# Talk:Square

Add to categorizations that a square is a rectangular rhombus. 75.117.226.44 (talk) 23:35, 10 November 2023 (UTC)

 Not done: Not quite sure how you want this done... there currently isn't a rhombus category. Liu1126 (talk) 00:04, 11 November 2023 (UTC)

A quadrilateral is a square if and only if it is any one of the following:
The layman might not catch the implication of if and only if that if one definition is true then all are; I would prefer language like The following definitions of a square are equivalent. —Tamfang (talk) 07:18, 28 June 2024 (UTC)

Please see the admins' noticeboard thread "Indefinite protection of Square about the protection status of this article. Anyone can comment there, regardless of their admin status. Graham87 (talk) 09:52, 31 January 2025 (UTC)

@David Eppstein, thanks for working on this article, including the lead section. I'm a bit concerned that the detailed list of random topics in the lead section seems a bit arbitrary and may be confusing or overwhelming for readers. Topics such as the inscribed square problem, the square of squares, etc. don't really seem essential to the concept of a "square", and I don't think we really need to mention all of them in the lead, even if they are discussed later on. I'd recommend we try to pare the lead down to the most fundamental topics and not necessarily try to make the lead a complete summary of everything mentioned in the article. –jacobolus (t) 06:33, 17 February 2025 (UTC)

It was intended as a rough summary of the rest of the article. See MOS:INTRO: "The lead section should briefly summarize the most important points covered in an article, in such a way that it can stand on its own as a concise version of the article." Note that this is part of WP:GACR#1, so if we hope to reach GA status we need a proper lead, not just a brief paragraph. I tried to include material from most sections of the article, but some calculation-heavy parts were difficult to summarize briefly and readably. —David Eppstein (talk) 07:07, 17 February 2025 (UTC)
It is not required that every topic discussed in the article must be mentioned in the lead. I think mentioning these somewhat niche/obscure topics seems like a non sequitur and is not really helpful for many expected groups of readers who might skim the lead section without bothering to read further into the article (and frankly it seems like a bit of an NPOV problem; these topics are definitely not given such prominent place in the full range of published literature involving squares). –jacobolus (t) 07:30, 17 February 2025 (UTC)
"The published literature involving squares" mostly involves kindergarten mathematics textbooks, at least if one focuses on works directly about squares rather than covering them in passing. Is that what you think we should emulate? As for the topics you think are niche: they are topics that are directly about squares (and not about orthogonal repetition, a different topic) for which we have articles. I think we should discuss those topics in the main article on squares. Almost everything in the packing and quadrature sections is merely a brief summary of material covered in more detail at the linked articles; I think the only exception is the very recent proof of NP-hardness for square packing problems. We have a separate article on square tiling, where I think what you want to cover better belongs. It should be mentioned here, but not be the main focus of this article. And it is mentioned here, in a paragraph-length summary, like the other topics in these sections. But the current state of the square tiling article is pretty dire leaving little that can be summarized here. Your proposal to add an entire section on it here is premature until summarizing what is there would take an entire section. Also, re your opinion that square packing in a square or the square peg problem are niche: both have been the subject of publications by Fields medalists, suggesting that maybe there is more depth to them than you might have suspected. —David Eppstein (talk) 08:23, 17 February 2025 (UTC)
You don't need to get testy. The published literature involving squares is millions of items in a tremendous range, of which only a trivially tiny proportion is kindergarten books. But sure, the material from the kindergarten books is essential and must be covered, whereas topics such as inscribing squares in arbitrary curves, making squares from arrangements of smaller squares, and packing circles into a square are more or less mathematical curios, not essential to the concept of "square" and not important enough to be the first things we tell someone trying to learn the basics about squares. –jacobolus (t) 15:37, 17 February 2025 (UTC)
Says you. The Fields medalists disagree. —David Eppstein (talk) 17:48, 17 February 2025 (UTC)
Whether a problem was of interest to Fields medalists is not equivalent to whether a problem is of fundamental importance (either to mathematics or in particular to the concept of squares). [However, if you had a Fields medalist saying something like "one of the most important things about squares is that there are some packing problems ...", that opinion would be worth weighting.] –jacobolus (t) 17:58, 17 February 2025 (UTC)
I reorganized paragraph 3 so that it starts with square tiling (ubiquitous, easy to visualize) and then goes to squaring the circle (pretty famous) before getting into unsolved and comparatively obscure topics. I like having the latter in the lede, actually. It spices up mathematics, in a way, by showing that a simple idea like "square" is one step away from a question that nobody can answer yet. XOR'easter (talk) 17:59, 17 February 2025 (UTC)
Thanks, I think that was an improvement. –jacobolus (t) 18:27, 17 February 2025 (UTC)
I'm mostly OK with the introduction. I think the formula for the area is paragraph-1 material, so I added a sentence about that, thereby introducing it before we get to an unsolved problem. The sentence about ubiquitous squares could perhaps be split into a two-sentence paragraph. XOR'easter (talk) 17:26, 17 February 2025 (UTC)
Thanks. I agree that some of the formulas are lead-worthy; I just wasn't sure about how to work them into the lead without overwhelming it with technicality. —David Eppstein (talk) 17:49, 17 February 2025 (UTC)

It seems to me that Wikipedia overall doesn't have particularly good discussion about square and rectangular grids. We currently have articles about Square lattice, Lattice graph, Square tiling, Checkerboard, Graph paper, Regular grid, Analytic geometry, Coordinate system, Cartesian coordinate system, Projected coordinate system, Grid (graphic design), Grid plan, UV mapping, Bitmap, Grid (a disambiguation page), etc., but most of these are relatively short and incomplete, and there's not really any place with a solid overview of basic concepts and tools, the range of applications, a clear comparison or list of trade-offs with other types of coordinates or structures for various types of data, etc. It's probably worth having a new article called Square grid (currently redirects to Square tiling which doesn't seem quite right) or maybe more generally Rectangular grid with square grids as a prominent section, but in any event to have a complete article about Square, it seems to me a significant early section should discuss square grids, since many (most?) of the applications and points of interest of squares have more specifically to do with square grids, and square grids have become really fundamental to the way modern society organizes all kinds of information and even thinks about mathematical concepts
I haven't done any kind of literature survey, but I bet there are some nice sources discussing square grids at a high level, maybe including some kind of philosophical considerations etc.
Edit: here are a few sources that pop up in a very brief search:

https://books.google.com/books?id=bpOUDwAAQBAJ&pg=PT109
https://books.google.com/books?id=ms--K3jipt4C
https://dspace.mit.edu/handle/1721.1/74743
doi:10.1207/S1532690XCI2103_03
doi:10.1007/978-3-319-72523-9_7
–jacobolus (t) 06:55, 17 February 2025 (UTC)

I disagree that most applications and points of interest about grids. They are important, but really a separate related topic. Most of the applications are about things with the shape of a (single) square. We should have an article (or two) about square and rectangular finite arrangements of points, though. Square grid is a natural title, but it points to something else. —David Eppstein (talk) 07:10, 17 February 2025 (UTC)
The applications mentioned here include tiles, square coordinates, graph paper, city grids, bitmap images, square-grid game boards, QR codes, etc. All of these are really applications of square grids in particular, more than the square shape for its own sake. I agree this is a separate related topic which should have its own article; I just think it's worth summarizing the topic here as well, since it is ubiquitous in (especially modern) human culture, including the basic structure of many areas of modern mathematics. –jacobolus (t) 07:25, 17 February 2025 (UTC)
Almost everything you mention is part of a single paragraph of a multi-paragraph section. That paragraph focuses on grids. The only exceptions in your examples, not from that paragraph, are "city grids", which are not mentioned at all in the article (they are mostly rectangular rather than square in my experience), game boards, which are primarily mentioned because the boards themselves are square and only secondarily because of the square grids some of them contain, and QR codes, where we do not even mention the grid layout of the pixels (it would be redundant to the first paragraph) and instead focus on the square overall shape and nested-square pattern of the alignment marks.
Taking a wider view, the intent of this section is to convey "squares are all around you in many familiar things", not "when you use square shaped things you are only allowed to place them in a grid". —David Eppstein (talk) 08:15, 17 February 2025 (UTC)
I feel like you are deliberately missing my point, and I'm not quite sure why. I am not talking about changing the "applications" section, which seems fine, though it could certainly keep accumulating examples if anyone wanted. I'm suggesting that this article is substantially incomplete (and Wikipedia's coverage of the topic more broadly is incomplete) insofar as it does a very poor and limited job discussing square grids.
"Squares are all around you" in large part because they fit into a grid, whether that's square kitchen tiles, square sidewalk sections, squares on a Go board or computer game grid, pixels in a bitmap image, square city blocks, squares as a unit of area, squares on a military map, etc. Other shapes (say, regular heptagons or non-rectangular trapezoids) are much less common as an organizing principle, because they are significantly less convenient for making a regular pattern with cleanly separated but equivalent directions, easily addressed by coordinates, etc. Just as triangles are culturally important to a significant extent because they are stable in a truss, squares are important because they are the basis for one of the most common types of human organizing structure. –jacobolus (t) 15:52, 17 February 2025 (UTC)
Your point in a nutshell, as it comes across to me, is, we should stop talking about these square things and instead talk about things that are periodic in square lattice patterns. Which is a fine topic for an article but to me is not really the topic of this article. —David Eppstein (talk) 17:50, 17 February 2025 (UTC)
Okay, well I'm doing a terrible job expressing myself, because no that's not it at all. What I am saying, in a nutshell, is that we should (a) have a separate article called something like Square grid, and (b) have a top-level section of this article called something like "Square grids", since that subtopic is extremely relevant and important here, but is not currently described very clearly or completely. Reframing the article titled "Square" to be entirely centered on a separate topic would be nonsensical. –jacobolus (t) 18:00, 17 February 2025 (UTC)
I agree that square grid is a reasonable topic for a separate article. (Having it redirect to square tiling as it does now doesn't quite fit.) I'm not sure that a top-level section with the heading "Square grids" would be the right way to organize the text in this article.  I'll admit that the discussion in this thread has left me a little confused. It looks like a dispute over whether a chessboard should be seen as a square grid or as a grid of squares. XOR'easter (talk) 18:17, 17 February 2025 (UTC)

I think it goes against MOS:LEDE (in spirit if not explicitly) to state things in the infobox that the article does not elaborate upon. Currently, the infobox is generated by {{Regular polygon db}}, which dumps in a pair of Coxeter–Dynkin diagrams, two properties that the article does not define (isogonal and isotoxal), and the statement that the square is self-dual. This seems less than optimal. Defining all these terms in the article might bloat it unacceptably, but dumping unsourced and unexplained terminology into the intro for a basic shape isn't great either. XOR'easter (talk) 19:38, 18 February 2025 (UTC)

I think it would be preferable to remove the Coxeter diagrams from the infobox than to try to explain them in the article. It's just not a very significant topic for a shape of such low dimension and it's too technical for the most front-facing parts of this article. We can mention squares being isogonal and isotoxal in the symmetry section but I'm still not convinced they belong in the infobox either. —David Eppstein (talk) 21:10, 18 February 2025 (UTC)
{{Regular polygon db}} doesn't seem to offer any flexibility, so I guess we should switch to {{Infobox polygon}}. XOR'easter (talk) 21:24, 18 February 2025 (UTC)

@David Eppstein, inre:

There was a reason I wrote it in the more constrained way I did. I searched for sources that described complex-number squares in other ways and didn't find them. I hope your searches are more successful but otherwise this material may need to be removed. Additionally, it seems over-detailed for readers unlikely to care about complex nos.
After searching I agree that a lot of basic properties of squares with lattice (or Gaussian integer) vertices – or more generally, coordinate squares / squares in the complex plane – seem surprisingly difficult to come by in a skim-search of published literature, which is pretty fragmented. Perhaps some observations were considered too obvious to write down by mathematicians talking about number theoretic topics; weren't noticed by high school teachers or school curriculum designers; and the people who would most care such as programmers or artists don't bother publishing such observations in papers. I'll explain what my thought process was here:
(1) The most obvious way to characterize a particular shape of square in the complex plane (or in a square lattice in general) is using a vector or complex number representing the side, rather than the half-diagonal. This goes along with the general practice since ancient times of characterizing a square by its side. It's of course possible to instead characterize a square using the half-diagonal (directed circumradius), effectively getting an arbitrary square by scaling and rotating the one with corners at 
  
    
      
        {
        1
        ,
        i
        ,
        −
        1
        ,
        −
        i
        }
      
    
    {\displaystyle \{1,i,-1,-i\}}
  
; taking this square as a prototype is logical enough in the context of general regular polygons: the vertices are the 4th roots of unity, this square could be considered a unit-"radius" orthoplex, and it is the central cell of the lattice of "odd" Gaussian integers, congruent to 1 modulo 
  
    
      
        1
        +
        i
      
    
    {\displaystyle 1+i}
  
. However, in general this origin-centric square is much less common to consider as the basic prototype than an origin-vertex "unit square" with corners 
  
    
      
        {
        0
        ,
        1
        ,
        1
        +
        i
        ,
        i
        }
      
    
    {\displaystyle \{0,1,1+i,i\}}
  
 (or coordinates in 
  
    
      
        {
        0
        ,
        1
        }
      
    
    {\displaystyle \{0,1\}}
  
). In all sorts of contexts related to geometry on grids (space groups, tessellations, computer graphics with pixel grids, data discretization, building Zometool models, working with self-similar fractal curves, ...) tiles are usually more fluently described from a perspective of vertices and edges rather than centers and radii/half-sides/half-diagonals.
(2) There are quite a lot of elementary sources mentioning squares with vertices on a square lattice, e.g. from school materials using geoboards, discussions of the Pythagorean theorem and pythagorean triples, residue classes of division in Gaussian integers, and so on. For example there are a lot of middle/high school level puzzles/activities about counting the number of tilted squares that can be made using a square lattice of some specific rectangular dimensions. So it might be worth mentioning these discrete squares specifically, not only ones of completely arbitrary size / position in a two-dimensional continuum.
(3) The easiest way to characterize arbitrary squares in the (complex) plane, including squares with lattice points or Gaussian integers for vertices, is by taking the "unit square" and then scaling/rotating and translating it to its final position, in terms of complex numbers this transformation is 
  
    
      
        z
        ↦
        a
        z
        +
        b
      
    
    {\displaystyle z\mapsto az+b}
  
 with scale/rotation 
  
    
      
        a
      
    
    {\displaystyle a}
  
 and translation 
  
    
      
        b
      
    
    {\displaystyle b}
  
. Every pair of Gaussian integers 
  
    
      
        a
        ,
        b
      
    
    {\displaystyle a,b}
  
 determines a square this way.
(4) But hmmm... there's already this discussion of characterizing squares in terms of a center and half-diagonal, so maybe we should discuss that for lattice squares as well / contrast with the vertex + side characterization. Well, the parameters now can't be claimed to be Gaussian integers because they aren't necessarily. One basic obvious fact about this characterization that has come up repeatedly in my own investigations is that the center and half-diagonal always either both have integral coordinates or both have half-integral coordinates, depending on the parity of the squared norm of the side length. I assume(d) that will be trivial to find mentioned in the existing literature.
(5) To make this comprehensible, it's probably best to include a picture.
So really there were 2 main motivations: (1) mention a vertex + side characterization instead of only mentioning a center + half-diagonal characterization for squares, (2) discuss squares with their vertices on the lattice. I think trying to satisfy both of those is important, but it could probably be tightened up. Neither of these motivations is really specific to complex numbers, and transforming a unit square or lattice can also be done with other tools, though complex numbers are convenient.
I'm still fairly convinced that there must be more explicit discussion of this in sources somewhere. I'll list some of the ones I looked at when I get a chance, but I have to go for now. –jacobolus (t) 21:19, 22 February 2025 (UTC)

Re your point (2): the problems of counting squares in lattices are already covered and sourced in Square § Counting.
As for it being more natural to define complex squares by two consecutive vertices rather than center and one vertex: I thought so too until I tried to source it and found only origin+vertex in the sources. I couldn't even find sources that talked about squares with arbitrary centers. —David Eppstein (talk) 21:52, 22 February 2025 (UTC)
No, § Counting discusses a different problem, of counting axis-aligned squares that can be drawn from the lines in a grid, not the problem of counting squares with corners at arbitrary lattice points. –jacobolus (t) 23:21, 22 February 2025 (UTC)
Read it again. The second paragraph. —David Eppstein (talk) 23:24, 22 February 2025 (UTC)
Oh fair enough. Would be worth adding a picture maybe: I completely missed that sentence on multiple skim-throughs. There are a good number of sources about squares on a geoboard (some of which mention this square counting problem), e.g. JSTOR 27959243, JSTOR 41191097, JSTOR 27960144, JSTOR 27968166, JSTOR 27960751, JSTOR 27963170, JSTOR 41198968, JSTOR 10.5951/mathteacher.105.7.0534. Here's one discussing "off the peg" squares whose sides are lines between lattice points JSTOR 10.5951/mathteacher.111.3.0232. –jacobolus (t) 23:47, 22 February 2025 (UTC)
I replaced the picture with one that shows both versions of the counting puzzle. —David Eppstein (talk) 07:39, 21 March 2025 (UTC)

@David Eppstein I notice you put Clifford torus in the see also. It might be worth including a section about different ways of associating the edges of a square to get different topologies, e.g. identifying two opposite sides to get a finite cylinder, identifying the opposite sides in reverse orientations to get a Möbius strip, associating both pairs of opposite sides to get a flat torus (doubly periodic square), associating both pairs of opposite sides one with reversed orientations to get a Klein bottle, associating both pairs of opposite sides each with reversed orientations to get a topological projective plane, associating pairs of adjacent sides oriented toward their shared point to get a right-triangular dihedron (topological sphere), etc. (Cf. Fundamental polygon § Examples of Fundamental Polygons Generated by Parallelograms.)
There is some literature having to do with metric squares with other topologies, such as this one about packing circles onto a square flat torus doi:10.1007/s13366-011-0029-7. There are also sources discussing dynamical billiards in a square, which can be analyzed by unfolding the square-with-reflective-edges onto a flat torus. –jacobolus (t) 00:53, 23 February 2025 (UTC)

The paper bag problem would fit this topic. And maybe we could find sourcing about square Klein bottles. But in general I think we should list material here only if it directly pertains to squares as distinct shapes from rectangles. If it is something that sources only discuss for rectangles, without saying anything specific about the square case, we should not commit original research by saying something specific ourselves that does not come from the sources.
The Möbius strip example is particularly problematic because there is a limit on the aspect ratio of rectangles that can be smoothly twisted into a Möbius strip and the square is beyond this limit. Note that the word square does not appear anywhere in the Möbius strip article. If there is anything specific to say about square Möbius strips that does not apply to rectangular Möbius strips more generally, I don't knw what it is.
The Wikipedia sourcing requirements may be annoying sometimes but they also serve as a limit on editors wanting to go into excruciating detail on personal hobbyhorse topics that are only vaguely related to the main topic. I think that going too far in this direction would be an example of that, which is in part why I have held off on adding the Clifford torus and paper bag problem already. Another reason is that I am not entirely happy with the state of the Clifford torus article. It is very focused on Clifford algebra and on a specific (but important) 4d embedding of the square flat torus. I think that the square flat torus (and separately the hexagonal flat torus) are important enough to have standalone articles that are about those tori as geometric spaces and not about their embeddings. In the same way, the square Klein bottle (or square Möbius strip) are abstract geometric spaces but not embedded surfaces. —David Eppstein (talk) 01:30, 23 February 2025 (UTC)
I think flat torus could be its own article, with sections about square and hexagonal examples (and more generally rectangle / parallelogram shape), to which Euclidean torus, Square torus, etc. should redirect. The article Clifford torus should focus on the embedding into 4-space. While we're at it we do not have an article Periodic interval, nor do we really have any article about the 2-infinite-ended cylinder (Cylinder barely mentions it). –jacobolus (t) 01:38, 23 February 2025 (UTC)
Hilbert & Cohn-Vossen (1950) use squares explicitly here p. 309, and then elaborate a bit later (p. 328) discussing putting together squares on the plane to make the universal covering of the torus, describing how the fundamental group of the torus is the same as the group of translations of the square lattice onto itself. Though this book doesn't talk too much about metrical properties of the square flat torus or other surfaces based on a metric square. –jacobolus (t) 20:12, 1 March 2025 (UTC)
(Here's a paper about packing squares into a square flat torus: doi:10.1088/1742-5468/2012/01/P01018.) –jacobolus (t) 01:36, 23 February 2025 (UTC)
I ended up adding a paragraph on square-tiled surfaces including the Clifford torus to the packing and tiling section. I didn't include the torus variation of packing squares into squares because I think that variation is the sort of detail that belongs in the square packing article rather than overloading the main square article with it. —David Eppstein (talk) 07:02, 21 March 2025 (UTC)

Still trying to make sense of it (not reading Russian, so needing to go through OCR + machine translation for any page I want to read), but Boris Kordemsky and N. V. Rusalev wrote a nice 1952 book Удивительный Квадрат [Amazing Square] discussing various topics related to squares, but which mostly seems focused on dissection of a square and rearrangements into other shapes, with a lot of specific examples. Many of them are a bit puzzle-like, as in a square can be rearranged into X other shape using only N cuts, etc. Does anyone around read Russian? Alternately, it might be worth trying to find other sources about these topics. There's some relevant material in English in Frederickson's book about dissections.  –jacobolus (t) 18:59, 25 February 2025 (UTC)

The important statement in this area is the Wallace–Bolyai–Gerwien theorem according to which any two polygons of equal area can be dissected into each other. We have a link in see-also about one specific example of this, of a dissection of one square into three smaller squares, and the equilateral-triangle-to-square dissection may also be independently notable, but for the most part there is not much to say here that is very specific to squares.
For the same reason I am somewhat reluctant to incorporate Monsky's theorem into the main article text, because it generalizes to all parallelograms and all centrally symmetric polygons, so the specific connection to squares rather than to more general shapes is only based on the history of its discovery rather than having any continued mathematical significance. —David Eppstein (talk) 06:59, 21 March 2025 (UTC)

I don't think the picture showing the symmetries of various quadrilaterals is very legible or particularly easy for most readers to interpret. I'm considering making a semi-interactive diagram, related to discussions from Wikipedia talk:WikiProject Mathematics/Archive/2025/Jan § Calculators! and Wikipedia:Wikipedia_Signpost/2025-01-15/Technology report. (And an image showing the relation between types of quadrilaterals could be perhaps added to a previous section.)
I'm considering a diagram showing the square broken into 8 triangles, roughly along the lines of File:Wallpaper group diagram p4m square.svg but a bit less ornate and with less excessively vibrant colors, and then trying to add buttons around it for rotations by 1/4 turn anticlockwise, 1/2 turn, or 1/4 turn clockwise, as well as reflections across the vertical, horizontal, and two diagonal axes. Does anyone know if we can safely use CSS transformations along with the calculator feature? If so, we could actually make the square appear to do rotations or 3d flips between states; if not we could at least do instantaneous transitions between static images.
I was thinking we could also include a table, below the image, showing the multiplication table of the dihedral group, perhaps color coded to match the interactive square. It seems worth actually including concrete details of these symmetries instead of only mentioning them in passing in a sentence or two, since they seem to me quite important to the concept of a square.
Does this seem worthwhile? If so I can try to spend some time on it, but if not, I won't waste my time figuring out the implementation. –jacobolus (t) 04:20, 28 February 2025 (UTC)

I also think the diagram is not very legible, but my experience is that the calculators do not work on mobile apps, so you would be making the result worse for a large fraction of readers. I think the effort would be better spent on coming up with a way of showing the important information from the diagram more legibly in a static image. —David Eppstein (talk) 06:53, 21 March 2025 (UTC)
Okay, in that case what if we use a static image like:

And then put underneath a multiplication table of the symmetry group. I was thinking we could just directly use the reflected or rotated letter F, set in the matching color, as a symbol for each group element. Someone who wants to know more about the group's possible generators, Cayley graph, etc. can click through to Dihedral group of order 8. –jacobolus (t) 06:47, 25 March 2025 (UTC)
I tend to think the subgroup structure of D4 and its connection with the type of quadrilateral that each subgroup corresponds to is more important than presenting the composition of symmetry operations as a giant table of 64 unrelated cases, among other reasons because it emphasizes the position of the square as most symmetrical, not merely because it has a big number of symmetries but because they subsume the symmetries of all the the other quadrilaterals. That would be lost in your proposed approach, I think. —David Eppstein (talk) 06:54, 25 March 2025 (UTC)
The diagram showing the various types of quadrilaterals is inscrutable, poorly explained, and seems mostly off topic. It would be more useful here to put a diagram showing a classification of quadrilaterals, at some other place in the article, and leave detailed discussion of the different symmetries of various quadrilaterals to the article quadrilateral. It would probably also be helpful to do a much better job discussing subgroups (including some geometric interpretations) at Dihedral group of order 8. –jacobolus (t) 06:56, 25 March 2025 (UTC)
I agree that it is a bad to read graphic, but I think the point it is trying to make is an important one, not off topic. And the classification by symmetry is very relevant to the topic of the symmetry of the square (because all other symmetries of quadrilaterals come from symmetries of a square), but the classification of quadrilaterals in general is a big mess that is best relegated to the quadrilateral article rather than reiterating here. —David Eppstein (talk) 06:58, 25 March 2025 (UTC)
It seems to me that there are quite a few other points we could make about the symmetry group D4 at this article that seem more important trying to delve into its subgroups. This feels like an undue emphasis on one subpoint while skipping a lot of the meatier core of the topic. (As an aside, I'd also cut or try to rewrite for clarity and concision "More strongly, the symmetries of the square and of any other regular polygon act transitively on the flags of the polygon, pairs of a vertex and edge that touch each other. This means that there is a symmetry taking each of the eight flags of the square to each other flag." which seems overly technical and somewhat inaccessible.) –jacobolus (t) 07:01, 25 March 2025 (UTC)
It both strengthens and unifies the other two transitivity properties on sides and vertices. Unlike the others it is a free and faithful group action. Maybe it could be expressed more concisely but I think it is important to mention, and flags are necessary to describe in some form since they are the things being acted on in this way. —David Eppstein (talk) 07:08, 25 March 2025 (UTC)
This comes pretty near the top of the page, and I would not expect most readers of this page to necessarily know the correct senses of the words "transitively", "flags", "symmetry". This is just one of several possible ways of describing what the symmetries do to the square (for example, we could say that if we break the square into 8 small triangles along its lines of reflective symmetry, then the eight symmetries take one of the triangles to each of the others). Overall this whole subsection is extremely jargon-heavy and technical ("isogonal figure", "congruence transformation", "point group", "dihedral group", "period lattice", etc.) while the subject it is discussing can be straight-forwardly described without any special jargon to schoolchildren, though it requires a bit more extensive explanation. It's nice to link to relevant other wikipedia pages describing the related technical topics in detail, but there must be ways of covering this section so that e.g. an average high school student can somewhat fluently read it. –jacobolus (t) 07:28, 25 March 2025 (UTC)
"Giant table" seems like an exaggeration. I'd call it a very small 8x8 table with one of the following symbols in each space: F   F   F   F   F   F   F   F   (or perhaps alternately some 2ish-letter abbreviations describing what the transformations do, expanded below). –jacobolus (t) 07:34, 25 March 2025 (UTC)
The point is less how compactly the table can be presented while still making sense to those who already understand what it is presenting,  and more that it is describing the composition of symmetries in a very non-conceptual way, as "look up each of these 64 compositions of symmetry by another to find the third" rather than conceptually such as for instance making the point that a composition with an odd number of reflections produces another reflection while a composition with an even number of reflections can only be a rotation or the identity.
Do you think students better understand multiplication (note: not are they able to calculate better, a different thing) by having to memorize decimal multiplication tables? What if we made it sexagesimal. It's a similar issue. Or maybe worse, since memorizing decimal multiplication tables is actually useful despite its conceptual vacuousness (it helps people perform arithmetic) whereas this table would neither convey understanding nor help perform a calculation people are actually going to want to calculate. —David Eppstein (talk) 07:45, 25 March 2025 (UTC)
I'd be fine if we also spend 2 paragraphs on describing "conceptually" what the symmetries are and how they compose. But the point of a table is not to be a substitute for an explanation or a tool for memorization. The point of a table is to make the full structure explicit, so that people can see the many patterns inside.
I think memorizing a 10x10 digit-by-digit multiplication table (as my 3rd grader is currently being asked to do in school) is an utter waste of time (in particular since my kid has already learned all of the individual digit multiplication facts in the course of working nontrivial word problems, solving numerical puzzles, etc., that happen to involve multiplication as a step). I would likewise urge people not to try to memorize the composition table of the symmetry group of the square – which would be similarly pointless. But that doesn't mean there's no value in ever looking at such tables, which contain a ton of very interesting structure.
If you think it's clearer, some other abbreviations of prose phrases could be substituted as an alternative to graphical images, though I think you underestimate the value of the direct graphical images per se. –jacobolus (t) 08:02, 25 March 2025 (UTC)
I don't think anyone is going to see any patterns in a table of d4 multiplication and that it is off-topic for squares. If I were reviewing this for Good Article it would go on my list of GACR 3b violations.
Tables can be useful computational tools. But if you have to resort to a table of cases to explain something then the explanation is a failure. They have no explanatory value.
Additionally, the focus on the group composition operation is misplaced here. It's an important topic, for group theory, and should be briefly mentioned here as it already is, but composition of symmetries is not an important part of understanding the ways that squares are symmetric. —David Eppstein (talk) 17:13, 25 March 2025 (UTC)
Counterproposal: We use something like this.
 —David Eppstein (talk) 17:39, 25 March 2025 (UTC)
This is much better than the previous picture. It would probably be worth putting textual description of these subgroups into the text (and possibly tightening the caption). For example, we could say that the symmetries of a square can be generated by two reflections across lines one eighth of a turn (45°) apart, which compose to a quarter-turn (90°) rotation. A kaleidoscope with the same symmetry can be made from two mirrors at a dihedral angle of one-eighth turns, and the fundamental domain of the symmetry group is a triangular sector of the square between these two mirrors. The rhombus and rectangle have symmetries of the digon, generated by two reflections across lines a quarter turn apart, giving them also half-turn rotational symmetry. The non-rectangular isosceles trapezoid and non-rhombic kite have only a single reflective symmetry with no rotational symmetry, and the parallelogram has only half-turn rotational symmetry but no reflective symmetry.
More explicitly describing the symmetries of the square and their composition structure is also independently valuable, of core importance to an article about squares, and this table is easily sourceable to a wide variety of sources aimed at various audiences discussing squares, symmetry, etc. Including a table is not "resorting" to anything; tables complement rather than replace explanation (and "no explanatory value" is an absurd exaggeration), and the purpose of such a table is not mainly to be a "computational tool", and describing it as such is missing the point. It might even be worth explicitly indicating these subgroups in a table, or explicitly writing sub-tables, though substituting textual description is probably also sufficient. Bell (1966), p. 128 uses ⁠
  
    
      
        I
        ,
        r
        ,
        
          r
          
            2
          
        
        ,
        
          r
          
            3
          
        
        ,
      
    
    {\displaystyle I,r,r^{2},r^{3},}
  
⁠ ⁠
  
    
      
        h
        ,
        v
        ,
        
          d
          
            1
          
        
        ,
        
          d
          
            2
          
        
      
    
    {\displaystyle h,v,d_{1},d_{2}}
  
⁠ as his names for the specific group elements. Fass (1975), p. 297 uses ⁠
  
    
      
        I
        ,
        
          R
          
            1
          
        
        ,
        
          R
          
            2
          
        
        ,
        
          R
          
            3
          
        
        ,
      
    
    {\displaystyle I,R_{1},R_{2},R_{3},}
  
⁠ ⁠
  
    
      
        H
        ,
        V
        ,
        P
        ,
        N
        
      
    
    {\displaystyle H,V,P,N\!}
  
⁠ in a book for primary school teachers. I came across a more explicit set of symbols in a different book aimed at a high school audience a couple weeks ago, but now I'm not finding it. Open University (2007), p. 7 has a nice picture showing each of the symmetries as a separate small picture, which is probably better (more explicit) than trying to show them all in one square. –jacobolus (t) 18:24, 25 March 2025 (UTC)
Again, I think we should focus on the individual symmetries, not on the groups. The group composition operation is important in group theory, but it is not important for squares, and discussing the topic at the level of groups rather than individual symmetries makes it unnecessarily technical. Introducing complicated subscripted notation schemata for the group elements is also a way of making the subject unnecessarily technical.
There is a time and place for all this material about abstract groups and their group operations but this article is not it. The actual symmetries of a square, their rotations and reflections, are a very basic concept that does not require university-level abstract algebra to present. We can present it at a level appropriate to schoolchildren, and so we should. —David Eppstein (talk) 18:56, 25 March 2025 (UTC)
We don't need "university level abstract algebra" to talk about composition of symmetries. This is a topic sometimes discussed in books for elementary school students. –jacobolus (t) 19:26, 25 March 2025 (UTC)
Abstract group theory is part of abstract algebra, a sophomore-level mathematics-major university level subject. Likely some bright students have seen some of it earlier depending on their curriculum but it is an unnecessary added level of abstraction. —David Eppstein (talk) 19:32, 25 March 2025 (UTC)
Group theory can be made as abstract as you like, but composition of symmetries is as concrete and tangible as it gets. We have a physical object which we can literally flip around in our hands and examine. (Indeed, taking elementary tangible subjects which can be straightforwardly described to anyone with no special prerequisites and obfuscating them as "advanced" topics requiring many years of prerequisites hidden by layers of abstraction and jargon is one of the biggest problems with mathematical writing both on Wikipedia and more generally.) –jacobolus (t) 19:34, 25 March 2025 (UTC)
The flipping is the important part. Those are the individual transformations. Why is it important, to someone trying to understand squares, to figure out exactly which flip would replicate a different combination of flips, beyond the bare fact that if you perform multiple steps that each preserve the square shape then they all preserve the square shape? Why do we need to step up another level from symmetries to groups of symmetries, and then another level beyond that from individual groups to group homomorphisms, etc etc? There is mathematics to be done that way but each step away from squares comes at a cost of being less immediately relevant to squares. —David Eppstein (talk) 20:11, 25 March 2025 (UTC)
Thinking about which flips of the square compose in which ways is not inherently less helpful to "someone trying to understand squares" than the isoperimetric inequality for quadrilaterals, a formula relating the circumradius and the four distances to the vertices from an arbitrary point in the plane, squares inscribed in a triangle, packings of squares inside squares, etc. Indeed, I would say the composition of symmetries is much more directly about the structure inherent in a square than most of the topics discussed on this page (which also seem fine to discuss). Sure we could just say to the reader "the symmetries form a group, go work the details out for yourself or click through to our much more abstract and difficult pages about that topic if you want to learn more", but it's probably even better to be explicit about it, as some readers may find the details interesting or useful, and the rest can skim past. –jacobolus (t) 21:19, 25 March 2025 (UTC)
There are lots of things in this article that we're not explicit about because the details are off-topic for squares. For instance, the "Definitions" section discusses rectangles, rhombi, etc., without separately defining what each of those things is. The "Measurement" section discusses area and perimeter without going into detail about real analysis and how it may be used to put those concepts on a rigorous footing. Making the article itself more abstract and difficult is not an improvement over suggesting clicking through to another more abstract and difficult article; that way lies an encyclopedia where every article contains the entire content of the encyclopedia, not helpful. We have separate articles for separate topics for a reason. —David Eppstein (talk) 21:28, 25 March 2025 (UTC)
These comparisons seem like non sequiturs. I'm talking about a small amount of concrete, explicit detail (neither abstract nor difficult) of the structure inherent in a square, and you're comparing that to the content of a year-long college course because someone might consider studying it to clarify the nature of measurement in general, or something. –jacobolus (t) 21:44, 25 March 2025 (UTC)
Yale (1968), p. 17 has an additional table showing the permutations of the sides and vertices induced by various symmetries. –jacobolus (t) 19:35, 25 March 2025 (UTC)
There is a reason we no longer print big books of log tables. Tables encode the information they encode but they do not explain it.
My preferred conceptual description of the symmetries of a hypercube, btw, is a signed permutation acting on points whose coordinates are all ±1. You can describe any face of the hypercube by its centroid (setting some of the coordinates to 0) and the signed permutations act in the same way on those. No tables needed! Just negate and permute coordinates.
But I think this is also too abstract for the article on squares. —David Eppstein (talk) 21:34, 25 March 2025 (UTC)
We don't print the previous big books of log tables because those log tables were used as a computational aid which is now obsolete, but we still are happy to encode the same information in a graph, picture and description of a slide rule, concrete computer algorithm (or code) for computing logarithms, etc. Because the decimal digits of logarithms are not closely related to the quantities they express, the big book is just an inscrutable list of effectively random numbers which is only useful as a reference when trying to look up a single entry. Tables of a more discrete character like a multiplication table, Pascal's triangle, truth table of logical operations, etc. are still discussed and examined all the time because they serve as a fruitful summary of a structure, not only a computation aid. –jacobolus (t) 21:51, 25 March 2025 (UTC)
We still don't make copies of Pascal's triangle within every Wikipedia article that happens to use a binomial coefficient somewhere. I'm not saying that we shouldn't have a multiplication table of D4 in the article Dihedral group of order 8. I'm saying we shouldn't copy it into every Wikipedia article that happens to describe an object with that symmetry group. This article is already too long. We don't need to pad it with off-topic cruft to make it longer. We should be looking for overdetailed sections we can tighten, not for vaguely-relevant material to add. —David Eppstein (talk) 22:44, 25 March 2025 (UTC)
These symmetries are typically explicitly called "the symmetries of the square", and their group is "the group of symmetries of the square". The more generic name "Dihedral group" is a shorthand for "the group of symmetries of a regular ⁠
  
    
      
        n
      
    
    {\displaystyle n}
  
⁠-gon allowing for reflections"; "square" is a synonym for "regular tetragon".
We do and should include a copy of Pascal's triangle in articles where it is relevant, including at least Pascal's triangle, Binomial coefficient, Binomial theorem, and even Sierpiński triangle, Simplex, Factorization, Binomial distribution, Bernoulli's triangle, Proton nuclear magnetic resonance, Blaise Pascal, Yang Hui, History of algebra, and likely others. Including such a numerical table isn't an imposition on readers who can easily skim past, especially if it is floated to the side.
This article is already too long – Personally I think this article is rather too short to yet cover such a central topic. Readers would benefit if it were made more detailed and comprehensive, up to maybe double the current length.  –jacobolus (t) 01:55, 26 March 2025 (UTC)
Both WP:GACR #1a "concise" and #3b warn against prolixity. And this article is about squares, the geometric shape, not about other topics that happen to be named after squares. —David Eppstein (talk) 04:08, 26 March 2025 (UTC)
Saying prose should be concise is not the same as deciding which topics to cover.
We don't need to go into incredible depth (e.g. we don't need to make a list of every possible other mathematical object with the same symmetries, or list every known property of the symmetry group), and we should try to write clearly, but what happens when you take a square and flip it around is by no means an "other topic named after squares". It's a fundamentally important and basic aspect of the topic "Square", more central and relevant than at least half of the current content here. –jacobolus (t) 06:31, 26 March 2025 (UTC)
And we cover that aspect, in detail. What I am balking at covering is a different topic, "how to simplify a sequence of rotation and flip operations to a single operation by looking things up in a table". —David Eppstein (talk) 06:40, 26 March 2025 (UTC)
Rather than describing the symmetry group as a lookup table, I added another paragraph describing its representation by signed permutation matrices. Is representation theory more or less abstract than group theory? Whatever, I think it's a more conceptual explanation of how to perform the group operation, and one that generalizes easily to higher dimensions (unlike the table approach where you would have to make a table of size 48 x 48 just to handle the cube). —David Eppstein (talk) 21:27, 26 March 2025 (UTC)
This is both less explicit and also much less accessible. We should skip any discussion of matrices here. –jacobolus (t) 03:04, 27 March 2025 (UTC)
Less explicit? You must have a different meaning of explicit in mind than I do, because to me a concrete matrix of numbers (and a concrete description of how it represents a transformation) is much more explicit than just naming the transformations by symbols and filling in a table of case analysis for how to multiply them produced by some never-described inexplicit calculation procedure. It is explicit in the sense that it is directly about how to calculate with coordinates, and might actually be useful to anyone wanting to implement these symmetries as part of a piece of software. And it is explicit in that immediately tells you how to multiply the group elements. Beyond explicitness, it is conceptual and informative in that it immediately generalizes to higher dimensions rather than treating each dimension as a separate case with an increasingly-unwieldy table of answers, and in that it unifies the symmetries as being all the same kind of thing rather than consisting of two unrelated types of operation, reflections and rotations. —David Eppstein (talk) 04:05, 27 March 2025 (UTC)
Yes, less explicit, because you are leaving all of the work to readers (who are now expected to know about matrix multiplication) instead of directly showing how the symmetries compose. A bunch of matrices is only going to be of any value for a relatively small fraction of readers, and for the rest it will be a big barrier, introducing a completely unnecessary and frankly gratuitously cumbersome abstraction which, while common in higher mathematics, is not really a particularly natural or convenient way to represent reflections and rotations. (Moreover, the result of composing the symmetries is much more easily figured out by directly holding a square and performing the symmetry transformations and then inspecting the shape, rather than by translating each symmetry to a matrix via the prose lookup table you provided, doing gratuitous matrix arithmetic, and then translating back to a symmetry using the lookup table in reverse.) If you really want an arbitrary different representation, it would be better to use multivectors (cf. Hestenes (2002) doi:10.1007/978-1-4612-0089-5_1) which is significantly tighter and less computationally/conceptually wasteful for this application, but this also seems unnecessary for an article about Squares intended for a wide audience. I'll revert the addition of the matrices, unless some much more convincing reason can be provided for including it. Feel free to include matrix representations in the article Dihedral group of order 8, where they are less off-topic. A table requires the least pre-requisite understanding and is the most concrete and explicit, but if you want a more abstract and less accessible depiction, a Cayley graph is an alternative that has been abstracted a bit and made more visual. Personally I think it's fine to leave the Cayley graph to Dihedral group of order 8 rather than including it here, but it would be much less distracting than a list of matrices to add here. Either a table or Cayley graph would be best placed floating to the right, rather than taking up too much of the main prose flow, which we should try to keep concise as you noted previously. –jacobolus (t) 05:09, 27 March 2025 (UTC)
A table requires no understanding because it provides no understanding. It tells you what the result is without any information about how to get to it. A reader provided with a table can use it only to memorize or to look up. You might say that they can infer patterns, but I doubt it, and if there are patterns to infer we should say what they are without being coy and demanding that students provide the understanding that we refuse to give them ourselves. The direct relevance of these matrices to squares is that they give you a formula for what the transformation does to each point of the square, tied directly to the geometry of the square, something that giving something a label and pretending that your abstract label is talking about flipping pieces of paper in 3d does not do. I do not think your proposed direction for change is an improvement and if reverted I will certainly escalate. —David Eppstein (talk) 06:05, 27 March 2025 (UTC)
As for a Cayley graph: that only tells you how to multiply by generators of the group. Which is a useful thing to do in an article that is primarily about the group, in the context of presentations of groups by their generators and relations, but not for an article about squares. —David Eppstein (talk) 06:14, 27 March 2025 (UTC)
Adamson (1973) pp. 808 ff. describes the applications of the symmetries of the square to physical chemistry. –jacobolus (t) 19:43, 25 March 2025 (UTC)
Thinking about the diagram some more, if we want to cover the subgroups of the symmetry group of the square, I think it would probably be clearer to show a variety of squares (e.g. painted tiles) with different designs on them to represent various symmetries. The quadrilaterals have the problem that they only show certain symmetry subgroups, which seems somewhat arbitrary for a section about symmetry.
It might be helpful to have a dedicated section about squares as a type of quadrilateral, where this quadrilateral image would be a useful illustration. –jacobolus (t) 03:10, 27 March 2025 (UTC)

It might be worth mentioning that a mechanical linkage of square struts with pivots at the vertices can be manipulated into a rhombus of any shape, and that a square can be transformed into a rectangle of any shape by non-uniform scaling aligned with its sides, into a parallelogram of any shape via orthogonal projection (affine transformation), and into any convex quadrilateral via perspective projection. This has some practical implications. For example: any geometry problem about parallelograms in general relying only on affine features can have the parallelogram made into a square and then solved using coordinate geometry; in computer graphics textures are often squares or square tilings which appear in perspective when viewed from a (virtual) camera; in making perspective drawings it is often helpful to start by establishing the vertex positions of a square grid as a reference for placing other features. –jacobolus (t) 19:25, 25 March 2025 (UTC)

I added a paragraph on this to the symmetry section but now it needs sources. —David Eppstein (talk) 20:07, 25 March 2025 (UTC)

It might be worth mentioning Chladni figures of a square plate, which was historically important and for which we can find a plethora of sources. –jacobolus (t) 19:48, 25 March 2025 (UTC)

It might be worth noting that the word "square" appears nowhere within the linked article text. It talks about rectangular plates and guitar-shaped plates but not square plates. If this is so central to the topic of Chladni figures, shouldn't there be something there to summarize, first, before we consider summarizing whatever it is here? —David Eppstein (talk) 04:11, 26 March 2025 (UTC)
Yeah, there should be a separate article about Chladni figures, which could be improved significantly, including a bit of detail about various specifically shaped plates. A search for "square plate" chladni in Google scholar turns up 469 results. –jacobolus (t) 20:47, 28 March 2025 (UTC)
Perhaps this could more generally be part of a section about the (interior of a) square as a domain for arbitrary functions / differential equations, a subject with many applications and a huge number of sources. –jacobolus (t) 01:05, 29 March 2025 (UTC)

It might be clearer to discuss the square as an example of various classes of objects (regular polygons, hypercubes, cross polytopes, quadrilaterals, etc.) in a separate section further toward the end of the article. We could mention how specific formulas generalize, which features are shared in common, any interesting features unique to the square per se, etc.
Putting brief and incomplete asides about these in the symmetry section feels awkward. I think it somewhat breaks the narrative flow of that section, as well as making readers skip around from one section to another to hunt for information. –jacobolus (t) 20:56, 28 March 2025 (UTC)

Almost everything we say about the symmetries of the square is unspecific to the square, but true of a much broader class of objects. Breaking up this information and scattering the part where we say these properties about the square from the part where we say that all the same things about regular polygons, hypercubes, or whatever else seems repetitive and confusing. —David Eppstein (talk) 23:16, 28 March 2025 (UTC)
Yes, but most of the other things we say about squares (not just their symmetries) are also unspecific to the square, but can be generalized to many other shapes. Except we largely don't discuss that generalization in other cases. There's no particular reason why this one section should be unique in this regard. –jacobolus (t) 01:03, 29 March 2025 (UTC)
Sure there is. It's because we're discussing the symmetry properties that squares and these other objects share, rather than (as in the related objects section) merely describing the other objects that are in some way squarelike without detailing their properties. —David Eppstein (talk) 03:01, 29 March 2025 (UTC)
I changed my mind and moved the symmetries of higher dimensional things to the discussion of the higher dimensional things. —David Eppstein (talk) 18:44, 29 March 2025 (UTC)

On a quick skim through, I think this is a fact that is missing: the unit balls of the 
  
    
      
        
          ℓ
          
            1
          
        
      
    
    {\displaystyle \ell ^{1}}
  
 and 
  
    
      
        
          ℓ
          
            ∞
          
        
      
    
    {\displaystyle \ell ^{\infty }}
  
 norms are both squares (just rotated differently); a peculiar fact about two dimensions because the higher dimensional versions (cube and octahedron etc.) are different. In a way, this is of course just that regular polygons are self-dual, but for the square, this is related to widely used norms. —Kusma (talk) 05:49, 13 July 2025 (UTC)

The square as a unit ball is in Square § Other geometries, but the different higher dimensional generalizations are elsewhere, near the start of Square § Related topics. —David Eppstein (talk) 05:51, 13 July 2025 (UTC)
Searched for the wrong keywords. —Kusma (talk) 06:14, 13 July 2025 (UTC)

This review is transcluded from Talk:Square/GA1. The edit link for this section can be used to add comments to the review.
Nominator: David Eppstein (talk · contribs) 19:08, 12 July 2025 (UTC)
Reviewer: MathKeduor7 (talk · contribs) 07:55, 13 July 2025 (UTC)

The review will take some time (a month, I estimate). MathKeduor7 (talk) 07:55, 13 July 2025 (UTC)
I think you should tell the story of the impossible "circle-square" (a "round square": a circle that is also a square). And I think you should mention that the square is topologically equivalent to a circle. MathKeduor7 (talk) 08:08, 13 July 2025 (UTC)

Not sure what you mean by this. Squircle is listed in the see also section. Or do you mean illusions like https://divisbyzero.com/2016/07/06/make-a-sugihara-circlesquare-optical-illusion-out-of-paper/ ? I'm not sure that has enough reliable sourcing even to make a separate article, let alone to do so and then link from this one. —David Eppstein (talk) 19:56, 13 July 2025 (UTC)
It is a common theme for debates in introductory philosophy courses. Google Gemini says: "The concept of a "square circle" in philosophy refers to a logical contradiction, an impossible object, similar to the problem of squaring the circle in mathematics. The term "square circle" (or, in German, "viereckiger Kreis") was used by Gottlob Frege in his work "The Foundations of Arithmetic" and is also related to the problem of non-existent objects, discussed by Meinong and Russell." MathKeduor7 (talk) 22:05, 13 July 2025 (UTC)
Ok, added: Special:Diff/1300366485. —David Eppstein (talk) 22:43, 13 July 2025 (UTC)
This is off-topic, but it goes back (at least) to Hobbes. Tito Omburo (talk) 17:41, 15 July 2025 (UTC)
The source I used traced it to Aristotle. —David Eppstein (talk) 18:25, 15 July 2025 (UTC)
Ha! Tito Omburo (talk) 19:04, 15 July 2025 (UTC)
I'm gonna read the whole article tomorrow. MathKeduor7 (talk) 08:10, 13 July 2025 (UTC)
 Comment: So far all the sources I've read are backing the contents properly. I think this will be a quick GA pass review. MathKeduor7 (talk) 08:45, 17 July 2025 (UTC)

Isn't there supposed to be some kind of peer review involved here? You don't have any thoughts about ways the article could be improved? –jacobolus (t) 11:08, 17 July 2025 (UTC)
There were two improvements during the process. Professor David Eppstein did such a great job, that other than those, this article was already perfect in my view (and should be FA, not just GA). MathKeduor7 (talk) 12:31, 17 July 2025 (UTC)

Congratulations, David Eppstein! This article is of GA quality level. MathKeduor7 (talk) 08:56, 17 July 2025 (UTC)

According to Google Gemini: "The ancient Chinese civilization did venerate the square. The concept of a square earth and a round heaven, known as "Tian Yuan Di Fang," was a fundamental part of their cosmology. This idea was reflected in their architecture, philosophy, and even urban planning." Idk, it's just a chatbot answer, but I think it may be important. Anyways, the article is already GA status. MathKeduor7 (talk) 12:37, 17 July 2025 (UTC)

I'm talking about this because of the comment of jacobolus (t) above. MathKeduor7 (talk) 12:38, 17 July 2025 (UTC)
P.S. You can find some interesting photos of ancient Chinese coins on the Internet. MathKeduor7 (talk) 12:45, 17 July 2025 (UTC)
Yes, there is also some hint that the Chinese thought of the square as the ideal shape for a city, but I had trouble finding clear sourcing for that. —David Eppstein (talk) 16:59, 17 July 2025 (UTC)
The circle and square shapes (and the compass and square tools) have been symbolically important in many cultures. In China, as you say, the sky was associated with the circle and the earth with the square (see also Nüwa and Fuxi). Seidenberg (1981) "The Ritual Origin of the Circle and Square" JSTOR 41133635 is one source to mine for ideas of related topics. –jacobolus (t) 20:47, 17 July 2025 (UTC)
Here's a paper about the cosmology of the Anaguta people of Nigeria, who apparently consider the earth to be a square oriented to have its corners in cardinal directions, JSTOR 30022239. –jacobolus (t) 19:35, 18 July 2025 (UTC)
There are plenty of sources about how Maya cosmology considers the earth and underworld to be square. There are some nice weaving/embroidery/stonework patterns involving squares that we could use as an illustration. (Disclaimer: my godmother is a Tzotzil shaman.) –jacobolus (t) 19:45, 18 July 2025 (UTC)

I don't think this article includes enough topics after deep searching. During a little expansion,

I couldn't find the source for the first Thébault's theorem on squares and a quadrilateral
Magic squares and their generalization might be included in the counting areas
Other see-also articles might be included somewhere, like Monsky's theorem and Square trisection in the "Tilings and packing" section, if possible?
Polymino, Polyominoid, Tarski's circle-squaring problem, Punnett square? And more?
Dedhert.Jr (talk) 14:51, 1 October 2025 (UTC)