Talk:Descartes' theorem/GA1

GA Review
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Reviewer: Kusma (talk · contribs) 21:17, 11 March 2024 (UTC)

Planning to review, should not take more than a few days. —Kusma (talk) 21:17, 11 March 2024 (UTC)


 * Thanks! Don't feel like you need to rush; I am likely to be a bit busy with real-world work and slow to respond myself. —David Eppstein (talk) 22:26, 11 March 2024 (UTC)

Content and prose review
A nice article! I will comment on the GA criteria and have a look at some more of the sources in a bit. —Kusma (talk) 07:30, 16 March 2024 (UTC)
 * Lead: The "bends" are signed inverse radii; not sure whether this detail should be here.
 * The poem doesn't appear in the body; generally I prefer not to have extra stuff in the lead. Maybe you can have a little more of the poem in the body, perhaps something explaining about positive and negative curvatures or how "a fifth sphere in the kissing shares" in the 3d case.
 * History: Can't find the claim that Descartes' reasoning was "lost" in Coxeter, just that there is no complete proof in the letter. (From a quick skim read of the French original letter, it looks like there is very little proof there, if any).
 * Gosset extended the theorem and poem: would be nice to be told that is was just a year later (1937) and you could consider citing the relevant lines of the poem somewhere.
 * Actually, according to Martin Gardner, Gosset was not the only one to write an n-dimensional poem about this.
 * "killer" problem: Not a fan of the one sentence paragraph, and while it is interesting trivia, it is a bit out of place here.
 * Solving the quadratic: is there any reason the term under the square root should be positive? What if it isn't?
 * Locating the circle centers: The EquationNote template linking to the equations is nice, but the many bold faced equations are a bit distracting. Abbreviate as "eq." or take the word "equation" outside of the template so it is not in blue and boldface?
 * In the equation for z4, the roots are now complex; worth mentioning?
 * Integer curvatures: Just to clarify, is the root quadruple the one with the smallest signed curvature or the smallest positive curvature? (I.e. is everything inside one bounding circle? The image certainly suggests so).
 * Arbitrary four-circle configurations: perhaps spend a word or two explaining what +1 and -1 mean as "relative orientation between the ith and jth oriented circles".
 * The equation $$\mathbf{k}^\mathsf{T}\mathbf{Q}^{-1}\mathbf{k} = 0$$ isn't this worth attributing in text? Similar for some of the generalisations.


 * signed inverse radii – I think it's fine to skip 'signed' in the lead, which is explained clearly enough in context. reasoning ... lost – it might just be better to say that Descartes didn't fully describe his reasoning. is there any reason the term under the square root should be positive – In the event that the three circles are all mutually tangent, then yes it is always non-negative. The case where they are externally tangent has all of the ks positive. The case where two circles are nested inside another the product of the ks for the two smaller circles must be at least as large as the sum of products of each with the k for the circle enclosing them, with equality when the two internal circles each have half the radius of the enclosing circle. Then $2\cdot 2 - 2 \cdot 1 - 2 \cdot 1 = 0 $ and the two other "bends" are both $2 + 2 - 1 \pm \sqrt{0} = 3.$  I'm not sure how helpful it is to discuss this. root quadruple – judging from Apollonian gasket a "root quadruple" includes the bounding circle.  –jacobolus (t) 15:06, 16 March 2024 (UTC)
 * Thank you! I didn't stop to think why it is clear the term under the square root can't be negative. In any case this means the formula is always valid without further qualifications, so there is probably no need to do anything here. —Kusma (talk) 18:19, 17 March 2024 (UTC)

General comments and GA criteria
Just source checks left to do. —Kusma (talk) 08:02, 16 March 2024 (UTC) Spotchecks: Sources are fine with just tiny comments, no original research, no problems of copying or close paraphrasing detected. Will put on hold just so you get a notification that I'm done reviewing, not because I think it will take a long time to fix the little issues I found. —Kusma (talk) 10:28, 16 March 2024 (UTC)
 * No major prose issues.
 * Lead is perhaps a bit short, other than including the poem.
 * Layout is fine.
 * Article scope and focus are fine. No neutrality/stability issues detected.
 * Images are fine and relevant, with the only possible question whether File:Generalized_circles_in_the_hyperbolic_plane.png is helpful here.
 * All free and nicely captioned.
 * 6 fine,
 * 15a doesn't really say that it is now called "Soddy's hexlet"; perhaps use 47 again as in 15b?
 * 16 could be used to clarify that Gosset was not the only one to generalise to higher dimensions.
 * 19 fine
 * 44, 45 and similar could be separate footnotes instead of mixed with the citations, but that is definitely optional.
 * 45 I am wondering whether it is helpful to say when Caratheodory originally published this (he was long dead in 1954).
 * 46 ok
 * A few replies, not intended to cover the entire review:
 * Re gloss of bends in lead: added "signed".
 * Re including more of the poem: I think it is still under copyright, so extended quotes would be problematic.
 * Re Descartes' labors lost: reworded as "Descartes did not provide the reasoning".
 * Re Gosset: added "the following year" and mentioned that he was one of several to do so, but again the poem is likely under copyright. Added the original version of the Gardner ref (easier to find for those with JSTOR access).
 * Killer problem removed.
 * Jacobolus has explained already the sign of the square root. I'm not sure there's much we can expand on this in the article itself without sources.
 * —David Eppstein (talk) 01:50, 17 March 2024 (UTC)
 * For the poem, Soddy died in 1956 so it will become free on 1 January 2027. Citing four lines instead of two would still be perfectly acceptable though. Gosset died in 1962 so it is 1 January 2033 for his poem. Awaiting your responses to my other points (none of them showstoppers). —Kusma (talk) 18:11, 17 March 2024 (UTC)
 * Inre File:Generalized_circles_in_the_hyperbolic_plane.png, the reason I made an image like this is that it's not immediately obvious (at least to me) that the kissing "circles" in the hyperbolic plane can include not just circles but also horocycles, hypercycles, and geodesics, and that the resulting "bends" $k_i$ can be interpreted based on the inverse (Euclidean) diameter under stereographic projection (conformal disk model). It would be nice if we had an article or section somewhere on Wikipedia describing more clearly what curvature of a curve or generalized circle means in spherical/hyperbolic geometry, various ways of defining it and their relations, etc.; I don't consider myself any kind of expert in this topic, and I don't know any particularly clear descriptions in the academic literature.
 * I guess the image could perhaps seem a bit out of scope insofar as we don't give any examples of kissing (generalized) circles in the hyperbolic plane, which might make it seem like unmotivated detail in a section which is otherwise quite condensed. I still think it's helpful, but YMMV. –jacobolus (t) 17:19, 17 March 2024 (UTC)

By the way, since I don't see it anywhere here: jacobolus made significant improvements to the article in the lead-up to its GA nomination and should get credit for that if this passes GA. I nominated this myself but am happy to see jacobolus's continued involvement in the GA review process. —David Eppstein (talk) 20:29, 17 March 2024 (UTC)


 * (I don't need any awards. I'm just happy to see articles improved.) @Kusma thanks for taking the time to make this review. –jacobolus (t) 20:51, 17 March 2024 (UTC)
 * Thank you @jacobolus for helping with the article! Formal recognition or brag sheets are indeed secondary to article improvement (getting a thorough GA review is more important to me than having another green plus to display on my user page). But it can help slightly with your standing inside Wikipedia so it is not just a question of getting a reward sticker.
 * @David Eppstein: remaining points are mainly whether to use the poem in the body and a few possible clarifications that should not take too long to resolve. Additionally there is a WP:WTW, namely "unfortunately", which runs afoul of MOS:EDITORIAL. Better to drop it. —Kusma (talk) 09:23, 18 March 2024 (UTC)
 * Dropped. More when I have more time to respond. —David Eppstein (talk) 15:51, 18 March 2024 (UTC)
 * As for expanding our quote of the poem later in the article: I'm not entirely persuaded that we have a valid fair-use case for doing so, and I think the quote is part of a summary of later material already: it is used to point towards the discussion of the poem in the history section, but it is also a readable and non-technical summary of the theorem itself. I've started a discussion thread asking about this at . —David Eppstein (talk) 06:38, 19 March 2024 (UTC)
 * Personally I don't really see an issue with the current use of the poem in the lead. The line from the poem is mainly used as a convenient device for stating the theorem in English words instead of introducing a bunch of symbols in the lead section; the full text of the poem isn't really that important to the article, and is linked from the footnotes for any reader curious to read it. We could plausibly write a new prose statement of the theorem, but I doubt it would be any clearer. –jacobolus (t) 07:09, 19 March 2024 (UTC)
 * Re "perhaps use [barnes] again as in 15b?": done (causing footnote renumbering). Re "[gardner] could be used to clarify that Gosset was not the only one to generalise to higher dimensions": this was already done when the Gardner reference was added: "Gosset and several others extended the theorem".
 * Re potentially relegating renumbered footnotes [45] and [46] (consisting of both a citation and some text describing how the citation supports the cited claim) into a separate notes section and then splitting off the citations from these footnotes into footnotes-within-footnotes: see ongoing discussion at and my comments there "This binary distinction between footnotes that are just extended body text and footnotes that are purely citations to references is not supported by reality.")
 * Jacobolus found and added an orig-year for the Caratheodory cite.
 * I think that mainly leaves the lead length query. —David Eppstein (talk) 07:14, 19 March 2024 (UTC)
 * Personally I freely add text to "ordinary" Wikipedia footnotes any time it seems like the text would be helpful to readers but is distracting or out of scope for the main body text, on all sorts of articles, most of which have no separate "textual notes" section distinct from the "bibliographic detail notes" section. I think it would be a mistake to force a uniform format choice across the project (cf. MOS:VAR). –jacobolus (t) 13:59, 19 March 2024 (UTC)
 * Sure, there are many ways to do this, and we do not have to observe the paper-based dichotomy of "explanatory footnote directly on the page, citations referring to bibliography at the end" (it would be nice to have to have support for this distinction for printed Wikipedia articles, but I don't think it exists). I did say I consider this "optional" above, and I mean it. —Kusma (talk) 21:00, 19 March 2024 (UTC)
 * I'm not sure which "paper-based dichotomy" you mean. Books and research journals have an even wider range of practices than Wikipedia articles. Many freely mix bibliographical information with textual notes, using either footnotes, sidenotes, or endnotes. Some use a combination of footnotes and end notes for different types of textual notes. Others use numbered or author/year parenthetical references referencing a bibliography list along with textual notes using some other format. Some use per-chapter endnotes while others put all the notes at the end of the book. Some bibliography lists are very compressed auth. (yr.) journ. abbrev., vol.: pg., with no paper title. Others include detailed annotations about each bibliography entry. Etc.
 * If someone wanted Wikipedia articles to render better for printing it would be possible to make with CSS, but a significant challenge to handle all of the range of possible edge cases. (It would be great to see a same-page footnote version, with the notes repeated any time they are referenced again and haven't appeared within the current or previous printed page.) But I'd probably recommend starting by fixing the crummy new default skin to improve rendering on screen.
 * (Sorry if this sounds argumentative or seems off topic. I guess my main point would be that I don't think GA criteria should push for change between otherwise acceptable citation formats unless there's some very obvious issue causing trouble for readers.) –jacobolus (t) 00:21, 20 March 2024 (UTC)
 * @David Eppstein, I think our remaining difference of opinion is not actually something that should stop the article from passing, so I will do that now. I guess one can argue that the article does not violate MOS:LEAD's "significant information should not appear in the lead if it is not covered in the remainder of the article" as the poem is covered, even if without quotes. That said, I do hope there will be more of the poem in the article once it becomes PD. —Kusma (talk) 17:47, 23 March 2024 (UTC)