Talk:Square

Did you know nomination

The following is an archived discussion of the DYK nomination of the article below. Please do not modify this page. Subsequent comments should be made on the appropriate discussion page (such as this nomination's talk page, the article's talk page or Wikipedia talk:Did you know), unless there is consensus to re-open the discussion at this page. No further edits should be made to this page.

The result was: promoted by AirshipJungleman29 talk 16:32, 23 August 2025 (UTC)

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• Comment or view
• Article history

)

• ... that when viewed in perspective, a square can look like any convex quadrilateral?

• Source: Francis, George K. (1987). A Topological Picturebook. New York: Springer-Verlag. p. 52.

• • ALT1: ... that baseball diamonds and boxing rings are square despite being named for other shapes? Source: Baseball: Battista, Michael T. (April 1993). "Mathematics in Baseball". The Mathematics Teacher. 86 (4): 336–342. doi:10.5951/mt.86.4.0336. JSTOR 27968332. See p. 339. Boxing: Chetwynd, Josh (2016). The Field Guide to Sports Metaphors: A Compendium of Competitive Words and Idioms. Ten Speed Press. p. 122
  • ALT2: ... that the square shape of a go board symbolizes the earth, with its 361 crossings representing the days of a year? Source: Lang, Ye; Liangzhi, Zhu (2024). "Weiqi: A Game of Wits". Insights into Chinese Culture. Springer Nature Singapore. pp. 469–476. doi:10.1007/978-981-97-4511-1_38. ISBN 9789819745111. See page 472.
  • ALT3: ... that squares in graphic design convey a sense of stability, symmetry, and order? Source: Nyamweya, Jeff (2024). Everything Graphic Design: A Comprehensive Understanding of Visual Communications for Beginners & Creatives. Bogano. p. 78. ISBN 9789914371413.
  • ALT4: ... that that the surfaces that can be tiled by equal squares include the biscornu, a pincushion formed by sewing the corners of one square to the edge midpoints of another? Source: Seaton, Katherine A. (2021-10-02). "Textile D-forms and D 4d". Journal of Mathematics and the Arts. 15 (3–4): 207–217. arXiv:2103.09649. doi:10.1080/17513472.2021.1991134.
  • Reviewed: Template:Did you know nominations/Rohard the Elder

Improved to Good Article status by David Eppstein (talk). Number of QPQs required: 1. Nominator has 189 past nominations.

—David Eppstein (talk) 07:10, 18 July 2025 (UTC).

• FWIW: I don't really like ALT1. A "diamond" is a very common name for a 45° tilted square (or more generally a rhombus with its vertices aligned up–down/left–right – like the card suit ♦), so not an "other shape" at all. (I don't like the claim in the article either.) The source only says that 'A major league baseball "diamond" is officially defined as a 90-foot square.' Which doesn't really support the claim that the word "diamond" is supposed to mean something else. The New English Dictionary has as a definition for the word "diamond" "5. A diamond-shaped figure, i.e. a plane figure of the form of a section of an octahedral diamond; a rhomb (or a square) placed with its diagonals vertical and horizontal; a lozenge. (In early use, a solid body of octahedral or rhombohedral form.)", with quotations going back 500 years. –jacobolus (t) 16:04, 18 July 2025 (UTC)
  • ALT1A ... that boxing rings are square despite their round name? —David Eppstein (talk) 16:45, 18 July 2025 (UTC)

    That one would be fine. –jacobolus (t) 17:10, 18 July 2025 (UTC)

    • Regarding ALT1, though: to me, "diamond" as a shape frequently refers to vertically oriented rhombi, with no implication that they are actually square. See e.g. the Black Diamond Equipment logo. —David Eppstein (talk) 18:35, 18 July 2025 (UTC)

      The name comes from the octahedral shape of diamonds, and has been routinely used for centuries to include squares oriented that way (as indicated by the NED definition I quoted). For example the original French name of the cards suit was carreau (lit. "square" or "tile"), and it was presumably adapted into English as "diamond" because of the orientation; historically the suit's shape was just a tilted square. –jacobolus (t) 19:06, 18 July 2025 (UTC)

• Boxing rings being square is more a curiosity about boxing rings than about squares. I am going to help think of an ALT5 for square. Give me a bit of time. Meanwhile, I implore you to take a look at my inline skates nomination. I picked this "square" entry to help, because it's geeky, long, and has been through a long process of enhancement to get to GA-hood, just like "inline skates" - the exact type of articles mainstream Wikipedians do not care about. We need to help each other out. Cheers. Fred Hsu (talk) 03:54, 18 August 2025 (UTC)
• I've been thinking about this topic for more than a day now. I can't think of a better hook from the article itself. But if the square article talks about Flatland, then you can use a hook like this:
  • "Did you know that a 2D square becomes a line in a 1D world, and can have its "inside" breached by a cube from a 3D world?" Fred Hsu (talk) 23:59, 19 August 2025 (UTC)
    • Every DYK hook must be based on sourced article content, not vague rewordings of other content from popular mathematics books. —David Eppstein (talk) 01:01, 20 August 2025 (UTC)
      • For sure. I was suggesting that the square article could have a paragraph or section on Flatland, as its main protagonist is a proper square itself. That's assuming that people even like my weak proposal of a new hook to begin with. Cheers. Fred Hsu (talk) 01:03, 21 August 2025 (UTC)
• This nomination still needs an actual review, none of which is provided by the above discussion. —David Eppstein (talk) 01:03, 20 August 2025 (UTC)

General: Article is new enough and long enough
New enough: Long enough:

Policy: Article is sourced, neutral, and free of copyright problems
Adequate sourcing: Neutral: Free of copyright violations, plagiarism, and close paraphrasing:

Hook: Hook has been verified by provided inline citation
Cited: Interesting:

QPQ: Done.

Overall: Approve Alt1a Graywhales (talk) 03:27, 21 August 2025 (UTC)

Comments

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)

cluebot Reverted potential vandalism but not sure if this is a false positive

Is this really vandalism in this link here from this article that’s been reverted by cluebot or is this a false positive from cluebot? WikiGrower1 (talk) 22:00, 27 December 2025 (UTC)

You mean Special:Diff/1320126894. It's not vandalism but it's unhelpful and clueless, adding information about the other kind of square (the product of a number with itself) rather than the topic of this article (the shape with four sides), without a source. It should have been reverted, but maybe not with that edit summary. —David Eppstein (talk) 04:57, 28 December 2025 (UTC)

Now that makes sense WikiGrower1 (talk) 13:58, 28 December 2025 (UTC)

It looks like it’s been true the information the editor put but should I report a false positive? WikiGrower1 (talk) 20:37, 13 January 2026 (UTC)

It wouldn't hurt to do so. —David Eppstein (talk) 20:48, 13 January 2026 (UTC)

Infobox image choice

The current thumbnail for this article is an unfortunate choice. The first thing you see viewing a thumbnail for the page, or looking at the infobox is "circle and square", or even "circle, square, or kite", when all that's needed is "square". I realize that there's a polygon series of images, but the transparency around the circle has too much contrast with the background in both light mode and dark mode, and it doesn't really work well for this article.

In contrast, the rectangle article has a straightforward thumbnail.

The square thumbnail should just show a square with some angle markers to show the 90° corners and indicators to show the equal lengths. Showing more than that can be left to subsequent images in the article. The current implementation is too clever for its own good and it ends up being more visually confusing and weird than helpful. Before I look into trying to replace it, I wanted to discuss here since it's been in the article for a long time. I'm thinking something like commons:File:Square Geometry Vector.svg scaled appropriately (same creator as the rectangle image), but I would remove the letters if starting from that SVG. Daniel Quinlan (talk) 22:22, 23 January 2026 (UTC)

Another alternative

One advantage of the current image over your choice is that it conveys the idea that a rotated square is still a square (not a kite, a diamond, or something else). That idea doesn't seem to come through in your critique of this image and is definitely not present in your suggested replacement. A better alternative might be File:Carre.svg. —David Eppstein (talk) 23:08, 23 January 2026 (UTC)

The article discusses rotation. I just don't think we need or want that complex of an image for the infobox and thumbnail. We could add a dozen more lines, arcs, circles, angles, etc. to illustrate different concepts, but the core requirements and concepts are illustrated by the simpler image. We could rotate the image, but an unrotated image is far easier to visualize as having 90° angles because we're using rectangular devices with rectangular windows. An image that cleanly and simply illustrates It has four straight sides of equal length and four equal angles. is ideal. Daniel Quinlan (talk) 23:28, 23 January 2026 (UTC)

What is complicated about the alternative I listed? It shows a square, its right angles at the corners and its right angles at the center, with labeled corners. The one you suggested shows a square, its right angles at the corners and tick marks that with some thought maybe one can identify as suggesting that the sides are equal, with labeled corners. It is not in any way simpler. —David Eppstein (talk) 00:51, 24 January 2026 (UTC)

I agree rotation is an important property of squares, but that doesn't mean we should inject that idea into the first infobox image. It's actually counterproductive because an unrotated square matches the rectangular frame of screens and page layout, making its right angles and equal sides easier to recognize for someone learning about squares.

In addition, only one corner is marked as being a right angle when all four should be marked. And there's no indicator that the sides are equal in length. Substituted for those core features is unnecessary complexity from the lines transecting the square and the interior right angle. Those are secondary properties and they should be discussed or illustrated later in the article.

Overall, it's best to focus on a few core concepts, using thick, dark lines, and avoiding faint lines or light and bright colors that are less likely to reproduce well when scaled down or on smaller screens:

Size	Suggested	Alternative
75px
100px

Daniel Quinlan (talk) 02:08, 24 January 2026 (UTC)

The letters are illegible blobs at that size and as I already said the equal-size tickmarks are far from obvious in meaning. Ok, another alternative: File:Square symmetry.svg. Everything legible at tiny sizes, it shows the symmetry lines that only a square can have, and it's much simpler than everything else above. Although I have to say I think your concerns about legibility are overblown. On my phone in the mobile app the image first shows up huge (entire screen width) and then easily big enough to read in the infobox). —David Eppstein (talk) 02:38, 24 January 2026 (UTC)

I agree about the labels, which are unnecessary anyway. As I mentioned above, I think it's better without the labels (like what was done at es:Cuadrado) and I just finished modifying the suggested image to remove them (see below). The definition of a square doesn't start with symmetry lines so it's better to leave them out of the infobox image. That would be best illustrated after the lead with a more detailed image where those concepts are discussed further. We don't need an initial image more reminiscent of a paper fortune teller or paper airplane instructions.

Simple vector image of square

Daniel Quinlan (talk) 02:54, 24 January 2026 (UTC)

It would be helpful to get a few more opinions from other people on what works best as an infobox image, especially for thumbnail usage and first-glance clarity. I'm happy to wait for additional feedback before making any changes. Cheers. Daniel Quinlan (talk) 03:05, 24 January 2026 (UTC)

Infoboxes are not thumbnails. The images in infoboxes are not thumbnail size, even on mobile.

And you have still not addressed my main complaint with your image: one of its most prominent features is not understandable. —David Eppstein (talk) 03:16, 24 January 2026 (UTC)

The image in the infobox doubles as a thumbnail for various page preview gadgets and scripts, search engines, etc. I believe it’s easier to understand than the current image and the other alternatives discussed so far. A similar image is on rectangle and it looks great. Adding a background color to the entire shape might help a bit, I updated the image to show that (open to other colors). We can add some short text underneath as done some other shape articles. Anyhow, it seems like it's time for more input and there's no hurry. Daniel Quinlan (talk) 03:38, 24 January 2026 (UTC)

Basically a square

If the disputed problem is about the annotations in a square, can we just use a plain square only without annotation of internal angles, dotted lines resembling its diagonal, not inscribed in a circle, and marks to denote equal length for all edges? Dedhert.Jr (talk) 04:17, 24 January 2026 (UTC)

The only issue is that people will think it a frame and wonder where the actual image is. I was thinking to be slightly less boring we could have more than one square with different rotations. —David Eppstein (talk) 05:26, 24 January 2026 (UTC)

Family of rectangles or rhombi

Umm... crop the square in the members of the rectangle and rhombus' family made by @Jacobolus? A slightly boring illustration might be adding a square border, slightly darker than the region, and placing two short black marks at the adjacent edges and at a right angle at the vertex. But it is still interesting. Dedhert.Jr (talk) 05:31, 24 January 2026 (UTC)

Quadrilateral has a similar infobox image (one I quite like), but I think the image needs a little something which is why I took my cue from the rectangle article and the It has four straight sides of equal length and four equal angles. sentence from the lead paragraph. The only change beyond that was adding some color. Daniel Quinlan (talk) 05:47, 24 January 2026 (UTC)

I think it's fine to put one square in the top image. We could have another image later with multiple squares in it at different orientations and sizes. –jacobolus (t) 07:10, 24 January 2026 (UTC)

I could imagine an alternative to the current top image using some arbitrary rotation (as a marker of generality) and leaving out the circumscribed circle; however, none of the proposals so far is better, and the current image also seems more or less okay. Labels don't seem helpful, and diagonals are also probably not necessary. I'd limit how many little marks are included, and I think one right-angle marker suffices. –jacobolus (t) 07:12, 24 January 2026 (UTC)

It's somewhat easier to discern that opposing sides are parallel when they're vertical and horizontal (and aligned with the screen and elements on the page). There's a reason why that's the default way teachers draw squares on a whiteboard (well, okay, it's easier to draw them that way too). I'd be fine with something more like the current image if it was rotated 45°, the diagonals were removed, and the circle was removed. Daniel Quinlan (talk) 07:37, 24 January 2026 (UTC)

You seem very insistent that the first thing people see as representing a square must be an axis-parallel square, to the point where you are now making that your primary objection to the current image. To me that reflects a misconception. Squareness is not about being axis-parallel. Confirming familiar misconceptions is not encyclopedic. To address both this point and an earlier point you made: its main purpose is to illustrate some of the defining properties of a square: it has right angles at its vertices, right angles at its sides, and (like other regular polygons, from which this series of images comes) can be inscribed in a circle. The 45° tilt is incidental, but valid. It is not any less a square for being tilted. Why are you so focused on rotations that, for you, only an untilted square can be used here? Put another way, why do you think a tilted square must only have the purpose of illustrating rotation, rather than being as much of an arbitrary choice as an untilted square would be? —David Eppstein (talk) 07:54, 24 January 2026 (UTC)

I'd prefer if we stayed focused on the image rather than each other. It's only one concern and it was a response to a specific comment, as indicated by the threading. I ended up on this talk page after reading the shape article and previewing (and then reading) many of the articles linked from it, and this image was the one that stood out as leaving room for improvement. I think the infobox image should prioritize clarity and focus on the key points in the lead definition. To that end, focusing the image on the 90° angles and equal sides would be an improvement, and making those features easier to recognize at a glance would better serve readers of all levels. I've proposed some images based on that.

As I mentioned before, I think we could use some fresh eyes and additional opinions since we seem to be going in circles. I'll post a short note in a WikiProject or two. Daniel Quinlan (talk) 09:31, 24 January 2026 (UTC)

"There's a reason why" – Sure, this is an essentially arbitrary cultural practice which has some historical inertia. Such practices can change over time or differ between places; for example, in the middle ages diagrams of arbitrary "rectangles" were typically drawn as square, and diagrams of arbitrary "triangles" typically showed an equilateral triangle; to the modern eye such a choice is misleading because it implies a false specificity, and I'm sure such misconceptions were reasonably common, but skilled contemporary readers learned to look past the diagram to understand the general situation. It's a similar situation here. –jacobolus (t) 17:16, 24 January 2026 (UTC)

Fresh eyes and additional opinion, here. I agree that the current image is suboptimal for having both a square and a circle. Of all the options presented, I think we should use something akin to what es:Cuadrado has. Almost everyone knows what a square generally looks like, so the best infobox image will be one that makes it clear what makes a shape a square. Showing the four right angles and equal length sides is ideal for that.

I understand David's argument that a rotated version might be preferrable, but I'm not convinced that it's that important to show that the shape can be rotated in the infobox. To me, the aesthetic faux pas of having a tilted shape outweighs the concern of a reinforcing some hypothetical misconception. And why can't we just show examples of tilted squares elsewhere nearby instead? MEN KISSING (she/they) T - C - Email me! 10:24, 24 January 2026 (UTC)

I don't understand how this is an "aesthetic faux pas". A faux pas is what happens when you make an embarrassing social mistake, usually based on a cultural misunderstanding. To the extent that Wikipedia readers have a unified "aesthetic culture" it certainly isn't one that punishes rotated shapes; if you personally were raised in such a culture, that's frankly really weird. –jacobolus (t) 16:25, 24 January 2026 (UTC)

I personally can't stand a tilted painting on a wall. I don't see why that's weird. MEN KISSING (she/they) T - C - Email me! 19:30, 24 January 2026 (UTC)

We shouldn't based our choices about illustrations on one Wikipedian's misplaced pet peeves. –jacobolus (t) 02:00, 25 January 2026 (UTC)

Yes, it's only if a larger consensus of Wikipedians agree that we should go through with the thing I am suggesting. Otherwise, a different thing should be implemented. I'm fine with that. MEN KISSING (she/they) T - C - Email me! 02:05, 25 January 2026 (UTC)

"Aesthetic faux pas" just means it looks weird. People use the phrase that way. "...if you personally were raised in such a culture, that's frankly really weird..." is rude, verging on name-calling. Please stay civil in the discussion. Joyous! Noise! 23:16, 25 January 2026 (UTC)

I find the use of the slash-dash-stroke thingies to indicate "equal length" to be distracting and potentially confusing. A reader has to remember the convention they maybe saw in a geometry class in order to parse them, and they look like part of the shape, rather than indicators about the shape. Lines in the drawing that aren't part of the square proper should be dotted, dashed, thinner, in a color that shows up as gray instead of black when printed grayscale, etc. Stepwise Continuous Dysfunction (talk) 01:45, 25 January 2026 (UTC)

To add a fresh pair of eyes, here are my thoughts on the current situation and my opinion of the image. 1. The image should be simple and only show the primary geometric meaning of the square. 2. As a diagram in geometry a square consists of four equal lines with four equal angles. 3. The standard notation of showing equal sides is having tick marks and equal angles is having small squares 4. Rotations are decorative and not significantly meaningful for a lead image nor infobox 5. The alternative suggested by David is notation for a kite not a square 6. I personally prefer colorless, but color helps highlight the information 7. the blue square that Daniel suggested with the appropriate notation is the current best option that has been suggested. EulerianTrail (talk) 13:44, 25 January 2026 (UTC)

This is factually incorrect and portrays a misunderstanding of the subject so severe that it calls your judgement on it into question. A 45-degree-rotated square is still a square. A kite is a different class of quadrilaterals. —David Eppstein (talk) 19:44, 25 January 2026 (UTC)

How about something like this one?

–jacobolus (t) 20:14, 25 January 2026 (UTC)

I'm happy enough with that. I'm not insisting on a rotated version, by the way; instead, I'm strongly rejecting demands by others here that it not be rotated. Some people here seem to think that it's important to demonstrate its squareness through equal side lengths rather than through perpendicular diagonals, I think through a belief that one of these two equivalent definitions is "the" definition. Again, that is dogmatic and incorrect; there are many equivalent definitions.

One thing I don't like about some of the versions that have been proposed is that the right-angle-corner markings and side-length-tick-marks are in similar colors and line thicknesses to the square itself making it non-obvious that they are annotations to the square and not part of the square itself. Yours does have the same line thickness for the square corner. Here's another one that doesn't (not rotated, because as I said above I'm not insisting on a rotated version). I deliberately used doubled and offset tick marks because placing them at right angles at the midpoint of each edge makes it hard to tell that they are tick marks rather than midpoint markers. And the annotations are colored, again to make it clear that they are not part of the square itself.

—David Eppstein (talk) 20:56, 25 January 2026 (UTC)

Using different colors for the markings definitely seems like an improvement, and the angling of the tick marks is probably an improvement too. I recommend making them somewhat lighter so they're more distinguishable from the black. It might also help to add a light shade to the entire square, perhaps a light gray given the blue and red. I don't think it's necessary to offset or double the tick marks, though.

Regardless of the final image, I recommend we add a caption or text under the image explaining any additional features such as:

A square has four right angles (each 90°) and four equal sides, as shown here with blue right-angle markers and red tick marks.

Daniel Quinlan (talk) 23:22, 25 January 2026 (UTC)

David, I like this image with the black square with the blue angle marks and red side marks. Maybe a caption could be "A black square with colored notations indicating equal sides and angles." EulerianTrail (talk) 00:19, 26 January 2026 (UTC)

This is another interesting illustration. Each color has its own purpose, although the double slash symbol is unfamiliar to me at the very beginning, usually with vertical or horizontal marks. If this image is included in the infobox, a caption regarding the marks as an annotation would be very helpful. Dedhert.Jr (talk) 01:19, 26 January 2026 (UTC)

I like this diagram, I'm not a big fan of the colors and the offset slanted tickmarks. I tried my hand at making my own version in Inkscape, this is how it turned out:

I kept the tick marks doubled, and tried to make them lighter so it's harder to mistake them as being midpoint indicators. I also made the square itself a solid grey color, do we have any thoughts on what color the square should be? MEN KISSING (she/they) T - C - Email me! 04:23, 26 January 2026 (UTC)

Okay. That's much better. But grey color looks unalive. Just pick any color that you want to imagine, but you have to remind yourself about WP:COLOR. Dedhert.Jr (talk) 05:02, 26 January 2026 (UTC)

Aw, I've always been a big proponent of light grey. How about purple?

MEN KISSING (she/they) T - C - Email me! 05:21, 26 January 2026 (UTC)

I would say do thicker black lines and increase the contrast by making the purple lighter. EulerianTrail (talk) 09:23, 26 January 2026 (UTC)

David's image is fine, and answers all of the complaints. I think we're done here. –jacobolus (t) 09:54, 26 January 2026 (UTC)

I'm not a big fan of the offset red tick marks on David's version. Let's not rush a conclusion, I'd like to make one more based on Eulerian's suggestion. MEN KISSING (she/they) T - C - Email me! 10:15, 26 January 2026 (UTC)

While David's image addresses many of the concerns, there was some feedback since his latest. There's also still room to explore alternative designs. Daniel Quinlan (talk) 10:19, 26 January 2026 (UTC)

@EulerianTrail:

Hows this? MEN KISSING (she/they) T - C - Email me! 10:21, 26 January 2026 (UTC)

I think this version is good. EulerianTrail (talk) 10:28, 26 January 2026 (UTC)

The only room for improvement might be to make the annotations darker. EulerianTrail (talk) 10:30, 26 January 2026 (UTC)

I prefer to keep them light so that it's harder to mistake them as being part of the square instead of annotations of the square. In any case, I have to head to bed. No more squares from me until tomorrow. MEN KISSING (she/they) T - C - Email me! 10:54, 26 January 2026 (UTC)

I was thinking about accessibility. You can make it appear as though it not part of the square by using color. EulerianTrail (talk) 11:13, 26 January 2026 (UTC)

This one is inferior. The contrast is lower and the tick marks are confusing. –jacobolus (t) 17:27, 26 January 2026 (UTC)

Does the gray-on-purple of the right-angle indications even meet MOS:CONTRAST? Accessibility is not optional. —David Eppstein (talk) 18:06, 26 January 2026 (UTC)

Well, per the letter of it, yes. It does meet MOS:CONTRAST

In the section linked there, there's one relevant point (emphasis added):

The visual presentation of text and images of text should meet a sufficient contrast ratio. Contrast is the difference in color that makes elements distinguishable from one another. A higher contrast ratio improves readability.

This is really the only part of the guideline that actually prescribes to use a sufficient color contrast. There's no text in my diagram, so it probably doesn't apply, and that checks out with common sense. We're dealing with a geometric diagram that will certainly be easier to interpret with lower contrast.

That is just the letter of it, though. The spirit is another matter, and regardless I'd be willing to come up with a different version with a different color scheme. In the mean time, I'm still not quite happy with your offset tick marks, could a compromise be had regarding those? MEN KISSING (she/they) T - C - Email me! 19:14, 26 January 2026 (UTC)

I am extremely unhappy with tick marks that are perpendicular to the square and centered. They look like they might mean something else. Markers for the midpoints of edges, maybe, or partial symmetry lines, or parapets for the floor plan of a square tower. It is not obvious that they are tick marks. —David Eppstein (talk) 19:20, 26 January 2026 (UTC)

What about tick marks that are centered, like what I have, but are askew, like what you have? MEN KISSING (she/they) T - C - Email me! 19:23, 26 January 2026 (UTC)

They're not inaccessible, they're just not very attractive. This discussion is now wasting a ton of time. @EulerianTrail and @MEN KISSING – I'm going to restore @David Eppstein's square image now. There seems to be a consensus that it successfully answers the various criticisms leveled at the previous lead image. It's good enough for now. Feel free to workshop alternatives between yourselves and make a new proposal when you think it's ready. Your current direction doesn't seem promising to me though. –jacobolus (t) 19:24, 26 January 2026 (UTC)

To be clear, you're talking about the one with right angle markers and tick marks, yes? That one is certainly an improvement, and I'd be satisfied to see it in the article over the previous version, but I do think it can be improved.

I'm not sure how much further I want to pursue having a better square up there, but I'm experimenting with vector graphics and image design for Wikipedia, and I'm having fun doing it. I don't think that's a waste of time. MEN KISSING (she/they) T - C - Email me! 19:35, 26 January 2026 (UTC)

Could you hold back on phrasing like "...portrays a misunderstanding of the subject so severe that it calls your judgement on it into question..."? That's pretty uncivil. Joyous! Noise! 23:12, 25 January 2026 (UTC)

Thanks, Joyous! David, I agree that a rotated square is a square. But a square has to have all four sides be equal and all four angles be equal. EulerianTrail (talk) 00:16, 26 January 2026 (UTC)

Having one right angle and perpendicular diagonals is sufficient to specify a square. You can derive the equality of the sides and angles from that information (or vice versa). –jacobolus (t) 00:23, 26 January 2026 (UTC)

Jacobolus, you are mistaken. Here is an image to show why the diagonals being perpendicular and corner is not enough to be a square.

Not A Square fixed

EulerianTrail (talk) 01:14, 26 January 2026 (UTC)

Looks like a square to me, but you basically tilted in any angle, if it's about the perspective angle. Dedhert.Jr (talk) 01:21, 26 January 2026 (UTC)

Ah you are right. Now you can see why the current top image includes a circle (which constrains the diagonals to be the same length). Another right angle in one of the other corners would also suffice. –jacobolus (t) 03:36, 26 January 2026 (UTC)

Very close, the two right angles have to be adjacent to each other. Else you could make a kite that is not a square by reflecting a right triangle across its hypotenuse. EulerianTrail (talk) 11:12, 26 January 2026 (UTC)

Unless I'm misunderstanding, I think this construction does always result in a square. MEN KISSING (she/they) T - C - Email me! 13:42, 26 January 2026 (UTC)

Here is an non example where the diagonals are perpendicular and there are two right angles. EulerianTrail (talk) 17:10, 26 January 2026 (UTC)

Oh, true! I was assuming an isosceles right triangle for some reason haha. MEN KISSING (she/they) T - C - Email me! 19:16, 26 January 2026 (UTC)

On top of being insufficient, as illustrated by Eulerian, it's not about what's sufficient to define a square. It's more about what a canonical definition of a square is. Just because you can represent it that way, doesn't mean you should. The reader shouldn't need to derive that information, it should be plainly obvious. MEN KISSING (she/they) T - C - Email me! 01:25, 26 January 2026 (UTC)

There is no "canonical definition". There are lots of equivalent characterizations. —David Eppstein (talk) 02:04, 26 January 2026 (UTC)

More than a "canonical definition", I mean a definition that is more in line with what the fundamental, important properties of a square are. We shouldn't use diagram that sort of just coincidentally produces a square, or one that produces a square in a sneaky way. Such illustrations may not necessarily be unwelcome in the article, but they are probably less ideal for an infobox. MEN KISSING (she/they) T - C - Email me! 02:16, 26 January 2026 (UTC)

@MEN KISSING. I can't comprehend your point. Can you tell the diagram you mentioned? Can you tell what "canonical definition" are you refer to? Dedhert.Jr (talk) 03:36, 26 January 2026 (UTC)

Ah, the darn aside images make it a bit hard to tell what image is introduced in what comment.

The "canonical definition" is the one mentioned in the second sentence of the lede: it's a shape with four sides of equal length, and a right angle at every corner. I'm in favor of diagrams that demonstrate these properties, i.e. the ones with right angle marks on every corner and tick marks indicating equal length on all four sides. I'm not in favor of the proposed diagram that shows two perpendicular diagonals and a single right angle, like the one here Talk:Square#c-Jacobolus-20260125201400-David_Eppstein-20260125194400.

Maybe I should take a crack at making a diagram? I'm sure I could, but I've never worked all that much with vector graphics. MEN KISSING (she/they) T - C - Email me! 03:49, 26 January 2026 (UTC)

@MEN KISSING. Oh, I see. I think the illustration by David Eppstein meets the definition you have considered as "canonical". From my perspective, every corner has a blue right angle symbol, and every side of a square has a red double slash, which indicates their length's equalness $=$. We just have to write a caption to annotate those colorful symbols. Pointing out his opinion about the dogmatic perspective because of choosing squareness by equal length instead of perpendicular diagonal lines and a 45-degree-rotated square considered as a kite (or rhombus) is something that the public cannot well-perceive geometrical illustration. It does not matter a square is portrayed by any rotation, but it is preferable to follow what most textbooks do. Dedhert.Jr (talk) 04:02, 26 January 2026 (UTC)

I don't love the image that's in the article now, mainly because the circle in the image isn't mentioned at all in the entire lead. So, let's say I wander in and I see the square. I see it surrounded by a circle. I think "huh? why circle there?" so I read the caption and the paragraphs beside it, and at the end of the lead, I am probably still thinking "huh? why circle there?" Or worse "Is that what 'squaring the circle' means? I thought it said that was impossible..." Its significance isn't explained until the end of Part 2. I think the image immediately above this comment, the one with the differently colored offset tick marks and angle annotations, is pretty clear, and doesn't overwhelm the reader with too much information. As far as the orientation, I prefer either the "typical" horizontal/vertical-sided view, or a 45-degree rotation. I don't think the lead image should provoke "Why in the world does it look like THAT?" in the reader. Joyous! Noise! 22:58, 25 January 2026 (UTC)

I don't think the lead image should provoke "Why in the world does it look like THAT?" – Provoking such a response sounds like a great feature: it might prompt some thinking in a world dominated by mindless passive consumption. Getting readers to ask and answer their own questions should be one of the highest goals of any expository mathematics. –jacobolus (t) 23:40, 25 January 2026 (UTC)

It's a square, Jacob.

This link is likely relevant: WP:ASTONISH MEN KISSING (she/they) T - C - Email me! 02:54, 26 January 2026 (UTC)

That guideline is about avoiding links that confuse the reader by pointing to bizarre places, not about avoiding images that are interesting. Stepwise Continuous Dysfunction (talk) 22:19, 26 January 2026 (UTC)

MOS:LEADIMAGE is the better shortcut. It makes some points along the lines of what I believe MEN KISSING is trying to say. Daniel Quinlan (talk) 22:26, 26 January 2026 (UTC)

That is how I've seen most people link it, but the section does talk about the general principle of not trying to astonish the reader. I will admit, it does seem fairly localized to guidance regarding text, though.

I'd also like to take this opportunity to explain "It's a square, Jacob", too. I... had no idea how to respond to that comment, but I felt it needed a response. It strikes me somewhere between an attempt to right great wrongs, a misunderstanding of what Wikipedia's purpose as an encyclopedia is, and a justification for having astonishing content, but the comment is also not really any of those very strongly at all, and it's not that big of a deal in the grand scheme of things. I more or less just wanted to use a bit of humor to let jacobolus know I think their comment came across as kind of silly and out of left field. I hope they know I didn't mean anything too negative.

Also, thank you Daniel. That is more in line with what I'm trying to say. MEN KISSING (she/they) T - C - Email me! 22:31, 26 January 2026 (UTC)

I'm least unhappy with David Eppstein's image (the one in the article at the moment). The lines that are about the shape but not part of the shape are distinguished by color and thickness. It expresses four right angles and four equal sides, and the caption makes the meaning of the symbols clear for anyone who has forgotten their school geometry. I think offsetting the tick marks is a nice touch that makes the picture a little more (at the risk of sounding like an art critic) "dynamic". Stepwise Continuous Dysfunction (talk) 22:26, 26 January 2026 (UTC)

Since there was still active discussion, I think replacing the image in the article was premature. While David's image is an improvement over the previous infobox image, the markings don't have enough contrast with the black, especially the red marks. Here's a version based on David's that lightens the markings to increase contrast. It also centers the tick marks for comparison:

Daniel Quinlan (talk) 23:37, 26 January 2026 (UTC)

They still look like symmetry markers rather than edge length markers in that centered position to me. Maybe find some other kind of thing that you can arrange symmetrically where it would be more constructive? —David Eppstein (talk) 23:43, 26 January 2026 (UTC)

Symmetry marks are usually a series of dotted lines. EulerianTrail (talk) 01:17, 27 January 2026 (UTC)

The left image has better choice of colors, though we're really getting to the level of trivial and subjective nitpicks at this point. The ones on the right should be darker and less colorful. Whether the tick marks are centered or not seems like a matter of personal preference; I'd personally make them centered, as a matter of personal style, but the offset version is also completely fine. –jacobolus (t) 23:50, 26 January 2026 (UTC)

Incidentally, the color palette I used for the left image (and most of my illustrations) is one I got long ago from a Wikipedia discussion that I no longer have a pointer to, in which it was put forward as a set of colors that could be easily distinguished by color-blind people. I have also checked it with a color-blind friend. I have no reason to believe that the colors on the right are problematic in that respect but I also have no reason to believe that they are good for that purpose. —David Eppstein (talk) 23:58, 26 January 2026 (UTC)

I appreciate your color-blind testing and these seem like good hues. While the current red (#bc1e46) meets the WCAG AA minimum of 3:1 for graphical objects at 3.41:1 contrast with black, it's barely over the threshold (4.5:1 is a better place to be, especially for lines and other small or thin features). Shifting the color to #e02655 would put the contrast ratio at 4.57:1 against both white and black while retaining the hue. Higher contrast improves accessibility for low-vision users and helps convey information more quickly. The current blue of #0081cd is okay, but #007ac3 would be a small improvement (increasing the contrast from 4.18:1 to 4.59:1 against white). I updated the alternative image to use those adjusted colors. Daniel Quinlan (talk) 01:08, 27 January 2026 (UTC)

It is also good practice to make things less thin when there is no reason to make something so small and thin. I would also like to state that there are different types of colorblindness. EulerianTrail (talk) 01:20, 27 January 2026 (UTC)

Differentiating red and blue is generally not a problem for common forms of color blindness, but the current contrast between the black and the markings is low and could be improved for better accessibility. Daniel Quinlan (talk) 01:28, 27 January 2026 (UTC)

In this context there is no need to differentiate between blue and red (just black and annotation color). EulerianTrail (talk) 01:40, 27 January 2026 (UTC)

The varying line thickness was also a deliberate choice, to differentiate the annotations from the square itself (another part of why I think some of the alternatives proposed above were bad: you can't tell which lines are part of the geometry being illustrated and which are annotations because they all look the same as each other). —David Eppstein (talk) 03:16, 27 January 2026 (UTC)

Hey, I just noticed the background of the square itself isn't white, it's transparent. I would imagine the shape itself ought to have some sort of solid fill. Is this an intentional choice? MEN KISSING (she/they) T - C - Email me! 04:10, 27 January 2026 (UTC)

Wikipedia renders the background of SVGs as a very light grayish color. Adding an explicit background color is fine but generally unnecessary, and whether to add a fill to the square is something best left to the artist's discretion. –jacobolus (t) 04:15, 27 January 2026 (UTC)

Measuring contrast against black is a nonsensical red herring. The line is not on a black background. How distinguishable two different marks are from each-other when placed on the same background is an entirely different question vs. how legible complex symbols (such as words) are, given foreground and background colors. If you want to know how distinguishable marks are from each-other, you need both a different metric and a different standard for judgment. Lightness contrast vs. the background is significantly more important than lightness contrast vs. the other marks in the same diagram. (As an aside, the WCAG contrast metric is a very poorly designed tool which gives almost no useful information, and should never be used for any purpose unless there is some kind of legal contract requiring it.) –jacobolus (t) 04:12, 27 January 2026 (UTC)

But a bit more to the point: this is all trivial nitpicking at this point, and a waste of time. Readers (including readers with severely compromised vision) are not going to have trouble with this particular diagram (at least on account of the color and line style choices). If you want to work on color accessibility of images in Wikipedia, this is in like the bottom 1 %ile of images to be concerned about, and not worth having an extended discussion about. –jacobolus (t) 04:21, 27 January 2026 (UTC)

Every image is important to make accessible. There is no reason to purposely make an image inaccessible. EulerianTrail (talk) 06:43, 27 January 2026 (UTC)

Making these sorts of incremental, minor improvements to an article is hardly something that can be considered a waste of time. Would you rather we simply make our minor adjustments to the image directly without talking about it here first? MEN KISSING (she/they) T - C - Email me! 21:38, 27 January 2026 (UTC)

Unlike article text, WP:OWN does not apply to images. Changes to images should only be made by or with the agreement of the image creator. If you want to make a derivative image, make a derivative image with a different filename. —David Eppstein (talk) 22:38, 27 January 2026 (UTC)

I'll be sure to use a new image with a new filename. It's an easy enough sort of diagram to recreate, just a square with some marks haha. MEN KISSING (she/they) T - C - Email me! 00:59, 28 January 2026 (UTC)

MEN KISSING, what do you think about the most recent alternative image I posted above? Daniel Quinlan (talk) 08:28, 28 January 2026 (UTC)

Oh, I can't believe I never actually replied, I thought I already had! Thank you for asking!

I prefer the version on the right. For a minor improvement, I think I would also prefer if both markings were blue, instead of being blue and red. But that's just me.

I have also mentioned that I prefer non-askew markings, but David seems to like them askew based on that they could be otherwise misconstrued as symmetry markers.

Aside from those two things, I think that image is perfect. MEN KISSING (she/they) T - C - Email me! 08:36, 28 January 2026 (UTC)

I agree I think that the annotation marks should be blue, not blue and red. Symmetry lines are also very different, so there should be no confusion. EulerianTrail (talk) 11:35, 28 January 2026 (UTC)

Spend as much time as you want making whichever new image(s) you want. When you have settled on something you really like, come back and propose it. I expect to disapprove, and I'll be happy to tell you why. But it's a total waste of time to have 5 people spend hours discussing each trivial artistic decision along the way. –jacobolus (t) 01:07, 28 January 2026 (UTC)

Jacob, as much as I would be happy to do this on my lonesome, I would prefer the company and input from other editors who are interested in the same image. Even if the matter is as trivial as you insist (and I don't entirely disagree), this is my first experience helping to create images for use on Wikipedia in discussion with other editors, and I'm learning some valuable things along the way. I really can't think of a more appropriate trial run for learning this sort of thing than a low importance, highly simple diagram.

I understand your perspective that a lengthy discussion should not be had over a minor thing, and I do intend to slow down my commenting overall here. At the same time, you're starting to sound less like you want editors to focus their time on more important matters, and more like you're just irritated with us. I'm sure you have the best of intentions, but it comes across as rude, injecting heat into an otherwise peaceful and productive discussion. I don't want to be anyone's nuisance, and certainly not yours, Jacob.

If you're upset about the notifications, then you can unsubscribe from the thread. I'll be sure to courtesy ping you for your input on any new proposal. MEN KISSING (she/they) T - C - Email me! 07:40, 28 January 2026 (UTC)

Interesting missing property

We are missing the property that if P is an arbitrary point inside a square and we draw straight lines from the midpoints of its sides to P to form 4 quadrilaterals A,B,C and D (labelled in clockwise alphabetical order) then area(A)+area(C)=area(B)+area(D) (proven by splitting each quadrilateral into 2 s.t we get 4 pairs of triangles of equal height and width and hence equal area). Overlordnat1 (talk) 10:03, 2 February 2026 (UTC)

@Overlordnat1. Can you tell the name of the property, statement of the theorem? Can you provide some sources? Dedhert.Jr (talk) 15:14, 2 February 2026 (UTC)

I don’t know the name of it but it’s proven here Overlordnat1 (talk) 17:43, 2 February 2026 (UTC)

This statement is true (of any parallelogram, not just a square), but doesn't really seem that important to me. –jacobolus (t) 18:22, 2 February 2026 (UTC)

I agree with both parts. If we get a reliably published source we could add it to the parallelogram article but a YouTube video is not a reliably published source by Wikipedia standards (despite the evident truth of the claim and validity of its proof). —David Eppstein (talk) 18:53, 2 February 2026 (UTC)

To elaborate about 'not that important', there is a very long list of possible characterizations of a parallelogram (or whatever shape) similar to this, and I think we should try to be selective about which ones we include. Even having a reliable source (e.g. a peer-reviewed paper) that mentions it somewhere doesn't necessarily give a good indication of importance/interest. –jacobolus (t) 19:32, 2 February 2026 (UTC)

With all due respect I completely disagree with you. This property is obviously important if you have 3 areas and want to work out the fourth. I begrudgingly accept that a ‘reliable’ source is needed to satisfy the demands of Wikilawyers but saying we shouldn’t mention this property even with one is for the birds. Encyclopedias are meant to inform. Overlordnat1 (talk) 15:24, 5 February 2026 (UTC)

How often have you come across a non-contrived situation (in any context, whether practical or for answering some other mathematical question) where you had a parallelogram and an arbitrary point in its interior, and the parallelogram was partitioned into four parts by connecting the interior point with the four midpoints of the sides, and then you knew 3 areas but needed the fourth? I posit that this is a situation that will never come up outside this specific contrived school exercise from the YouTube video. Furthermore, there are hundreds of similar contrived situations that could be cooked up, and covering the comparable identity for every one of them turns the associated article into a gigantic list of decontextualized trivial formulas, which readers will have great trouble making any sense of. (We have plenty of math articles which are already far too much like this; what our articles need is more narrative prose, not just more lists of formulas.) –jacobolus (t) 16:48, 5 February 2026 (UTC)

Absolutely. There are countless thousands of properties of geometric figures like this, many of which may be interesting, but Wikipedia is not a repository for extensive lists of interesting but unimportant little facts, and I see no reason whatsoever to regard this one as more significant than countless other little facts. JBW (talk) 23:33, 8 February 2026 (UTC)