Talk:Square

Semi-protected edit request on 10 November 2023

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)

if and only if

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)

Admins' noticeboard thread about semi-protection of this article

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)

Lead section as summary seems a bit too niche for intended audience

@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)

Would be great to have a top level section about square grids

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)

Things that are in the infobox but not the article

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)

Lattice squares and characterizing squares in a coordinate plane

@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 $\{1,i,-1,-i\}$; 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 $1+i$. 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 $\{0,1,1+i,i\}$ (or coordinates in $\{0,1\}$). 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 $z\mapsto az+b$ with scale/rotation $a$ and translation $b$. Every pair of Gaussian integers $a,b$ 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)

Metric squares with alternate topology

@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)

(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)