Talk:Euclidean space/Archive 1

message: See Talk:Coordinate space for why I disagree. Melchoir 19:06, 25 January 2006 (UTC)
thread_title:  Anon's merge proposal 

message: This article is written by mathematicians, for mathemeticians.  The lay reader gets nothing from it.  Look at Britannica's introduction to the topic here: http://www.britannica.com/ebc/article-9363979?query=euclidean&ct=
Compare that to the ridiculous run-on sentence which introduces this article ("In mathematics, n-dimensional Euclidean space (also called Cartesian space or n-space) refers to the space of ordered n-tuples of real numbers along with the associated operations of component-wise addition and scalar multiplication which make it into a vector space, and the dot product which makes it into an inner product space and a normed space with the Euclidean norm. ").
You guys should have taken an English class or two.  I got more out of the Britannica intro in 30 seconds than I got in ten minutes trying to decipher this nonsense. —The preceding unsigned comment was added by 66.208.37.3 (talk • contribs) 04:08, July 3, 2006.
:I couldn't agree more. It's one of the two worst articles I've ever seen on Wikipedia. (The other one being Quantum State, in which the opening sentence is "In quantum mechanics, a quantum state is any possible state in which a quantum mechanical system can be." Gee, that's helpful.)
:I'm going to add an appropriate "cleanup" tag to this article. 24.6.66.193 23:49, 8 July 2006 (UTC)
replies: text: I think the new introduction (the 1st paragraph) is great: accessible and well written. The rest of the article still needs work though. Hermajesty 16:22, 27 August 2006 (UTC)
replies: text: Perhaps "accessible and well written" but unfortunately wrong. The physical space we live in is not assumed to be Euclidean! I've tried to fashion a quick fix, but the intro still needs work. For example the new text added by PAR introduces some redundancy into the intro which still needs to be resolved. Paul August ☎ 06:17, 28 August 2006 (UTC)
replies: text:  I've restored the old introduction with a small modification. Look, we can nitpick this to death. Strictly speaking a flat piece of paper is not a Euclidean space on the microscopic level. The only precise way of stating things is to remove the first paragraph entirely and go to the mathematical introduction. Then we have to reinsert the insufficient context tag. This new introduction, while more precise is less helpful to a new reader. I'm not saying the old introduction was perfect, but lets not wander back to the pure mathematical introduction of the old one in search of perfection. PAR 14:15, 28 August 2006 (UTC)

text: I like your first sentence, although I would say geometric instead of geometrical. In your second sentence, the word theory should be replaced with axiom (if you're talking about the parallel line postulate -- but then there are clearer ways to say it) or theorem (if you're talking about a proven statement). Regards, Joshua Davis 14:03, 10 September 2006 (UTC)

text: I like that first sentence (of Raikinili's) too, with suggestions by Joshua Davis. Look, people coming to this page may likely have just come from manifold, one of the most foundational objects of mathematics, which deserves to be accessible to those outside of math. Both history and technical jargon should be postponed. The primitive idea of Euclidean space is that it's flat, and is communicated with the examples of line and plane (and point, perhaps next), and suggestions of higher analogs, NOT a space with a "Euclidean Norm," or a space that fulfills a given (context-free) list of postulates. Let's get the idea across. MotherFunctor 00:23, 24 April 2007 (UTC)
thread_title:  Obnoxious article 

message: We in no way live in Euclidian space - pleeeeeaaaasee someone fix this!  The most once can say is that someone who took high school geometry and never went on to higher math may blieve we lie in Euclidian space, virtually no ene else does. Slrubenstein   |  Talk 17:31, 28 August 2006 (UTC)
thread_title:  intro 

message: You can always give wrong information if someone else is saying it.  For example:
"A layman example of Euclidean space in the third dimension may be the everyday world we live in, with its width, height, and depth.  However, this is not strictly correct." 
This shows that normal people can think of it that way, but, as it says, that way of thinking is not strictly correct.  It's like saying "for most practical purposes", but by ending it with the "however" statement, I'm bringing attention to the fact that there IS a difference, and the "strictly" points out that the difference doesn't matter to most people. I think it's also more friendly than some of the suggestions, except for the word "layman" (and I'm not sure of the names of the three dimensions). After those two sentences, you can expand on the differences or have an in-page link to a (new) section explaining the difference between 3D Euclidean space and what we know of as space. --Raijinili 08:57, 16 September 2006 (UTC)
replies: text: I don't like in-page links, but that's a minor point. In my opinion, it is more helpful to say that the idea of a Euclidean space is based on the "layman's understanding" of the real world as a 3D space, rather than simply give it as an example of a Euclidean space. We don't really need to focus much on the fact that Euclidean space is not a perfect model for reality. Apart from that, your sentence focuses on the three dimensionality, and says nothing about what makes a space Euclidean. It is true that this is a more difficult concept, but the term "Euclidean space" is useless until we are considering other sorts of spaces. JPD (talk) 11:03, 16 September 2006 (UTC)
replies: text: What about "intuitive" explanation having its own section? --149.4.211.70 14:38, 21 September 2006 (UTC)

text: I agree that the "encoded" bit is counterproductive. The intro has several other defects ("very carefully expressed" --- as opposed to slightly carefully expressed?). On the other hand, if I remember correctly Euclid's axioms are not really concrete. Also, we cannot define Euclidean space to be two- or three-dimensional because higher dimensions are also allowed. By the way, if you're interested in contributing, I encourage you to sign up for a Wikipedia account. Joshua R. Davis 20:44, 11 September 2007 (UTC)
thread_title: A point to keep in mind

message: Under "See also", a link has been added to "Orders on the Euclidean space". I have two objections. First, the linked content is trivial and inane. Second, it is about as relevant as linking to "Words beginning with the letter 'E'". (And, of course, there is no such thing as "the Euclidean space"; there is only "Euclidean spaces".) I have removed it once, and want it removed permanently. Based on painful prior experiences, I have negligible respect for the editor in question, so would like to hear opinions from third parties. --KSmrqT 13:36, 20 May 2007 (UTC)
replies: text: I agree with your conclusion, KSmrg, although I would argue it differently. The order material is about general Cartesian products of sets, not about Cartesian products of the real numbers under the \ell^2-norm, which is what this article is about. Of the three orders given, it seems that only the product order is relevant to this article, since it is the only one that's continuous with respect to the topology? And it generates the topology, right? Then I support actual discussion of the product order in this article, and I support removal of the order link from See also. (By the way, this article has much bigger problems for us to worry about.) Joshua R. Davis 16:14, 20 May 2007 (UTC)
replies: text: Perhaps we need a separate page on Rn then?--Patrick 21:55, 20 May 2007 (UTC)
replies: text: I don't think so, given that most things about Rn as a set are probably covered in general articles about Cartesian products and so on. Why should an article about Euclidean spaces, or Rn, link to Total order. OK, it links to a particular section of total order, which might be appropriate if the section was about total orders on Rn, but it's more general than that. The best way to link the article to more directly relevant material would be along the lines of mentioning the Cartesian product in the Real coordinate space section. JPD (talk) 10:46, 21 May 2007 (UTC)
thread_title:  Orders on the Euclidean space 

message: In the section Real coordinate space, the article states:
:"For any non-negative integer n, the space of all n-tuples of real numbers forms an n-dimensional vector space over R, which is denoted Rn and sometimes called real coordinate space."
Shouldn't this be:
:"For any positive integer n..."
Or do we need to explain what R0 is? - dcljr (talk) 15:22, 4 June 2008 (UTC)
replies: text: A mathematician would certainly acknowledge R0 = {0} as the 0-dimensional Euclidean space, but it's possible that the omission (or at least delay) of this case would make the article clearer. If we do not omit it, then yes we should explain it. Joshua R. Davis (talk) 16:20, 4 June 2008 (UTC)
thread_title:  R^0? 

message: Im all for making math articles accessible, but the current lede "around 300 bc.." is poor form. -Zahd (talk) 23:13, 30 July 2008 (UTC)
replies: text: Are you saying that it's improper because it's "BC" instead of "BCE"? Or because it's "around" instead of "circa" or "floruit"? Please explain. Better yet, go ahead and edit it. Mgnbar (talk) 23:39, 30 July 2008 (UTC)
thread_title:  improper lead writing style 

message: the article says:
"Our universe, being subject to relativity, is not Euclidean. This becomes significant in theoretical considerations of astronomy and cosmology, and also in some practical problems such as global positioning and airplane navigation. Nonetheless, a Euclidean model of the universe can still be used to solve many other practical problems with sufficient precision."
but does being subject to relativity, or the universe being non-eucildean have any importance when it comes to airplane navigation or global positioning? it certainly does have to do with the fact earth's surface is non-euclidean, but that would also be true for a sphere in a perfect euclidean (newtonian) world, and has little to do with the nature of the universe, or does it?--Ghazer (talk) 21:13, 18 February 2010 (UTC)
replies: text: Search Google for GPS and relativity together and you get many hits --- for example,  or . Because the satellites are at high altitude, they are subject to weaker gravity, and this affects the pace at which their clocks tick, which affects the GPS calculations. But you are right that the article needs to cite sources. Mgnbar (talk) 22:18, 18 February 2010 (UTC)
thread_title:  ariplane navigation? 

message: My linear algebra book simply defines a Euclidean space as any inner product space that has a finite number of dimensions. To me, this seems much easier to grasp than the explanation at the beginning of the article. Attys (talk) 19:38, 9 December 2010 (UTC)
replies: text: An Inner product space is usually considered as a generalization of Euclidean space, so defining Euclidean space using the notion of inner product space isn't usually done. And only the finite dimensional inner product spaces over the reals are the Euclidean spaces. Paul August ☎ 20:46, 9 December 2010 (UTC)
thread_title: More concise definition

message: Let's change the lead back to what it was. It was better than it is now. At the very least, let's exchange the second and third sentences, so that we finish the task of saying what Euclidean space is, before we bother with the historical trivia of for whom it is named. Mgnbar (talk) 18:39, 11 October 2012 (UTC)
replies: text: "Euclidean space is the Euclidean plane and three-dimensional space of plane geometry" ?! Gibberish! —Tamfang (talk) 19:54, 11 October 2012 (UTC)
replies: text: Well, you're right. At the risk of offending the editor in question, I'm going to edit the lead now, just enough to restore it to competence. Mgnbar (talk) 22:18, 11 October 2012 (UTC)

text: A basic ambiguity in the title: The Euclidean view of space versus an affine geometry that is also a metric space. Since space is three-dimensional, the first interpretation has limited dimensionality. The second interpretation of the title includes the case of four-dimensional geometry that came into view with quaternions about 170 years ago. Weakness in the article is seen in References which are books on topology! This article gets about 800 readers a day; it has 147 editors watching it currently. Work is needed.Rgdboer (talk) 21:05, 11 October 2012 (UTC)
replies: text: This article has always been (at least in my memory) about the modern mathematical concept called Euclidean space, not about space-as-viewed-by-Euclid. The latter topic might deserve an article, but it's not this article (at least, not without complete rewriting). Given that we're talking about the former, then, a topology book is exactly the sort of reference you'd expect for this topic. Mgnbar (talk) 22:18, 11 October 2012 (UTC)

text: Also, there has been a great deal of contention and argumentation over the into to this article in the past; see the "Intro redux" section above. So let's propose major changes here, before implementing them in the article. Mgnbar (talk) 22:23, 11 October 2012 (UTC)
thread_title:  Today's lead change 

message: Substituted at 06:04, 24 July 2016 (UTC)
thread_title: Assessment comment

message: Note about three related (and important) discussions:
* WT: WikiProject Mathematics/Archive/2013/May #Space Wars
* talk: Euclidean subspace
* talk: Vector space #Diagrams
Incnis Mrsi (talk) 13:14, 27 April 2013 (UTC)
thread_title:  Related discussions 

message: For some reason I initially inserted this section just after “Euclidean group”, but before “shapes”. Has it much sense, with all reasonings about coordinate hypersurfaces, before the section about shapes? Incnis Mrsi (talk) 13:14, 27 April 2013 (UTC)
thread_title:  Non-Cartesian coordinates 

message: Must be redirected to Norm (mathematics). Jumpow (talk) 19:04, 7 August 2013 (UTC)
replies: text: Done.  An alternative could have been Lp-space, but Norm (mathematics) seems more straightforward.  — Quondum 20:00, 7 August 2013 (UTC)
thread_title:  Wrong redirection from 2-norm.  

message: WP:MOSMATH #Common sets of numbers prefers  over , and the majority of mathematical articles which use HTML formatting follow this. NevilleDNZ, is there some specific reason to ignore the MoS? The  is an astral character and is missing from many fonts (including those used for  by several advanced editors. I definitely oppose to inclusion of U+01D53C, and do not see any merit in replacement of R with ℝ. Incnis Mrsi (talk) 05:57, 28 April 2013 (UTC)
replies: text: BTW, [ de-italicizing] of the variable  is something not only going against all customs and guidelines, but a change which disrupts consistency of notation through all the article. When I’ll do my next edit to the article, I would drop NevilleDNZ’s changes without further notice. Incnis Mrsi (talk) 06:06, 28 April 2013 (UTC)
replies: text: Without bothering to find a policy specifically addressing it, I strongly agree that n should be italicized. Mgnbar (talk) 12:42, 28 April 2013 (UTC)

text:  I agree with both of Incnis Mrsi's points. WP:MOSMATH #Common sets of numbers is unambiguous here and WP:MOSMATH#Variables recommends that variables be italicized. In the linked diff above, the n is a variable, indicating an arbitrary number of dimensions. --Mark viking (talk) 23:09, 29 April 2013 (UTC)
replies: text: While I do not disagree with the above comments, I would like to note that WP:MOSMATH #Common sets of numbers might not be that unambiguous: it makes the reference "see blackboard bold for the types in use", which many would interpret as implying that BB is also regarded as a boldface type and suggested. A rewording of the guideline might be appropriate to eliminate this implication being read into it.  — Quondum 20:08, 7 August 2013 (UTC)
thread_title:  '''R''' or ℝ? 

message: I put in a group name (footnote) for the existing footnotes and put in a simple reflist so ordinary inline references can be easily added to a References section by other editors. To start the ball rolling, I added an inline reference.
Collieuk (talk) 13:13, 12 December 2013 (UTC)
thread_title:  Use of a different syntax in the markup 

message: It is an off-topic, but the discussion already happened. One can embed the Minkowski space into , but this way gives botched, geometrically abhorrent compactifications of pseudo-Euclidean spaces (possibly,  will write someday about their good compactifications, but here it’s a far off-topic). Generally, one should think of the projectivization’s elements as of lines (or vectors up to scalar multiplication, so the projectivization is defined formally). Each line passing through the origin intersects the unit sphere in two points. In Euclid there is one unit sphere, but in Minkowski there are two separate unit spheres: one for time-like directions, and one for space-like directions. There is no sphere for light directions, only cones. One could combine projectivizations of all 3 flavours of vectors at a single manifold (that is ), but the construction for time-like directions separately is the simplest because it gives a hyperbolic 3-space immediately; see hyperboloid model for details. This 3-space is a 3-ball in the said . One should deal with each of 3 flavours of directions with its separate rule; the pseudo-Euclidean space article explains this matter rather comprehensively. Incnis Mrsi (talk) 20:37, 17 February 2014 (UTC)
replies: text: You express yourself strongly on what appears to me to be a mathematically consistent compactification; indeed, the symmetries that it implies may suggest the framing of some interesting questions if it is taken as a global model for cosmology. This is off-topic here though, as you say. As for the rest, I have no issue with your characterization. —Quondum 16:55, 19 February 2014 (UTC)
thread_title: Projectivization of 4-vectors

message: Even now, the phrase "Minkowski space, where rotations correspond to motions of hyperbolic spaces" has the analogue "Euclidean space, where rotations correspond to motions of elliptic spaces" and is likely to sound like mumbo-jumbo to many unless clarified a bit, and probably best dealt with by referring to a free-standing example of the projectively obtained geometry that is more fully developed in its own right.  —Quondum 21:50, 16 February 2014 (UTC)
replies: text: Doesn’t the current text say that Euclidean rotations are direct motions of the -sphere? Apparently it doesn’t, but it is a mishap. Similar statements are already scattered over Wikipedia, e.g. there and there ( count only sections written by myself), and the article “elliptic geometry” also says something about it. Get these pieces and compile the necessary paragraph for this article. Incnis Mrsi (talk) 07:28, 17 February 2014 (UTC)
replies: text:  I seem to feel that the reader needs a little more guidance than you do. There are two ways to obtain a Euclidean geometry from the same vector space: by "forgetting" the origin, or by using it to produce a projective geometry and then deleting a hyperplane (making it affine). In both cases one then introduces a metric.  The results differ in dimensionality.  How one gets to an elliptic/hyperbolic geometry has a similar choice: form a projective space, and constrain the group of actions (elliptic) or delete everything on and outside a conic (hyperbolic), or else start with a Euclidean/Minkowski space and find the space of ideal points (hyperbolic: also delete a bunch). Because of choice of the route to a hyperbolic geometry, and that some people may be familiar with only one or none, a clarification such as "Minkowski space, where rotations correspond to the motions of the hyperbolic space of its ideal hyperplane", might be helpful? This avoids producing duplication of material available elsewhere. —Quondum 15:24, 17 February 2014 (UTC)
replies: text:  What do you mean under the same vector space, a space with certain dimension but without any supplementary structure? It would be rather pointless. And  do not understand the aim of your run through the projective geometry and then back to an affine space. We discussed the point that projectivizations of Euclidean/Minkowski/Galilean structures on vectors give us elliptic/hyperbolic/Euclidean structures correspondingly on points, didn’t we? It is one step only. If you want to end with an elliptic geometry, then you should start from the Euclidean one, with one extra dimension. But if you want to end with a Euclidean geometry, then you should start from the Galilean one, with one extra dimension. The projectivization won’t make Euclidean (on points) from Euclidean (on vectors): the transformation groups mismatch. Incnis Mrsi (talk) 18:48, 17 February 2014 (UTC)
replies: text:  Yes, I meant starting without supplementary structure, since otherwise it would not work. And I was only trying to make the point that someone who has heard of an affine geometry as a projective geometry with a hyperplane removed, a tortuous route might be what comes to mind. But nevermind, my purpose is not to run through constructions of geometry, only to suggest that a few more words might help the reader to make the connection between the rotations in Minkowski space and "motions of hyperbolic spaces", since  I expect that such things are not as obvious to everyone as they may seem to you. Do you have comment on the wording that I suggested in my previous post? —Quondum 19:30, 17 February 2014 (UTC)
replies: text:  "The hyperbolic space of its ideal mumbo-jumbo-plane".  start to understand your thinking pattern… that’s why you make two steps where  need only one. When you departed from a flat space and arrived to vectors (a.k.a. the group of translations), you needn’t elements at infinity any more. It is generally a poor idea to think about the projectivization of a vector space as about the hyperplane at infinity. It is not abhorrently bad at Euclidean→elliptic run (that  tried to discuss in this new subsection), but in the case of Minkowski→hyperbolic it is bollocks; see #Projectivization of 4-vectors subsection above. Drop the hyperplane: it is geometrically incorrect in the context of vector projectivizations at all. Sorry,  have to sleep now. Incnis Mrsi (talk) 20:37, 17 February 2014 (UTC)
replies: text:  Well, okay – let's try to determine whether we are talking about the same thing. By "Minkowski space, where rotations correspond to motions of hyperbolic spaces", do you mean something like: "Rotations in a Minkowski space act on its projectivization (which corresponds to the space of 4-velocities) such that these are the motions of a hyperbolic space"? —Quondum 00:16, 18 February 2014 (UTC)
thread_title: Motions of elliptic spaces

message: Much of this article describes Real coordinate space and not Euclidean space. Euclidean space has no notion of coordinates - it is simply a space endowed with Euclid's axioms. 67.252.103.23 (talk) 20:30, 11 March 2014 (UTC)
replies: text: of references to Euclidean plane, Euclidean 3-dimensional space, and rational numbers, is not a “clarification”. These concepts were, historically, among the first ones associated with Euclidean spaces (or Euclidean geometry, if you like it more), regardless of whatever do you think about them. Incnis Mrsi (talk) 22:10, 11 March 2014 (UTC)
thread_title:  Euclidean space 

message: I have now rewritten the part oc the article related to the definition of Euclidean spaces, including the stuff related to isometries. My main deal was to have a coherent presentation that includes the various definitions and do not include circular reasoning (using notions, such as angles, that are commonly introduced by using Euclidean spaces). I hope this will be understandable by readers without a high mathematical background. Be free to fix the mistakes that I have certainly done.
Some work is also needed on other sections. 
* I have already removed the section on shapes. A part of it (segments, lines, subspaces, ...) duplicates the beginning of the article. The remainder seem to be to be misplaced here: nobody will come here to find information about polytopes and root systems. Moreover concepts that are developed inside Euclidean spaces are too numerous  for being listed here, and there is no reason for such an emphasis on root systems, which are a very specialized topic.
* The section on non-Cartesian coordinates is confusing. IMO, it should be removed, possibly replaced by a link in See also section.   Not removed, but moved and completely rewritten. D.Lazard (talk) 08:24, 15 September 2019 (UTC)
replies: text: Who does require Euclidean space #Applications to list all applications? As for shapes, my writing and stuffing with pictures in 2013 was arguably an excess, but the most important things, such as regular polytopes, should be at least briefly mentioned. Topology? We forgot to write that a closed Euclidean ball is compact, and IMHO this addition would be enough. Incnis Mrsi (talk) 14:09, 7 September 2019 (UTC)
replies: text: In the previous list, I forgot "updating the lead". This is done. D.Lazard (talk) 18:12, 8 September 2019 (UTC)
replies: text: I have finished what I am able to do for improving this article (). D.Lazard (talk) 15:00, 18 September 2019 (UTC)
thread_title:  Rewriting the article 

message: The article lead sentence currently claims that "" Klein's Erlangen program granted this honour to Projective space over a century ago, and it is echoed for example by Johnson in Geometries and Transformations, Cambridge, 2018. Euclidean space was of course the first to be described and may perhaps be seen as the fundamental space of Cartesian analysis (though not of course of homogeneous analysis), but that is a far weaker claim. What is this lead sentence really supposed to mean? — Cheers, Steelpillow (Talk) 12:26, 12 November 2019 (UTC)
replies: text: I agree. Currently it reads as POV. I suspect that the intent was something like "the most popular, plain vanilla space in geometry", but it's not clear how to word that encyclopedically. Unless ample citation can be provided, the statement should be replaced. Mgnbar (talk) 13:09, 12 November 2019 (UTC)
replies: text: I have fixed this by replacing "geometry" by "classical geometry". I do not believe thay anybody has ever included Erlangen program in classical geometry:-) D.Lazard (talk) 16:36, 12 November 2019 (UTC)
thread_title:  Fundamental? 

message: Shouldn't in this section be written: an origin?Madyno (talk) 20:33, 26 August 2020 (UTC)
replies: text: D.Lazard (talk) 07:37, 27 August 2020 (UTC)
thread_title:  Prototypical examples 

message: There is no formal definition of the associated vector space. Madyno (talk) 20:40, 26 August 2020 (UTC)
replies: text: The associated vector space of an affine space is formally defined in Affine space. D.Lazard (talk) 07:41, 27 August 2020 (UTC)

text: Now, it is defined. D.Lazard (talk) 10:36, 28 August 2020 (UTC)
thread_title:  Associated vector space 

message: Some editing is required to clean up for standard english sentence structure.  — Preceding unsigned comment added by 47.185.212.35 (talk) 23:20, 31 August 2020 (UTC)
thread_title:  Sentence Structure