Talk:Geometry

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Note 3 full citation is Greek and Vedic Geometry Frits Staal Journal of Indian Philosophy 27 (1/2):105-127 (1999)

A space is not always geometric

@D.Lazard: Revision https://en.wikipedia.org/w/index.php?title=Geometry&oldid=1144056819 added the text This enlargement of the scope of geometry led to a change of meaning of the word "space", which originally referred to the three-dimensional space of the physical world and its model provided by Euclidean geometry; presently a geometric space, or simply a space is a mathematical structure on which some geometry is defined. However, the word space can refer to mathematical structures that are not geometric, e.g., vector spaces over arbitrary fields. I'm not sure how it should be worded, since the term Geometry is itself murky. Shmuel (Seymour J.) Metz Username:Chatul (talk) 15:04, 12 March 2023 (UTC)

This depends of your definition of “geometric”. Currently, nobody pretends that algebraic geometry and finite geometry are not geometry, and vector spaces over a finite field belong to both areas. There is nothing murky in geometry. Simply, this is a scientific area and not a mathematical term, and, as such, it is not subject to a mathematical definition. D.Lazard (talk) 19:51, 12 March 2023 (UTC)

How is Geometry not a mathematical discipline? --Shmuel (Seymour J.) Metz Username:Chatul (talk) 13:47, 13 March 2023 (UTC)

We hear geometry-related words all the time: ‘what’s your angle?’ and ‘everyone should eat three square meals a day!’ and ‘she ran circles around me!’, often with little thought to how fundamental those shapes are to the discipline called geometry. Barrista hex (talk) 10:36, 20 December 2023 (UTC)

wanna learn from you... Barrista hex (talk) 10:32, 20 December 2023 (UTC)

Geometry just refers (except in very limited cases in NCG) to any set whose elements we can describe as "points" because in addition the set has some information about how its elements have a "position" relative to each other. "Space" is just a catch all term used to describe such structures, so I think its sort of tautological to say Geometry is the study of Spaces.

There's a more limited definition of geometry in the context of topology which refers to spaces with some particular kind of rigidifying geometric structure on them such as a metric, Riemannian metric, volume form, algebraic structure, etc. But I don't think that really applies to "Geometry" in the large. Tazerenix (talk) 23:09, 12 March 2023 (UTC)

I've never seen an Algebra text refer to the elements of, e.g., a vector space, a Fréchet space , as a point. --Shmuel (Seymour J.) Metz Username:Chatul (talk) 13:47, 13 March 2023 (UTC)

The requirement is not that a textbook refers to them as "points" but that there is a relation between elements which provides information about their relative position. In the case of a vector space, the relation is linear (you can specify when two elements lie along the same line). In particular there is an affine structure (and more, as there is a distinguished point at the "center", another positional relationship). Of course an algebra book will not think of vector spaces as spaces if its goal is to do algebra, but they certainly don't refer to them as "vector sets". Tazerenix (talk) 23:07, 13 March 2023 (UTC)

In Topology there is no concept of relative position. Does that mean that a topological space is not a space.? --Shmuel (Seymour J.) Metz Username:Chatul (talk) 08:58, 14 March 2023 (UTC)

Closeness is the basis of topology, and is a sort of relative position. However, although although Tazerenix's definition of points and spaces is ingenious, I am not sure that I completely agree with it, and it is WP:OR. So, it is better to say that space, point, geometry, geometric method, geometric space, etc. are what is so called by the community of mathematicians. These terms do not require to be formally defined as they are only used to provide an intuitive support to reasonnings, which otherwise would be more difficult to understand. For example, learning the axioms of vector spaces is easy, but understanding the richness of the concept cannot be done without considering the geometrical aspects of the concept. D.Lazard (talk) 10:31, 14 March 2023 (UTC)

See for example Kuratowski closure axioms in which topology is defined entirely using the concept of a point being "close" to a set. This is an example of information about the relative positions of points: If a point x is close to a set A and a point y is not, then x is closer to A than y! Tazerenix (talk) 22:58, 14 March 2023 (UTC)

Not so. None of the axioms refer to closeness. There is a derived concept of a point being close to a set, but none of the axioms use it. --Shmuel (Seymour J.) Metz Username:Chatul (talk) 00:23, 15 March 2023 (UTC)

If you define the relation "$x$ is close to $A$" as "$x$ is contained in $\mathrm {cl} (A)$" then the axioms of a topology can be specified as

1. No point is close to the empty set
2. Every point of $A$ is close to $A$
3. The points of $X$ which are close to $A\cup B$ are the points close to $A$ or to $B$
4. If a point $x$ is closeto the set of points close to $A$, then $x$ is close to $A$

A set $X$ with a relation between points and sets of "closeness" is equivalent to specifying a topology (precisely, define the closure operator $\mathrm {cl} :{\mathcal {P}}(X)\to {\mathcal {P}}(X)$ by $\mathrm {cl} (A):=\{x\in X\mid x{\text{ is close to }}A\}$). Tazerenix (talk) 02:23, 15 March 2023 (UTC)

Speaking as a topologist, I don't believe that every topological space ought to be described as geometric, however one might reasonably define the term. While there is, of course, a close connection between topology and geometry, I don't think topology is best described as a subset of geometry. Paul August ☎ 16:50, 13 March 2023 (UTC)

I would probably classify Topology as part of Geometry, although topologies not satisfying the separation axioms might be counter-intuitive. I could probably make an argument for considering it to be a part of Analysis, albeit a weak one. --Shmuel (Seymour J.) Metz Username:Chatul (talk) 17:19, 13 March 2023 (UTC)

Add IPA

How do I say this word? Please add IPA. 1.127.110.251 (talk) 11:03, 25 November 2023 (UTC)

See wikt:geometry. –jacobolus (t) 17:37, 11 June 2024 (UTC)

Geometry WikiProject proposal

View the proposal here. Writehydra (talk) 04:59, 2 February 2024 (UTC)

Should the main ideas section include notions important in Projective Geometry?

Shopuld the main ideas section include notions such as collineation, cross ratio, perspectivity, projectivity, that are important in Projective geometry? -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 17:22, 11 June 2024 (UTC)

What do you mean: removing the existing mentions of these topics or adding a new section? In the second case, what should be its title and where it should be added in the article? D.Lazard (talk) 17:41, 11 June 2024 (UTC)

I mean add them to Geometry#Main concepts. I don't see any existing mention of cross ratio, perspectivity or projectivity, and collineation only occurs in symmetry. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 18:12, 11 June 2024 (UTC)

Those are topics worth mentioning in a new section about projective geometry, but this section is already kind of awkward and overstuffed; I think adding a bunch more to it would be overwhelming. If someone has the time/energy it might be worth doing a more substantial reorganization and ideally rewrite of much of it. I don't think that Axioms, Constructions, Symmetry, and Rotation (among others) make sense as siblings in the same top level section. Overall this page is too focused on making a hierarchical list of random things, and not focused enough on telling a coherent narrative. –jacobolus (t) 19:58, 11 June 2024 (UTC)

Wrong statement

"The Pythagorean theorem is a consequence of the Euclidean metric." should be replaced by "The Pythagorean theorem is a consequence of the axioms of Euclidean geometry." Steamyer (talk) 11:01, 3 October 2024 (UTC)

Reversion of short description

D.Lazard, You reverted my edit of the short description from Branch of mathematics concerning properties of space, shape and position (72 characters) to Branch of mathematics with the edit summary "Reverted 1 edit by Pbsouthwood (talk): WP:SDSHORT and WP:SDNOTDEF". Please explain what part of WP:SDSHORT, and what part of SD:NOTDEF you think the new short description violated. Please take into account that there is other guidance on the content of short descriptions which must also be considered, in particular, how well the short description performs its WP:SDPURPOSE, which includes use as an annotation in lists of links, like see also sections and Index articles, where the current short description is not much use, and the revised one works much better. Cheers, · · · Peter Southwood ^(talk): 14:43, 17 November 2024 (UTC)

People should not ever use short descriptions in lists of links such as see also sections. The result is nearly always mediocre. If you need an annotation for a see also section, write it explicitly. –jacobolus (t) 14:49, 17 November 2024 (UTC)

This opinion is not a consensus opinion, and is unrealistic at best. Local annotations are notoriously unreliable, and seldom cited, whereas a reasonable short description is at least nominally backed up by a source in the home article. The amount of work needed to annotate an index with good annotations, each local, and each reliably sourced, is unrealistic, when a good enough substitute is available in annotated links. You need only look at how few links are annotated manually compared to templated versions, and when they are not good, fixing the short descriptions can improve a large number of articles simultaneously for very little work. Providing useful short descriptions is a gift that keeps giving. It aligns with the spirit of Wikipedia. Cheers,

· · · Peter Southwood ^(talk): 15:09, 17 November 2024 (UTC)

Short descriptions are only occasionally and incidentally "reasonable" for this purpose, and are subject to change at any time to become unsuitable for it, because the criteria for short descriptions are unrelated and largely contradictory to the criteria for good annotations in a list. I have never seen an example of a see also section annotated using {{annotated link}} where the results were not terrible. I'm not sure why people try to automatically populate these annotations in this way, but they should be encouraged to stop. –jacobolus (t) 15:49, 17 November 2024 (UTC)

Feel free to start an RfC any time you like, and please ping me if you do. Until then, it is an accepted and fairly common practice, as it is simple, efficient, effective and in all cases that I have seen, good enough and better than no annotation. It is easy enough to fix the bad short descriptions if no-one reverts you, which is almost always the case. YMMD. Cheers, · · · Peter Southwood ^(talk): 18:05, 17 November 2024 (UTC)

Can you provide links to a few of these see also sections with annotated links that are so terrible? Maybe we edit very different topic areas, and our experiences may differ for that reason. Cheers, · · · Peter Southwood ^(talk): 18:15, 17 November 2024 (UTC)

It's a rarely used practice with consistently bad results (worse than no annotation) which should be removed whenever it is found. When people add such lists in articles I pay attention to, and I feel even slightly motivated, I revert them, sometimes writing manual annotations. Often it's a good excuse to thin out the see also sections, which in such cases are often also overstuffed with links that don't belong.

Some people started doing this out of a misguided desire to "save work" by re-using one tool for a different job without ever thinking about the consequences, but the tool is not suited for the task.

Such additions were never the subject of any widespread agreement among Wikipedians, but most editors don't care enough to stop people from doing ineffective and reader-unhelpful things because they have better things to do, so plenty of such practices stick around even when they are clearly bad ideas. I urge you to tell editors employing the {{annotated link}} template to stop because they are wasting their own time and (marginally) harming the project. –jacobolus (t) 18:36, 17 November 2024 (UTC)

Thank you for clarifying your own opinion of what a short description should be. However it contradicts guidelines that have been established after a consensus of many editors. So, that is the guidelines that prevail here.

WP:SDNOTDEF says When visible on desktop or mobile, the short description immediately follows the article title, and should be comprehensible in that location. A short description is not a definition, and editors should not attempt to define the article's subject nor to summarise the lead. You added a definition of geometry, which is rather controversal and not useful in a short description, since everybody has some idea of what is geometry. So WP:SDNOTDEF applies fully.

WP:SDSHORT says Fewer than 3% of short descriptions are longer than 60 characters, and short descriptions longer than 100 characters will be flagged for attention. Short descriptions exceeding 40 characters may be truncated in some contexts. Because they are intended to be scanned quickly, longer, more specific descriptions can be less useful. As your short definition has 72 characters (if I remember correctly), WP:SDSHORT applies fully.

So, these are two different strong reasons for reverting your edit. D.Lazard (talk) 15:21, 17 November 2024 (UTC)

I am reasonably familiar with the guidelines, having been closely involved in the original drafting and several more recent discussions where they were fine tuned a bit due to some editors having difficulty understanding what they do and do not say. I notice that you quote only part of each of the two guidelines you invoked, and may have misinterpreted the statistics as meaning more than they actually say, so I will attempt to clarify.

the statistic that less than 3% of short descriptions were longer than 60 characters at the time of the analysis is a simple statistic, and does not indicate anything about the quality of the shorter and longer short descriptions at that time or now, so is irrelevant to this dispute. Short descriptions longer than 100 characters are indeed flagged for attention, as there is a good chance that most of them can be shortened, often without reducing their utility. I have done just that on many occasions. They often ended up between 80 and 100 characters long, and were better than the longer version, and still fit for purpose. If you read WP:SDNOTDEF carefully you will see that it explains that a short description does not have to be a definition, but if it is, that is quite acceptable. What is not acceptable is claiming that definitions are not acceptable as short descriptions as that contradicts the explicit meaning of the guideline. Note that where SDSHORT states be short – no longer than is needed to fulfill its functions effectively this implies that it should not be shorter than needed to fulfill those functions effectively.

If you disagree with the wording of the non-definition I gave, the appropriate response is to improve it without reducing its usefulness, not to revert to a less useful version.

You claim that "everybody has some idea of what is geometry"'. Can you cite a reliable source supporting that claim? (or even provide original research) Even if it is true, the most likely thing they will already know is that it is a "Branch of mathematics", so the current short description will not add much value for them, and will help very little on a category page for articles on mathematical topics.

You state However it contradicts guidelines that have been established after a consensus of many editors Please state exactly which guideline/s it contradicts. I was one of the "many editors" you mention in those consensus discussions, and drafted several of the consensus conclusions, so I would be interested to see how you came to this conclusion.

I will also mention that {{Annotated link}} is used on about 10 000 pages, and by its nature is often used multiple times on a page, sometimes more than a hundred times on a page, in large lists, which suggests that a fair number of editors consider it useful for annotation, and that the quality is not as dire as some may claim. I could link to examples if it would help you. There are even other templates to do similar things in slightly different circumstances, also using the short descriptions. For example category lists can display short descriptions as annotation, making them considerably more user-friendly, and it helps there too if the short description is usefully informative. Using short descriptions as annotation puts more eyes on them, and should lead to more editors improving them when they are poorly written., and like Wikipedia in general, some are poorly written, but can be improved. The fact that some mobile app/s truncate short descriptions at about 40 characters is a problem with the apps. We do not write Wikipedia for the convenience of some app coders who choose to truncate our work. Cheers, · · · Peter Southwood ^(talk): 17:55, 17 November 2024 (UTC)

The purpose for which short descriptions were created, and the only purpose for which they are well suited, is to disambiguate similarly titled articles in app search results. So yes we do write these "for the convenience of some app" and if you don't care about app search annotations you should leave the short descriptions alone.

Overall, short descriptions are a substantial waste of editor attention which we'd be better off not having at all – they cause page history spam and pointless edit wars and arguments and waste far more attention than their very marginal benefit is worth.

But at any rate we should strongly discourage their proliferation into other places than app search results, which causes people to start trying to optimize the text for several unrelated purposes in contexts with contradictory needs, edit warring about whether more extensive descriptions are sufficiently neutral and supportable by sources, arguing about which aspects of a topic are worth mentioning, etc. –jacobolus (t) 18:50, 17 November 2024 (UTC)

By the way, the short description I added that you called a definition, was just taken from the lead paragraph. It was not intended to be a definition, and if it is somewhat controversial, maybe the lead needs revision. · · · Peter Southwood ^(talk): 18:19, 17 November 2024 (UTC)

The main use of short descriptions, and the main reason they must be short, is to add a short gloss to search results in the mobile app, so that people using the app can tell which of several similarly titled articles is the one they want. Your elaboration does not help that purpose; it just makes the short description too long. —David Eppstein (talk) 21:22, 17 November 2024 (UTC)