Wikipedia talk:WikiProject Mathematics/Archive/2022/Jun

Jörg M. Wills
I recently created a draft for German mathematician Jörg M. Wills. Any help would be appreciated. Thank you, Thriley (talk) 03:28, 7 June 2022 (UTC)

"The most remarkable formula in mathematics" listed at Redirects for discussion
An editor has identified a potential problem with the redirect The most remarkable formula in mathematics and has thus listed it for discussion. This discussion will occur at Redirects for discussion/Log/2022 June 7 until a consensus is reached, and readers of this page are welcome to contribute to the discussion. By the way, I don't know the format for adding RfD to WikiProject Deletion sorting/Mathematics. Sorry … SilverMatsu (talk) 09:21, 7 June 2022 (UTC)
 * Update: Add "The Most Remarkable Formula In The World" to the list.--SilverMatsu (talk) 06:29, 8 June 2022 (UTC)

Lobachevsky
Could somebody with rollback rights revert the edits made to Nikolai Lobachevsky by ? (plain vandalism, but spread around enough that it's best not to undo by hand). jraimbau (talk) 05:12, 10 June 2022 (UTC)
 * Done. Rollback wouldn't help. For this sort of thing, the easiest thing to do is look back through the history for the last good version, edit that version, and save it over the vandalized versions. —David Eppstein (talk) 07:23, 10 June 2022 (UTC)
 * Thank you! that makes sense, i'll keep that in mind. jraimbau (talk) 13:09, 10 June 2022 (UTC)

Mathematician ITNRD nomination needing attention
There is an ITN RD nomination regarding the recent death of Aleksei Parshin which has not seen much input, perhaps due to the technical nature of some of the article content. The nomination may be of interest to this WikiProject, so your additional input is appreciated. Thanks. — MarkH21talk 08:47, 23 June 2022 (UTC)

Euler-alpha equations ?
I don't think Euler's formula are also known as Euler-alpha equations. So I think it needs to be retargeted, but I don't know the target. --SilverMatsu (talk) 04:24, 22 June 2022 (UTC)
 * Then take it to RfD. –LaundryPizza03 ( d c̄ ) 06:33, 22 June 2022 (UTC)
 * There appears to be something called the Euler alpha equations in fluid dynamics, some kind of perturbation of Euler equations (fluid dynamics), but they don't seem to be mentioned at that article. In any case that meaning, if it is notable, is unrelated to the current link target. —David Eppstein (talk) 07:43, 22 June 2022 (UTC)
 * Thank you for your comments. I will report when RfD is ready. --SilverMatsu (talk) 08:25, 22 June 2022 (UTC)
 * Done ! (Redirects for discussion/Log/2022 June 22) thanks !--SilverMatsu (talk) 08:43, 22 June 2022 (UTC)

Content dispute at Sine and cosine
The Sine and cosine article is now fully protected because I and MrOllie can't reach an agreement. It concerns an inclusion of this article. This is the difference between the version proposed by MrOllie and the version proposed by me. Please help us to resolve this dispute on the relevant Talk page. A1E6 (talk) 14:37, 22 June 2022 (UTC)

"Complex exponential" listed at Redirects for discussion
An editor has identified a potential problem with the redirect Complex exponential and has thus listed it for discussion. This discussion will occur at Redirects for discussion/Log/2022 June 23 until a consensus is reached, and readers of this page are welcome to contribute to the discussion. SilverMatsu (talk) 03:49, 23 June 2022 (UTC)

Malfatti circles
I've been trying to beat back a very stubborn and very promotional editor on Malfatti circles who insists that all previous papers claiming a certain theorem are somehow unsatisfactory, that all later papers referring to those previous papers as being rigorous solutions are incorrect, that only a brand-new publication from 2022 (presumably, by the editor in question) counts as a valid solution, that their own edits to other-language Wikipedias count as evidence for these assertions, and that more than merely citing this new publication among others claiming solutions we must proclaim it to be the only true solution in the text of the article. Assistance here would be welcome. —David Eppstein (talk) 17:20, 22 June 2022 (UTC)


 * Is there an openly readable version of this work somewhere? It’s paywalled, not in sci-hub, not yet in Google Scholar or other indexes, and I don’t feel like shelling out $40 to read it. With nothing more substantial than a link getting added to Wikipedia, it’s impossible to evaluate any claims the anonymous contributor makes here. –jacobolus (t) 19:22, 22 June 2022 (UTC)
 * I have subscription access to it and could email a copy if you want. (Obviously, I don't think it would be a good idea to make a copy public rather than merely emailing privately.) It is a published paper, clearly relevant, so I think it should be cited. It is the claims that all previous solutions were faulty and now is the first solution of the problem that I find overblown. —David Eppstein (talk) 19:27, 22 June 2022 (UTC)
 * That's okay. I am happy to take your word for it. –jacobolus (t) 21:50, 22 June 2022 (UTC)
 * The Wikipedia Library. --SilverMatsu (talk) 01:33, 23 June 2022 (UTC)
 * Having access to the article as well, I briefly looked through their claims. As far as I can tell, the numerical calculations that the 1994 paper rely on are involved and not pretty, but they are not "simulations" in any sense of the word which would detract from their propriety as steps in a mathematical proof. They seem to be just numerical approximations "to n decimal places", which is a precise fact with precise consequences. So I agree with David Eppstein on both counts: The new proof is clearly worth a mention, but not as the "first published proof" of the conjecture, at least unless future third party sources (ideally surveys or review articles) agree with IP's reading of the situation. Felix QW (talk) 20:05, 22 June 2022 (UTC)
 * Agreed. It seems to be like saying that it isn't rigorous to use a computer to say that $$\sin(12.8)>e^{-\pi}.$$ (I believe it is almost inarguably rigorous and formal to do so.) Gumshoe2 (talk) 04:06, 23 June 2022 (UTC)

Geometric algebra as a duplicate article of Clifford algebra
The article Geometric algebra is about the same object as the one covered in the article Clifford algebra, but presented using different notation and perspective. Should the two article be merged according to WP:OVERLAP? Should Geometric algebra be turned into something more clearly separate in scope like Geometric models of Clifford algebras?

Both article are quite long, but it looks like unnecessary material from both current articles can be easily cut to form a single well-written article from a quick glance. — MarkH21talk 09:46, 27 June 2022 (UTC)


 * As someone who is not a mathematical physicist, I cannot really judge the potential of a separate article on 'geometric models of Clifford algebras', but I certainly agree that the current state is untenable. So I would support a merger of what we have, if someone were willing to take on the requisite cutting of material. Felix QW (talk) 10:04, 27 June 2022 (UTC)
 * I think the way the object (the exterior algebra of a vector space) is talked about is very different from the two perspectives. The "Clifford algebra" language is all very geared towards describing spin representations and Dirac operators, whereas geometric algebra is very oriented towards describing the extended geometric constructions that an inner product on a vector space gives you. The fact that from one perspective you can understand spinors as certain objects in geometric algebra is interesting, but not really the way they get thought about (rather as certain representations of Spin groups, which the "Clifford algebra" language always emphasizes).
 * For example, I'm sure it would be very confusing for physics articles if the page on Clifford algebras spent half its time talking about unrelated geometric algebra constructions before completely shifting the language to describe the spin representations in the Clifford algebra (and similarly anyone looking for a fun introduction to geometric algebra would be very confused by all the detail about Spin/Pin groups and other language clearly set up for someone studying Dirac operators on manifolds).

A fresh perspective: a Clifford algebra is a mathematical structure, and abstract mathematicians seem to know exactly what they mean by the term. Keeping aside for the moment "geometric algebra", nominally the study of "geometric algebras", it seems to me that those that use the term "a geometric algebra" usually think they are talking about a structure, namely a Clifford algebra (usually over the field of real numbers). They do not appear to realize that they really seem to mean the use of a Clifford algebra as a representation of a geometry with its properties – that is, the correspondence of features of an algebra to model aspects of a geometry. For example, by a CGA is meant a specific mapping between elements of a Clifford algebra and points, circles, etc., and the transformations of a conformal geometry. As such, the subject area "geometric algebra" is the study of such correspondences and their application, which one could regard as belonging to applied mathematics. Given this perspective (which I do not claim to be able to source), the most valuable article Geometric algebra that we could have would deal with the application of Clifford algebras to express geometric problems (for which vector algebra, Pauli algebra, Dirac algebra, etc., are also used). An introduction of geometric algebra that addresses vector algebra problems alone would be very helpful to the lay reader – which is something that would not belong in Clifford algebra. With the intuition of bivectors (oriented areas) to replace pseudovectors, etc., this article could act as a reference for people who want to find out about what geometric algebras are good for. I agree with jacobolus that the current lead totally misses the right approach, however one looks at it. 172.82.46.195 (talk) 00:17, 29 June 2022 (UTC)
 * Whilst the actual object (exterior algebra of a vector space with inner product) is the same in both articles, I think the topic is different. I don't really think they should be merged unless someone can find a very good source which manages to explain in a way that isn't confusing to spin geometers/physicists or elementary geometric algebra people how the two subjects are the same. I seriously doubt the existence of such a source. Tazerenix (talk) 10:10, 27 June 2022 (UTC)
 * I see your point, but I do want to at least say that we shouldn't be writing the articles as a textbook-like fun introduction. An encyclopedic article can cover multiple perspectives in different sections. Also, re spent half its time talking about unrelated geometric algebra constructions before completely shifting the language, sections can be definitely self-contained and introduce new notation; this is quite normal in my experience (as long as it is made clear that there is a shift and a brief indication of why there is a shift). — MarkH21talk 05:24, 28 June 2022 (UTC)
 * Geometric and Clifford algebras are not the same object. In physics, the Hestenes-type of geometric algebra is a type of Clifford algebra with a real-valued vector basis. See the papers and  for a comparison of the two types of objects. Because geometric algebras in physics are a subset of Clifford algebras and as Tazerenix notes, they are typically used for different purposes in physics, I think it would be better to keep the topics as separate articles. There is already a section Clifford_algebra on geometric algebra in the Clifford algebra article, which I think is the right approach to linking them. Part of the challenge is that some people do talk of complex geometric algebras and so "geometric algebra" means different things to different groups. --  17:23, 27 June 2022 (UTC)
 * The point of view is quite different. Books about “Clifford algebras” start from a pure math grad student kind of audience (say, someone who has taken courses in linear and multilinear algebra, abstract algebra, complex analysis, differential geometry, Lie theory, ...), defining very general/abstract objects in terms of tensors and dual spaces using highly abstracted proofs, and typically starting from complex numbers as a scalar field; Clifford algebra are then seen as a niche special-purpose tool, just one among many others. Hestenes’s “geometric algebra” (Clifford’s own name for the subject, btw) can start as a subject aimed at a high-school-level audience building on the basic notions of vectors and introductory geometry (and even for more advanced audiences, eschewing abstract machinery to the extent possible), and always using “real” numbers as scalars; it considers geometric algebra to be a fundamental and unifying single language, in terms of which most (all?) other geometric tools can be built or described. Cf. “Reforming the Mathematical Language of Physics”, “Grassmann’s Vision” “Mathematical Viruses”. –jacobolus (t) 22:46, 27 June 2022 (UTC)
 * Aside: No offense intended to the authors, but the current lede for geometric algebra is incredibly unfriendly to a lay Wikipedia-reader audience, while also largely of missing the point of GA: In mathematics, a geometric algebra (GA) is another name for a Clifford algebra Cl(V, g) of a vector space V with a quadratic form g over a field of scalars F. It is an algebra over F generated by the vector space V.... The first few sentences here should have no mention of quadratic forms or fields (and the whole article should focus primarily if not exclusively on “real” scalars), and does not need letters or symbols. Vector division should probably be mentioned somewhere near the top. It might instead say something along the lines of “In mathematics, geometric algebra (also known as real Clifford algebra) is an extension of elementary algebra to work with geometrical objects such as vectors. Geometric algebra is built out of two fundamental operations, addition and and the geometric product. Multiplication of vectors results in higher-dimensional objects called multivectors. Compared to other formalisms for manipulating geometric objects, geometric algebra is noteworthy for supporting vector division and addition of objects of different dimensions....” The rest of the lede section that I didn’t quote here is way too long and link-heavy (most of it can be transplanted to later sections or removed), and the early sections of the article are too jargony, technical, and unfocused. –jacobolus (t) 00:46, 28 June 2022 (UTC)
 * For anyone from a mathy background who wants to learn about GA, let me recommend Chisolm (2012) “Geometric Algebra”. –jacobolus (t) 01:05, 28 June 2022 (UTC)
 * Thanks for all the replies! I'm not very familiar with the perspective given in geometric algebra, so this has been quite helpful. The suggestions of sound quite good. If it is indeed the case that geometric algebra is mostly used in RSes for Clifford algebras over the real numbers, then I think that the article (or at least the lead) should be revised so that this restricted scope is clear from the beginning. — MarkH21talk 05:18, 28 June 2022 (UTC)
 * I think it's important to distinguish a geometric algebra (the type of structure) from geometric algebra as a mass noun (presumably the equational theory of those structures, or some such). At least it's really important for Boolean algebras, because the structures differ in important ways, whereas there's only one equational theory.  I don't know nearly as much about geometric algebra(s), but if there's a distinction to be made then it should be made clearly, and the structures would likely merit a separate article, along the lines of Boolean algebra (structure). --Trovatore (talk) 06:55, 28 June 2022 (UTC)
 * I think it is important to elide/suppress this difference for the article geometric algebra, because (a) it is irrelevant and confusing to lay / nonspecialist readers and emphasizing it clutters the narrative, and (b) a focus on “a geometric algebra” as some particular precisely defined formal structure obscures the fundamental point that “geometric algebra” is a common unifying language which can be widely employed under multiple interpretations in different contexts, but using common algebraic identities/manipulations (revealing some commonality in apparently different situations), and (c) if you really want to you can think of any “a geometric algebra” as a sub-algebra of a single universal geometric algebra which encompasses it and all the others (though I think a discussion of this should be considered out of scope for the wikipedia article). –jacobolus (t) 07:14, 28 June 2022 (UTC)
 * As I say, I'm not an expert in this area, but I have trouble believing that it's ever irrelevant. A theory and its model are utterly different things.  If you want to talk about the language, fine, but the structures are something different. --Trovatore (talk) 07:16, 28 June 2022 (UTC)
 * To put it another way, you can just not talk about the structures in a particular article if you don't want to. But you can't "elide the distinction".  That's like eliding the distinction between words on the page and the things the words are talking about. --Trovatore (talk) 07:18, 28 June 2022 (UTC)
 * It’s more like eliding the difference between “language” and “a language”. –jacobolus (t) 16:16, 28 June 2022 (UTC)
 * No, it's not like that at all. --Trovatore (talk) 16:21, 28 June 2022 (UTC)
 * A closer analogy would be it's like conflating elementary algebra with the real numbers. --Trovatore (talk) 21:09, 28 June 2022 (UTC)
 * No, the analogy would be conflating “elementary algebra” (a language) with “the algebra of real numbers” (a formal structure relating objects on which that language can be used); both of these are different than a real number. A geometric algebra (a formal structure establishing a domain in which the language of geometric algebra can be applied) is not the same as an element of that algebra (typically called a “multivector”). –jacobolus (t) 22:09, 28 June 2022 (UTC)
 * I didn't say "a real number"; I said "the real numbers". --Trovatore (talk) 22:18, 28 June 2022 (UTC)
 * Okay, but here “the algebra of the real numbers” and “the real numbers” formally mean the same thing (I guess up to isomorphism). The distinction is certainly important and meaningful, and it’s fine to talk about a formal definition for a geometric algebra. It’s just not that helpful to belabor the point in an article aimed at newcomers. If you started an introductory algebra course for middle school students by first coveringthe material from an undergraduate level abstract algebra course to make sure they had their terms formally precise, most of them would be bored and confused. –jacobolus (t) 22:29, 28 June 2022 (UTC)
 * I'm not distinguishing between "the field of the real numbers" and "the real numbers". I'm distinguishing both of those from "elementary algebra" (meaning the symbolic manipulations).  This shouldn't even be a question; of course those are not the same thing, not even remotely comparable, and writing that confuses them is only going to lead to misconceptions. --Trovatore (talk) 22:59, 28 June 2022 (UTC)
 * I feel like you are missing my point. The subject of the article at geometric algebra should be geometric algebra (analogous to elementary algebra), not a geometric algebra (analogous to the field of real numbers, if you like). In the context of that topic, it maybe worth defining what a geometric algebra means somewhere (just as it might be worth defining, somewhere in elementary algebra, what the field of real numbers is – though note that article currently does not do so), but the point should be to describe the language of geometric algebra, and to that end belaboring the details of the formal definition of a geometric algebra is a distraction. –jacobolus (t) 23:12, 28 June 2022 (UTC)
 * As I said, if you want geometric algebra to be about the symbolic stuff, and not about the structures, that's potentially OK. In that case you probably need a separate geometric algebra (structure) article, or some such, and a hatnote pointing to it.  I don't have any strong opinion on that.  I have a very strong opinion that we must not confuse the two. --Trovatore (talk) 23:23, 28 June 2022 (UTC)
 * I think it is sufficient to leave Clifford algebra as a topic specifically about the formal structure. –jacobolus (t) 00:14, 29 June 2022 (UTC)
 * Mm, maybe so. As an aside, though, it's a little odd to me that you keep associating the word "formal" with the structures.  Aren't the equations more "formal"?  The structures seem more Platonic than formal. --Trovatore (talk) 00:26, 29 June 2022 (UTC)
 * But in any case, yes, I agree, it's the same as conflating elementary algebra with the field of real numbers ("field" is a better choice than "algebra" here; it's also "an algebra" in some sense but not a very relevant one) . And that is an absolutely unacceptable conflation!  We must not do that. --Trovatore (talk) 22:22, 28 June 2022 (UTC)
 * Seems reasonable. Though I would spend the first few sections primarily talking about the “vector model” of Euclidean / pseudo-Euclidean geometry in 2–4 dimensions, and put discussion of using GA to represent other kinds of geometric objects later (can later discuss projective geometry, affine geometry, the outermorphism of a general linear transformation, conformal geometry, multivector-valued functions on manifolds, etc.). –jacobolus (t) 01:22, 30 June 2022 (UTC)