Talk:Tessellation/GA1

GA Review
The edit link for this section can be used to add comments to the review.''

Reviewer: David Eppstein (talk · contribs) 00:12, 26 May 2015 (UTC)

First pass
I don't think this is ready for GA yet. Here are some detailed issues I have with it in its current state :
 * Thank you for the detailed and thoughtful reading. Chiswick Chap (talk) 07:05, 26 May 2015 (UTC)

The history section skips from the Sumerians to Kepler — is there really no history of the subject in between?
 * Added a mention of geometric mosaics from classical times, with refs.

In the history section, it says that Kepler "made one of the first documented studies" of this subject, but the citation is to Kepler's own work; can you provide a secondary source stating that Kepler was first?
 * Said it was an early study. Gullberg (p. 395) says that honeycombs and snowflakes were "both originally investigated by Kepler", which I'll take as evidence he was one of the first. Stewart notes that Kepler "came up with a pretty good explanation of the snowflake's sixfold symmetry."Chiswick Chap (talk) 08:03, 26 May 2015 (UTC)

In the overview, the claim that tessellation is a "branch of mathematics" seems dubious; it's a topic in geometry, not a branch itself.
 * Done.

What does "all corners should meet" mean? It is certainly not that all corners of all tiles meet at the same point, because then there would only be a single corner point and not a repeating pattern.
 * Said "no gaps between tiles".

The first paragraph of the overview section cites Mathworld as a source for the claim that there are only three regular tilings, which I'd prefer to avoid (because they have often been incorrect on mathematical details for other subjects); this should be easier to find a more reliable source for such as a published book by a notable mathematician, as you use for the surrounding references.
 * Done, ref Gullberg.

Later in the same section, the two sentences "A general method for identifying shapes which will tile the plane periodically without reflections is known as the Conway criterion.[13] However, mathematicians have found no general rule for determining if a given shape can tile the plane or not" appear to contradict each other; this needs further elaboration to explain why it is not a contradiction.
 * Done: good catch. Conway is sufficient but not necessary: there are tiles that don't meet the criterion. Chiswick Chap (talk) 08:10, 26 May 2015 (UTC)

And the sentence "the types of convex pentagon that can tile" should have a wikilink to pentagon tiling.
 * Done.

In "kinds of tessellations", I don't think the uniformly bounded condition is stated correctly. The stated condition that "a finite circle can be circumscribed around the tile and a finite circle can be inscribed within the tile" is satisfied by every tiling in which the tiles are topologically equivalent to disks. What you want to state is that a single circumscribing radius and a single inscribing radius can be used for all the tiles in the whole tiling.
 * Done.

Also, this section is kind of schizophrenic: its first few paragraphs are about classes of tessellations constrained to be nice in certain ways (normal, monohedral, regular, etc) but then at the end it instead talks about one specific tiling (the Penrose tiling) and one specific method of generating tilings (Voronoi diagrams); neither of these is a constraint on the shape of a tiling. I think it would be better to move the part about aperiodicity in the same section as you talk about periodicity (wallpaper groups), because an aperiodic tiling is just one that doesn't have any translational symmetry. There are also many other important aperiodic tilings that should be mentioned in connection with this subtopic.
 * Moved Penrose/aperiodic to Wallpaper groups, put Voronoi in separate subsection. Changed "Kinds of tessellations" to "Introduction to tessellations", hoping to suggest these are the well-behaved ones. Chiswick Chap (talk) 15:03, 27 May 2015 (UTC)

The caption to the four color theorem illustration has a typo: "repeat" should be "repeating".
 * Done.

The part about the torus is confusing, because when you use this tiling on the torus there is no repetition: there are only seven tiles, not infinitely many.
 * Said "exactly once"

It's not obvious how to turn the diagram into the intended repeating pattern in the plane, so it should probably be stated (use infinitely many rectangles, matching the arrows).
 * Done.

And the caption is not mathematically correct: it is possible to form a repeating pattern with this infinite tiling with only four colors, but the point is that to do that you have to use a larger fundamental domain than this rectangle. For more on this distinction between how many colors are needed if you stick to the original fundamental domain, and how many colors are needed if you use a larger repeating color pattern, see http://11011110.livejournal.com/71062.html
 * Said "with this fundamental domain".Chiswick Chap (talk) 07:05, 26 May 2015 (UTC)

In the section "Tessellations with triangles, quadrilaterals and hexagons", there is a line "Most hexagons with at least two parallel sides will tessellate": what does the word "most" mean here?
 * Removed.

In the phrase "basis vectors are the diagonal", fix the singular/plural conflict, and in any case a line segment is not a vector, so there's a mathematical problem here too. This should not have been the first place fundamental domains were mentioned; see the earlier section about wallpaper groups.
 * Done this section, and noted.

The "Tessellations in higher dimensions" section uses semicolons where it should be using commas.
 * Done. Chiswick Chap (talk) 07:50, 26 May 2015 (UTC)

What does the line about crystals taking certain shapes have to do with tiling?
 * Reworded, they are examples of three-dimensional tessellation. Chiswick Chap (talk) 07:50, 26 May 2015 (UTC)

Somewhere in here you should also mention the Kelvin conjecture and the Weaire–Phelan structure.
 * Done.

The Nature section might be a good place to mention Gilbert tessellations, a mathematical model of mud cracking and similar processes.
 * Done. Chiswick Chap (talk) 07:43, 26 May 2015 (UTC)

The see-also section is too long, has some entries that would seem to better fit a disambiguation page because they are similar more in name than in topic, has many entries that would be better expanded as subtopics within the text of this article, and appears to violate WP:SEEALSO by including links that are also included in the main text.
 * Done; and used the hyperbolic geometry links in a new subsection on non-Euclidean geometry, and a discussion of these in Escher's art. Chiswick Chap (talk) 09:43, 26 May 2015 (UTC)

Why are some references given a full citation within a footnote, and others listed in the "sources" section with a brief citation to them in the footnote? Is there a rationale for choosing which way to site each source? And why are some references that are already listed in the sources also given a full citation within a footnote rather than just a brief citation (e.g. footnotes 9 and 17 to Coxeter)? Please check that the capitalization of book titles is consistent (probably should be in title casing) and that the capitalization of journal/conference article titles is also consistent (probably should be in sentence casing). Some citations (e.g. Pickover 2009) appear to be formatted in citation style 1, using the cite series of templates; others (e.g. Fedorov 1891) appear to be manually formatted in an inconsistent style. In short citations (e.g. "Gullberg, 1997. p. 395") please use the harv or sfn series of templates to format the short citation in a way that makes an internal link to the complete citation, and use harv within the complete citation so that this link works correctly.
 * These comments, while sensible and logical, are outside the GA criteria; I am quite willing to work on them but perhaps, given the other comments, they are not the top priority. It is convenient to use short form references when a source is reused, and there is no requirement to use a particular template such as 'harv', or even to use a particular format.Chiswick Chap (talk) 07:05, 26 May 2015 (UTC)
 * There's definitely no requirement that you use citation style 1 (I prefer cs2 myself) or footnotes vs parenthetical referencing, etc. However, I think the references are currently not formatted consistently with each other, that this detracts from the workmanship of the article, and that choosing a style and formatting them to be consistently in that style shouldn't be particularly difficult. —David Eppstein (talk) 07:11, 26 May 2015 (UTC)
 * Yes, it's entirely fixable. But I think we should start by seeing if we can place the big tiles we absolutely have to have in the pattern, before we try fitting in the little ones. Chiswick Chap (talk) 07:43, 26 May 2015 (UTC)
 * OK, I've done all that for you. Chiswick Chap (talk) 14:32, 26 May 2015 (UTC)

Also, if you are going to cite mathworld (and I prefer you would instead just move these citations to the external links and find a better source), please cite them consistently using the mathworld template.
 * Done.

Is reference 30 Jaworski, J. "A mathematician's guide to the Alhambra" reliable? Was it published anywhere, or does Jaworski fall under the "recognized expert" clause of WP:SPS?
 * Jaworski has published many math. articles and is a recognized expert. Chiswick Chap (talk) 08:17, 26 May 2015 (UTC)

I'm pretty sure reference 36 (AJS Gems) is not reliable (it seems more about selling gems than about conveying information).
 * Replaced (Kirkaldy)

Reference 37 (Amethyst Galleries) appears to be a copyvio of https://books.google.com/books?id=wm_X_LzoN-cC&pg=PA107.
 * Replaced.

I'm also dubious about the reliability of reference 40 (spain.info), 42 (Jinny Beyer on quilting), 43 (Irena Swanson on quilting), and 48 (a book on flower bulbs that appears to be self-published).
 * #40 replaced; #42, #43 removed with their text; #48 replaced.

—David Eppstein (talk) 00:12, 26 May 2015 (UTC)


 * That's all done to date. I hope you think it looks more promising now. Chiswick Chap (talk) 14:32, 26 May 2015 (UTC)

Second pass
Ok, after mostly checking in the first pass that the content in the article is accurate and reliably sourced, let's check how thorough its coverage of the topic is by comparing it to three books that I happen to have on my shelves, Radin's "Miles of Tiles", Montesinos' "Classical Tessellations and Three-Manifolds", and Grünbaum and Shephard's "Tilings and Patterns". I am by no means suggesting that you should be required to use these as sources or cover the subject as in-depth as a book, only that their selection of topics might be useful to give us an idea of anything that might be missing here. Necessarily because of this choice of books, my comments will be directed more to the mathematical side of the subject than its other aspects. And because they're focused on what's not in our article rather than what's in it, they will necessarily be vague about how and where to add this material to the article. And including all of this material might be too much. But as I think these comments demonstrate, our coverage of some important aspects of the subject is more superficial (or even nonexistent) than I think it should be.


 * Noted. I suspect that this very thorough and welcome analysis essentially sets out the program for making this a Featured Article with "comprehensive" coverage. The criterion for Good Article is "broad" coverage of the "main aspects". I guess we agree that some of the topics listed and wikilinked below deserve mention, and that some may possibly be advanced or minor rather than main aspects. I propose therefore to provide brief mention of as many of them as possible, with wikilinks and references to provide broad coverage.

Radin is mostly about aperiodic tilings, represented currently by a short paragraph in the wallpaper groups section. He points to the Wang tilings (currently only in see-also) as the beginning of this area; they are aperiodic, but that was a side effect of some other fundamental properties, their ability to simulate certain computational systems such as Turing machines and as a consequence the fact that it is impossible for a computer algorithm to correctly determine whether a set of prototiles forms a tiling. Some mention of this complexity connection should be made in our article. Next (intro pages 8-9) Radin mentions Truchet tiles in connection with the construction of random tilings; Radin doesn't mention this part but Truchet tilings also have important applications in the visualization of arrays of binary data, and are not mentioned in our article. Starting around p.11 Radin discusses Substitution tiling, one of the classical methods of generating aperiodic tilings; our article mentions them briefly but in saying that "they have surprising self-replicating properties" seems to be confusing substitution rules in general with an important special case, the rep-tiles. We also mention Penrose tilings, pinwheel tilings, and substitution tilings as if they are the same sort of thing when instead Penrose tilings and pinwheel tilings are very specific tilings and substitution tiling is a general process that can be used to generate many types of tiling, those included. The last topic covered in the introduction is symmetry, where Radin makes the point that aperiodic tilings, while lacking in translational symmetry, do have symmetries of other types, both in the infinite repetition of any bounded patch of the tiling and in certain finite groups of rotations or reflections of those patches. The rest of the book is more technical and less applicable in finding missing coverage in our article, but it covers ergodic theory and its application to the theory of substitution tilings (Chapter 1), the physics of light diffraction, statistical mechanics, and their application to the X-ray tomography of quasicrystals (Chapter 2), topological spaces representing the ways patches of aperiodic tilings can fit together (Chapter 3), and the symmetries of the pinwheel tiling and of analogous three-dimensional tilings (Chapter 4).
 * Wang tiling, Turing undecidability: done.
 * Truchet tiles: done.
 * Substitution tiling: clarified.
 * Rep-tiles: mentioned in Puzzles section and in Aperiodic section.
 * Symmetry in aperiodic tiling: done.
 * Other topics: probably beyond our scope, but mentioned quasicrystals.

Montesinos is also a bit specialized, focusing on topological spaces whose points represent positions at which a tessellation can be placed in a higher-dimensional Euclidean space. (Even though the tiling is two-dimensional you get a three-dimensional topological space because a position can be specified by the location of a two-dimensional base point plus a rotation angle. Two such specifications represent the same point in the topological space if they represent the same placement of the tessellation.) I don't think there's much there that belongs in a general-audience article. However, he does have some coverage of the wallpaper groups of Euclidean tilings in terms of orbifolds, possibly worth mentioning, and he treats the cases of Euclidean geometry, spherical geometry, and hyperbolic geometry on an equal basis with each other, whereas the nominated version of our article didn't say anything about the non-Euclidean cases (some material was added after I wrote a draft of this part of the review). Relevant articles that we should probably say something about include Schwarz triangle and Uniform tilings in hyperbolic plane.
 * Orbifolds: mentioned in Wallpaper groups.
 * Schwarz triangle: mentioned (for sphere) in Higher dimensions.
 * Uniform tilings in hyperbolic plane: Done.

Grünbaum and Shephard start their introduction with a brief historical survey of tilings and patterns, which they say have been used in "every known human society". They mention the possibilities of varied tile shapes and use of colors in these decorative patterns. They go on to mention applications in manufacturing (fitting together shapes to be stamped out of sheet metal or other material without wasting material on gaps between the shapes), something we don't discuss. They distinguish between repetitive tilings from tilings formed by arbitrary partitions of the plane into regions without need for repeated shapes, and they mention the application of random tilings in the study of crystals in thin sheets of materials, biological cells in skin surfaces, leaves, and other two-dimensional membranes, and also in image processing. One method they discuss for constructing random tilings is by using Voronoi tilings of randomly placed points (we discuss Voronoi tilings but not in connection with random point placement). They survey mathematical developments but only briefly because they claim that this work (prior to their book does not provide "a satisfactory mathematical treatment" of the subject. At the end of the introduction they provide a list of dozens of books on tilings and patterns (in art and nature rather than in mathematics) that might also be useful as sources here.
 * Manufacturing: done.
 * Voronoi for random tilings: done.

G&S Chapter 1 is on basic definitions, and is mostly relevant for our "Introduction to tessellations" section. One thing it mentions that we don't is the still-unsolved problem of determining algorithmically whether a given shape can be the prototile of a monohedral tiling. In this context it also mentions monohedral tilings by polyominos and polyiamonds, something we mention only briefly in the see-also section. This section also gives a couple of examples of tilings that are not normal (the prototiles are non-disks or even disconnected), something that might be helpful for us to do. Additionally it discusses the obvious generalization of monohedral tilings to dihedral, k-hedral, etc. In this chapter's discussion of symmetry they distinguish between symmetries of the tile shapes and of colorings, and they discuss the connections between translational symmetries and lattices. They give examples of prototiles that are more symmetric than their tilings. As with monohedrality, they generalize isohedral tilings to k-isohedral tilings, and they also discuss transitivity classes for vertices (we only mention them for tiles) and the associated notions of monagonal and isogonal tilings. They provide a detailed classification of wallpaper groups (and one-dimensional strip groups), but in our case that might be too much detail, better left in the wallpaper group article. And they discuss the issue of monomorphism and k-morphism, the situation when a given prototile admits only one or a small number of tilings.
 * whether a given shape can be the prototile of a monohedral tiling:
 * monohedral tilings by polyominos and polyiamonds: In Overview and in Puzzle section.
 * other topics: noted as probably beyond our scope.

G&S Chapter 2 is on tilings by regular polygons and star polygons, and most closely overlaps the last two paragraphs of our "Introduction to tessellations" section. Where we state that there are only three regular tessellations, they state somewhat more strongly that there are only three edge-to-edge monohedral tilings by regular polygons (without any requirement of symmetry). They also classify the semi-regular tilings, which they give explicitly as a short list of vertex configurations (we only mention one vertex configuration without listing all the possibilities). They generalize to k-uniform tilings (k-isogonal tilings by regular polygons) and classify the 20 2-uniform tilings, and they also look at tilings with edge instead of vertex symmetries. They discuss non-edge-to-edge tilings by regular polygons and classify them into eight parameterized families. (For instance one of these is the family of Pythagorean tilings, parameterized by the size ratio of its two squares.) In this context they discuss the squaring the square problem and the (at that time unsolved, now solved) problem of squaring the plane. There is some discussion of tilings by star polygons which relates both to Kepler's work and to girih tilings. They discuss dissection tiling (which we don't seem to have an article on but maybe should) in which a given non-prototile is cut up into pieces that can tile the plane in the correct proportions, and the use of tilings to find solutions to dissection puzzles, and they discuss uniform colorings of uniform tilings (i.e. each vertex should have a color-preserving symmetry mapping it to each other vertex).
 * only 3 edge-to-edge monohedral tilings by regular polygons:
 * Pythagorean tiling: done.
 * Squaring the square, squaring the plane: done.
 * Dissection, dissection puzzle: Mentioned in Puzzle section.

G&S Chapter 3 is an in-depth study of what can happen when a tiling is not normal, and chapter 4 is on combinatorial and topological equivalence of tilings. They are probably too technical to be of much use for us, but we might at least mention some basics (e.g. the equivalence between the hexagonal tiling and the brick wall tiling). Chapter 5 is on patterns (off-topic for us), Chapter 6 is on classification of tilings (again, quite technical), and chapter 7 is on classification of patterns. Chapter 8 extends these ideas to colored tilings. So I think these are all safe to skip; we don't need to cover this material in our article to provide adequate coverage of the subject.
 * Noted.

G&S Chapter 9 discusses proper tilings by polygons, a special case of normal tilings (not mentioned in our article) in which the intersection of any two tiles is a subset of an edge of each tile (not necessarily edge-to-edge). It classifies 107 types of proper isohedral tilings by convex polygons (14 by triangles, 56 by quadrilaterals, 24 by pentagons, and 13 by hexagons), and proves that every combinatorial type of proper tiling can be represented by convex polygons in this way. It discusses the problem of pentagon tiling including some of Marjorie Rice's discoveries here, but without a complete list of pentagon tiles. It goes into some detail about monohedral tilings by polyiamonds and polyominoes. It concludes with a discussion of Voderberg's spiral tiling; it shows that with isosceles triangles one can construct analogous tilings with any even number of spiral arms, and it provides examples of prototiles and spiral tilings with odd numbers (one and three) of arms.
 * proper tilings by polygons:
 * pentagon tiling: done.
 * Voderberg tiling: was covered, now also wikilinked.

G&S Chapter 10 is on aperiodic tilings. It starts with a section on "similarity tilings", tilings in which the prototiles are all similar to each other instead of congruent, or maybe instead tilings in which the symmetries one considers involve similarities of the whole tiling instead of just congruences; this is where it also covers composition rules and rep-tiles. It has three sections on the Penrose tiling discussing several variants of the tiling, its local symmetries, and connections to line arrangements and Fibonacci words (which can be viewed as a prototypical one-dimensional aperiodic tiling), as well as sections discussing other aperiodic tilings by Robinson, Wang, and Amman. Chapter 11 continues the same topic and is entirely about Wang tiles, the ability of Wang tiles to simulate some of the other known aperiodic tilings, Wang's theorem that if a set of prototiles admits tilings of arbitrarily large finite patches then it tiles the whole plane, the use of Wang tiles to simulate computation devices and the undecidability of tiling with a given set of prototiles.
 * Aperiodic tilings and Fibonnaci words: done.
 * Wang tiling: see Radin paragraph above.

G&S Chapter 12 returns to the idea of non-normal tilings. It has some theorems (showing that certain types of tilings admit classification theorems without doing an explicit enumeration of the tilings) but I think its main purpose is to present lots of examples of border cases of things one might call tilings, as a way of challenging the definitions.
 * Probably beyond our scope for GA.

I know this reads more as a review of those other books, but I hope you can still find it helpful as a listing of other material that might be included to beef up the article and more completely cover the subject (Good Article criterion 3a).
 * Noted; see discussion at top of this "Second pass" section.

(BTW, one really minor change: we should authorlink Geoffrey Colin Shephard.)
 * Done.

—David Eppstein (talk) 23:11, 28 May 2015 (UTC)
 * OK, I've done all the marked items, which I'd suggest is about as far as we should go, unless we've missed something obvious. The article looks and feels more rounded, which must be a good sign. Chiswick Chap (talk) 15:21, 29 May 2015 (UTC)

Third pass
Almost there; a few minor things to fix.
 * "In 1619 Johannes Kepler made an early documented study of tessellations when he wrote about regular and semiregular tessellations, coverings of a plane with regular polygons, in his Harmonices Mundi": I think the "coverings" clause is missing a conjunction. Or maybe it's intended as a parenthetical clause that glosses the meaning of semiregular tessellations? In any case the grammar seemed unclear to me.
 * split the sentence; removed the "coverings" clause as unnecessary.


 * "striking patterns are formed, and these can be used to form physical surfaces": how is a physical surface formed by a pattern? I think rather than repeating the verb "form" the second one should be "decorate" or something similar.
 * done.


 * In "every tile is topologically equivalent to a disk", the wikilink to "topologically" is not particularly helpful. A good link for topological equivalence would be homeomorphism.
 * done.


 * "Pinwheel tilings are non-periodic, using a Conway construction": the term "Conway construction" is never defined nor wikilinked, so this is unhelpful. In any case the pinwheel tiling just uses a rep-tile construction, so it would probably best to repeat that phrase rather than giving another unfamiliar name for it.
 * done.


 * "Sometimes the colour of a tile is understood as part of the tiling, at other times arbitrary colours may be applied later.": please fix the comma splice.
 * done.


 * The claim that Conway discovered the Schmitt-Conway biprism is not supported by the source given for the claim (you linked the wrong MathWorld page), and is contradicted by Senechal and Kennard  who say that Schmitt found such a tile in 1988 and that Conway and Danzer improved the construction. This sort of inaccuracy is exactly why I earlier warned against using MathWorld as a source.
 * Removed claim, replaced ref.


 * The description of the Weaire–Phelan structure misses the point of the construction, which is that it uses less surface area to separate cells of equal volume than Kelvin's foam.
 * done.


 * "Squaring the square is ... Henle proved that this was possible": run-on sentence.
 * split the sentence.

—David Eppstein (talk) 03:15, 31 May 2015 (UTC)

Pass pass
1a. Good prose, not copied: yes.

1b. Section order; lead summarizes text; no weasel or peacock wording: yes.

2a. Well formatted references: yes.

2b. Claims are sourced and sources are reliable: yes.

2c. No original research: yes.

3a. All important aspects covered: yes.

3b. Stays focused on topic without unnecessary detail: yes.

4. Neutral and fair representation of all points of view: not really relevant, but yes.

5. Stable: This has undergone big changes over the course of the review but I don't see that as likely to continue, so yes.

6a. Illustrations are properly licensed: yes.

6b. Images are relevant and properly captioned: yes.

Ok, I will pass this as GA now. —David Eppstein (talk) 16:44, 31 May 2015 (UTC)
 * @David Eppstein: Thank you very much. It was more than usually obvious in this case that the reviewer was contributing as much work as the editor to the article, for which I am extremely grateful. It has been a long-standing ambition to get this article up to a decent standard, to join others on the broad topics of camouflage and patterns in nature. But those barely strayed into mathematics... Chiswick Chap (talk) 17:46, 31 May 2015 (UTC)


 * A wonderful job on bringing this to GA status, both of you! --Mark viking (talk) 18:31, 31 May 2015 (UTC)
 * Thanks! Now let's just hope that we can prevent this from getting choked up by the copied-and-pasted gallery after gallery of vaguely-related and unsourced hyperbolic tessellations and honeycombs that have afflicted some of our more specialized articles on this topic (an example). —David Eppstein (talk) 04:30, 1 June 2015 (UTC)