Talk:Euclidean space

Definition in the lead
The lead reluctantly gives out a definition only in the last paragraph. Definitions should appear in the opening sentence. The short description is equally vage: "Fundamental space of geometry". Can we anticipate some sort of definition, e.g., "Euclidean space is a real coordinate space equipped with the Euclidean metric"? If it's too technical, how about something more informal, like: "Euclidean space is the mathematical space obeying the Euclidean distance formula"? fgnievinski (talk) 03:38, 4 May 2022 (UTC)
 * The first sentence is This is not a definition in the mathematical sense, but it follows exactly the guideline MOS:OPEN (}; that is, it identifies clearly and unambiguously the subject of the article in a way that is understandable to everybody. The same applies to the short description, which is aimed to help to recognize the searched article in the result of a search query. This aim is clearly and perfectly fulfilled by the present short description.
 * Your sugggestions are not only much too technical for the lead, but are also mathematically wrong: A "real coordinate space equipped with the Euclidean metric" is certainly a Euclidean space, but the converse is wrong: A Euclidean space where coordinates are not defined is not a coordinate space. Similarly, in "the mathematical space obeying the Euclidean distance formula", the definite article "the" is wrong, as the linked article uses "a space"; also, "Euclidean distance formula" is not defined in Wikipedia, and as far as I know, it is not definied either in any reliable source. D.Lazard (talk) 11:21, 4 May 2022 (UTC)
 * The lede currently says that three-dimensional Euclidean space was the first example. The nearest citation does not support this claim. I suspect (without citation) that two-dimensional Euclidean space was the first example.
 * If I am right, then the claim that Euclidean space aims to model physical space needs to be clarified. Mgnbar (talk) 13:49, 4 May 2022 (UTC)
 * The citation is about the existence of Euclidean spaces of any dimension, not about the origin of the concept (I guess without having accedded to it). "Originally" refers to Euclid's Elements in which "space" was the three-dimensional space of both the physical world and the mathematical one (which were clearly distinguished only centuries later). The (Euclidean) plane was not called a two-dimensional space before 19th (or possibly 18th) century. So, the plane is far from being the first example of a space.
 * I agree that "model" is not convenient in the first sentence, as being somehow anachronic. I'll replace it with "represent". Also, I'll add a reference to Euclid's Elements. D.Lazard (talk) 14:19, 4 May 2022 (UTC)
 * The article Euclid's Elements makes it seem that books 1-10 deal with 2D geometry (with digressions into non-geometric topics) and books 11-13 deal with 3D geometry. Mgnbar (talk) 15:07, 4 May 2022 (UTC)
 * I know that, but this article is about space, and Euclid never used the terms "dimension", "2D" and "3D", and also never used "space" for dimension two. So it is definitevely true true that originally, Euclidean space (without article) denoted the three dimensional space, and that Euclid never considered plane (without? article, when used in the sense of the 10 first books) and planes (plural in space geometry of books 11–13) as specific spaces. (By the way, this is a difficult problem for a non-native English speaker to know when an article is needed before "space" and when not.) D.Lazard (talk) 15:58, 4 May 2022 (UTC)
 * This article is about the current mathematical concept of Euclidean space. There is essentially one example per natural number. The evidence suggests that the example corresponding to the natural number 2 is at least as old as the example corresponding to the natural number 3. Mgnbar (talk) 17:17, 4 May 2022 (UTC)


 * said "Euclidean distance formula" is not defined in Wikipedia, and as far as I know, it is not definied either in any reliable source. Given the existence of article and section Euclidean_distance, it's hard to assume good faith in anything they say. All I'm proposing is to invert the definition given in article Euclidean norm: the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points. To break circularity, the norm would be defined first in terms of Pythagoras' law. fgnievinski (talk) 17:26, 4 May 2022 (UTC)
 * That section doesn't define a unique "Euclidean distance formula", but instead gives several examples of formulae, all of which calculate Euclidean distance in different situations or given different information. XOR&#39;easter (talk) 21:30, 4 May 2022 (UTC)

Euclidean plane
This article seems pretty good in structure/content (albeit with insufficient references to external sources). Does anyone here want to take a crack at cleaning up Euclidean plane? That article should (at least as a summary) cover most of the same ground as this one, but then go into further detail about geometry that is specifically planar. –jacobolus (t) 02:59, 5 November 2022 (UTC)

Generalizations in lede
Today, a well-meaning editor added a one-sentence paragraph to the lede, stating that Euclidean space is a metric space. That's true, but it's just one of many such statements that could be added. I propose that the paragraph be either deleted or expanded --- perhaps to something like, &quot;Euclidean space is an important archetype for many kinds of spaces in mathematics, including manifolds, topological vector spaces, metric spaces, ...&quot;. Mgnbar (talk) 23:29, 12 February 2023 (UTC)


 * This probably does not need to be in the lead section, but it could be useful to add to a new section near the bottom of the page. Hilbert space is linked from the "see also" section, but that and other structures generalizing Euclidean space (e.g. Lp spaces) could profitably fill out at least one top-level section near the bottom. There is a little bit of relevant material in . Ideally any discussion would aim to be lay-accessible, leaving interested readers to click through for more advanced coverage.
 * While we are at it, the section is far too general and abstract and filled with jargon. This section should focus on isometries of Euclidean space specifically, describe them concretely and in lay-accessible language with concrete examples and link to more advanced (abstract) articles for anyone who wants a more expert-level discussion discussion. Readers would be better served by e.g. material about crystallography or inverse kinematics or about the changing conception of Euclidean space as the focus shifted from studying physical forces toward rigid motions for their own sake (concretely studied by Olinde Rodrigues in 1840 decades before the Erlangen program), rather than nitpicky formalities.
 * The same is to some extent true of this article in general. As many concepts/sections as possible should be pitched to be legible to e.g. high school students or laypeople. It should include more specific details about concrete geometrical figures (or at least obvious links to other articles where those are discussed) and less reliance on lofty abstraction and jargon.
 * It’s wild to me that this article does not discuss the concepts of spheres or conics/quadrics (or any other curves or surfaces) or simplexes (or any other polyhedra), only mentions the word "triangle" in the context of the triangle inequality and does not mention the word "trigonometry", does not describe what a glide reflection or screw motion is, has no discussion of differential geometry, only mentions 'analytic geometry' per se in a throwaway aside, barely mentions other coordinate systems (and what little it does is inaccessibly jargony/abstract), makes no mention of the geometry of oriented planes, etc. etc. The fundamental notions that make a "Euclidean space" are lines (and higher-dimensional flat subspaces), parallelism, angles and perpendicularity, and distance and spheres. Just as planar Euclidean geometry substantially focuses on lines and line segments, angles, triangles and circles (and then also conics, algebraic curves, ...), higher-dimensional Euclidean geometry substantially focuses on those same planar objects and higher-dimensional analogs like tetrahedrons, spheres, and quadrics.
 * We could in theory lean more on Euclidean plane for some of this, except that article has most of the same problems. –jacobolus (t) 01:45, 13 February 2023 (UTC)


 * You raise good points. If I recall history correctly, this article sat at a particular, heavily contested equilibrium for many years. Then, over the past year or two, User:D.Lazard and a couple of other editors broke the equilibrium and essentially re-wrote it (mostly for the better, in my opinion). My point is that the current text is, in a sense, new/immature and ripe for continued polishing/expansion. Mgnbar (talk) 02:39, 13 February 2023 (UTC)
 * This should need further discussion. However, in my opinion, most topics that would add to this article would be better placed (if they are not there) in Euclidean geometry, as this article is about the definition of the space of Euclidean geometry, not about that that can be done with it, which is Euclidean geometry. D.Lazard (talk) 09:22, 13 February 2023 (UTC)
 * But the title is Euclidean space, not one modern formal definition of Euclidean space. ;-)
 * Joking aside, I think this focus poorly serves many/most readers. Readers are not generally coming here because they want to find out the formal/technical details and logical ordering of the basic definition(s) Marcel Berger chose to use in his (very fine) textbook, which was convenient for his particular purposes of setting up a framework for the rest of the book, in a logically consistent order, intended for a 2-semester undergraduate course. Readers of Berger's book per se can just directly refer to the relevant definitions in context. Instead, many readers who come here are looking for a more accessible and/or broader view: what is the concept about, why does it matter, what implications does it have (Berger spends 2 volumes on this subject, only a few pages of which we are describing here), what is its mathematical context, how has the idea changed over time, etc. Or when it comes to formal definitions per se: why was this one chosen, which other definitions are equivalent, how does it relate to definitions of other structures, and so on. For instance, this article currently makes no mention of Cayley–Klein metrics, which could provide an alternate of Euclidean space. It also is entirely focused on the geometry of points in Euclidean space, ignoring Laguerre's dual geometry of oriented planes, which is equally much a fundamental part of "Euclidean space".
 * Even if we accepted the current structure this article is very hard for non-expert readers. For example the concepts of "inner product" and "affine space" are first mentioned as:
 * The standard way to mathematically define a Euclidean space, as carried out in the remainder of this article, is as a set of points on which a real vector space acts, the space of translations which is equipped with an inner product. The action of translations makes the space an affine space, and this allows defining lines, planes, subspaces, dimension, and parallelism. The inner product allows defining distance and angles.
 * This is mostly incoherent to someone without significant math background; we are more or less assuming someone has studied 2+ years of undergraduate level pure math courses before trying to approach this article.
 * I agree that some such material as I mentioned in my previous comment could also conceivably fit in an article titled 'Euclidean geometry'. But elaboration about such details mostly does not fit particularly well at the article Euclidean geometry as it is currently structured. That article is organized around meta-discussion, starting with a description of The Elements, then proceeding to discuss axiom systems, a (currently very incomplete and somewhat idiosyncratic) historical summary, an idiosyncratic smattering of sentences and pictures about applications, and concrete details mainly provided as examples rather than as the primary topic. It would also be great to dramatically improve the article solid geometry, which is currently a wreck. But even if we assumed that those articles were rewritten / dramatically expanded, the scope of this article should be broad enough to include a summary. –jacobolus (t) 20:51, 13 February 2023 (UTC)

Why was this moved from 'Euclidean space' to 'Euclidean n-space'
This seems like a serious regression. The name 'Euclidean space' is significantly more common and just fine for this topic. @Fgnievinski can you please try to start up conversations and reach some kind of consensus before making sweeping disruptive changes? –jacobolus (t) 06:32, 11 March 2023 (UTC)


 * Because the lead said (and still mostly says) so:
 * "Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any positive integer dimension, including the three-dimensional Euclidean space, the Euclidean plane (dimension two) and the Euclidean line (dimension one)."
 * I've just linked as written literally and inserted the number n:
 * "Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean n-spaces of any positive integer dimension n, including the three-dimensional Euclidean space, the Euclidean plane (dimension two) and the Euclidean line (dimension one). The qualifier 'Euclidean' is used to distinguish Euclidean spaces from other spaces that were later considered in physics and modern mathematics."
 * The renaming was necessary to make more explicit the scope of the present article. It is not specifically about the ordinary three-dimensional Euclidean space, it's about the general case of Euclidean spaces of any dimension. fgnievinski (talk) 06:45, 11 March 2023 (UTC)
 * I agree with Jacobolus, that this name change is an anti-improvement. Mgnbar (talk) 07:07, 11 March 2023 (UTC)
 * There should be an article at 'Euclidean space', it should focus on Euclidean space per se as its primary topic (but also including discussion of the relationship to non-Euclidean, pseudo-Euclidean, affine, projective, etc. spaces), and it should encompass both the basic geometry and more abstract topics, including most of what is currently on this page. Having Euclidean space (double-)redirect to Three-dimensional space is much worse in my opinion. –jacobolus (t) 08:36, 11 March 2023 (UTC)

I agree with the opponents to this move, and I have requested to revert the move at Requested moves/Technical requests‎‎, before reading this thread. D.Lazard (talk) 11:04, 11 March 2023 (UTC)

Splitting of section "Geometrical space"
Please comment at Talk:Geometric_space. fgnievinski (talk) 06:51, 11 March 2023 (UTC)


 * It’s fine to make a separate article (which can even be significantly expanded). However, all of the material currently on this page is easily in scope (indeed, the material on this page should be further expanded here), and removing it and replacing with a useless 'excerpt' is a major regression. –jacobolus (t) 08:40, 11 March 2023 (UTC)

Requested move 11 March 2023

 * The following is a closed discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. Editors desiring to contest the closing decision should consider a move review after discussing it on the closer's talk page. No further edits should be made to this discussion.

The result of the move request was: not moved. Favonian (talk) 16:43, 18 March 2023 (UTC)

Euclidean space → Euclidean n-space – This article has been the subject of moves, reverts and technical requests. Opening this RM to assess if there is indeed consensus to move it. Dr. Vogel (talk) 13:39, 11 March 2023 (UTC)


 * I disagree 88.111.123.182 (talk) 20:53, 12 March 2023 (UTC)

Procedural note: redirect Three-dimensional Euclidean space has also been nominated for discussion. fgnievinski (talk) 04:51, 12 March 2023 (UTC)

The discussion above is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.
 * Oppose. I've started this RM to see if we can formally document consensus. I'm personally against this move. Dr. Vogel (talk) 13:40, 11 March 2023 (UTC)
 * Notifying participants @Fgnievinski, @Mgnbar, @Jacobolus, @D.Lazard. Dr. Vogel (talk) 13:43, 11 March 2023 (UTC)
 * Oppose Don't see how it's an improvement. -Kj cheetham (talk) 14:55, 11 March 2023 (UTC)
 * Oppose. Clear disimprovement. See . D.Lazard (talk) 15:08, 11 March 2023 (UTC)
 * Oppose. There are many wiki-links to Euclidean space and the article three-dimensional space does not currently successfully define what that means, and makes a poor redirect target. (Aside: This article could probably also do a better job in the lead section of clearly describing what makes a space "Euclidean" per se.) –jacobolus (t) 16:44, 11 March 2023 (UTC)
 * Incoming links have mixed usage and would deserve disambiguation, for example: in Euclidean geometry: " Euclidean space is a good approximation..."; in Differential geometry: "in the three-dimensional Euclidean space ..."; in Cartesian coordinate system: "in an n-dimensional Euclidean space ..."; in Cube: "four-dimensional Euclidean space ..."; in Absolute value: "in Euclidean n-space is..." fgnievinski (talk) 03:55, 12 March 2023 (UTC)
 * Support. As the lead of Euclidean spaces says: "Euclidean space is the fundamental space of geometry, intended to represent physical space". I.e., Euclidean space is understood as the three-dimensional one. Otherwise, where should the redirect three-dimensional Euclidean space be targeted? Wikipedia serves a wide audience and titles should reflect the primary topic. Most people -- including scientists, engineers, etc. (mathematicians excepted) -- would equate "Euclidean space" with 3D, not n-dimensional spaces. (I'd like to thank for the arguments, pro or con, and note this is not a WP:VOTE.) fgnievinski (talk) 03:42, 12 March 2023 (UTC)
 * Oppose. The proposed title is unnecessarily verbose. It requires, but cannot offer, an explanation of the variable n. Moreover, I do not agree that "the" Euclidean space is the three-dimensional one. That is contrary to my sense of the math literature, where the word "space" does not imply three dimensions. The sentence, that Fgnievinski is citing from this article, should be changed. Mgnbar (talk) 04:24, 12 March 2023 (UTC)
 * That's a crucial point: should the article title reflect the usage in modern mathematical literature or the broader literature in physics, science, engineering, etc.? How about a less verbose version: "Euclidean spaces", in plural form? fgnievinski (talk) 04:59, 12 March 2023 (UTC)
 * Having both a singular and a plural form would be very confusing to most readers. The ambiguity between “the Euclidean space” (which is three dimensional) and “a Euclidean space” (which can have any dimension), is not an artefact of Wikipedia, but relies on the common use. So, it cannot be resolved by manipulating article titles. Apparently, you did not read the second sentence of the article, which is clearly intended for explaining this ambiguity.
 * Also, except for mathematical oriented physical literature, “Euclidean space” is not used in physics. Physicists use normally "three dimensional space”. This is because of these multiple uses of the word “space” that Three dimensional space and Euclidean space are different articles. Both leads could certainly be improved for better resolving the ambiguities, but playing with article titles is certainly not a solution. D.Lazard (talk) 10:49, 12 March 2023 (UTC)
 * In Newtonian mechanics space is implicitly Euclidean. In special relativity spacetime ("Minkowski space") is explicitly non-Euclidean. In general relativity space gets all wibble-wobbly. –jacobolus (t) 22:58, 12 March 2023 (UTC)
 * It should be singular, following prevailing practice in the majority of Wikipedia articles. It is trivial for authors of other pages to write  to generate Euclidean spaces if that name makes more grammatical sense in context.
 * Most of the time there's no problem figuring out from context which among Euclidean 3-space vs. any Euclidean space is intended. When there is some ambiguity authors can just explicitly say "Euclidean 3-space" or "any Euclidean space" or whatever. –jacobolus (t) 22:56, 12 March 2023 (UTC)