspect there is another reason for you saying this. Please enlighten me on what you mean?

I think we're talking at cross purposes. All I mean is that in an ideal article one only discusses a subtopic once; of course it may be necessary to name it in various other places in passing. I was not referring to anything you had written in the article or elsewhere.

2) I realize that we only can add information that can be referenced by reliable published sources. Since I started delving into this field these are the questions that have evolved, so guess I'm asking this community for help to see if someone here have any knowledge of additional sources about this?

• I don't have a formal higher education in mathematics, so I wanted to know if there, from a mathematical point of view, would be valid to add a small section for the equilateral five Girih prototiles in the "In mathematics" section.
  
• The issue is about the extended set of prototiles. As it contains non-equilateral prototiles I assume it would be defined differently from a mathematical point of view, which should be mentioned in this article too, but I assume it would require the article about it first.
  
• This would lead to the writing of this new article. Already today, I believe there is enough published references that would support such an article. I'll start to see if I can create something useful. Is there a way to share one's sandbox to others here in the Talk discussions?

Rixn99 (talk) 04:39, 17 September 2018 (UTC)

All sounds reasonable. You can simply provide a wikilink here to your sandbox article, and you could post a notice on relevant projects such as at Wikipedia talk:WikiProject Mathematics. Chiswick Chap (talk) 08:02, 17 September 2018 (UTC)

Split "Tessellation (mathematics)" ― or an alternative

Tessellation currently seems a broad-concept article (BCA), covering several closely related concepts. Its main section, Tessellation#In mathematics, is notable enough to deserve a separate article, Tessellation (mathematics). That section is so large that it has its own introductory section. A summary of the new article would be left in the old section, of course. fgnievinski (talk) 05:45, 23 May 2021 (UTC)

If you don't agree with the splitting proposal, that means Tessellation is not meant to be a BCA. Then, I assume it's supposed to be mainly about the mathematical concept and secondarily about its occurrence in other areas. In that case, the present article would need some restructuring. First, sections In art, In manufacturing, and In nature would need to be demoted as subsections of a new section Tessellation#Occurrence and applications. Secondly, section Tessellation#In mathematics would need to be promoted, by making its subsections first-level sections. fgnievinski (talk) 05:45, 23 May 2021 (UTC)

There was a previous discussion at #Creating a separate Tiling (mathematics) page. fgnievinski (talk) 05:47, 23 May 2021 (UTC)

You know it's currently listed as a Good Article, correct? So perhaps your "this is all wrong and everyone who thinks otherwise is also wrong" approach could use re-thinking. Or in other words, what's wrong with an article that covers both the real-world and mathematical aspects of a topic, in a single article? Where is the need to ghettoize all mathematics into its own separate articles to keep them away from all those nasty real-world applications? To put it more bluntly, I strongly disagree. I think that having both applications and their mathematical analysis in a single article is a good thing. —David Eppstein (talk) 07:18, 23 May 2021 (UTC)

I appreciate your concerns for ghettoization of mathematics. I believe it could be saned leaving a summary in the broad-concept article (BCA). That's how I and others have done at Surface and in Surface (mathematics). The intention here is more to give space for Tessellation (mathematics) to grow. Perhaps even have some equations (other than the few existing in-line equations), you know. But it looks like you prefer Tessellation not to be a WP:BCA, in which case the Mathematics subsections would be promoted and the Application sections demoted, as proposed in the second original option. fgnievinski (talk) 20:17, 23 May 2021 (UTC)

Or, we could, you know, keep the balance as it already is and not do what you say or else. The threatening language you used as the header for this thread is not promising as a way of building consensus for some kind of change. —David Eppstein (talk) 20:55, 23 May 2021 (UTC)

I apologize that my language made you feel threatened. That was not my intention, I value and expect civility in Wikipedia. So now I've rewritten the talk section title, replacing "or else" with "or an alternative". fgnievinski (talk) 23:28, 23 May 2021 (UTC)

Infinite tessellation with single shape

There are article in New Scientist, but behind a paywall:

• "Mathematicians discover shape that can tile a wall and never repeat". New Scientist. 2023-03-21.

jcubic (talk) 19:16, 2 April 2023 (UTC)