Talk:Euclidean space

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Contents

[edit] Early comments

Working on filling out this page over the next few weeks. Nevertheless, please feel free to be bold in making improvements. --Eddie 21:05, 16 Jan 2005 (UTC)

Hope I'm not treading on anybody's feet. Please edit, if you think it's off the mark. --Eddie | Talk 20:54, 8 Feb 2005 (UTC)

What is Euclidean Space? Why dosen't this article tell me? I know, it is because I don't know enough about mathematics, but surely there are lots of people like that? Shouldn't it explain it for them? Please? Furius 08:16, 20 Jun 2005 (UTC)

In the article R^n is defined as the set of all n-tuples of real numbers. However, the set of all 1-tuples of real numbers is not the set of all real numbers. Therefore (according to this definition) R^1 != R. This violates common conventions. I would therefore suggest, that the case n=0 is defined separately (since R^0 is pathological anyway and normally implicitly excluded when something is said over all R^n), and the R^(n+1) space is simply defined as RxR^n for all n >= 1, and R^1 := R.

Yes, but there is a obvious canonical isomorphism between R1 and R, so there really isn't a problem (this kind of thing happens all the time in mathematics). At any rate, it may be better to distinguish them since we are considering the former to be a real vector space and the latter to be a field. -- Fropuff 15:53, 2 October 2005 (UTC)
It's splitting hairs, and it's foolish. There's no canonical isomorphism; the two spaces are absolutely the same. I wonder, as I've always wondered, how, with this sort of nitpicking legion, the modern mathematician finds the time to do aught. R is a vector space. The fact that it is a special case with a special structure doesn't deny its rights as a vector space. Now of course you can view R as the set of all functions which solve the equation f(0) = x for any x in R. The question is, why? --VKokielov 15:14, 6 June 2006 (UTC)
I agree with the above. If we take R^n to mean RxRxRx...xR (with n R's in Cartesian Product), then R^1 simply becomes R.
Formally a Vector space is a Triple consisting of a set V and two functions +:VxV |--> V and *: RxV |--> V, and a Field is a Triple consisting of a Set K and two functions +:KxK |--> K and *:KxK |--> K. In the case of the canonical Vector Space and Field on R, they happen to be the same thing (that is, set). Therefore one can't distinguish between "them". Furthermore, since the space R0 is (here) defined through a special definition of n-tuples, which starts out from the 0-tuple {}, for every set S: S0 = R0 = {{}}. The set R0 therefore has nothing more to do with R than any other set.
WOW! upon reading through what's above I see there is a fundamental problem: no one can agree whether a foundational concept should be a) made an easy read, with the layman in mind; b) strictly formalized, with foundations in mind; c) defined with the simplicity and lenient language used in the math discourse. Is "R^1" equivalent to "R"? this is a problem for b) but not for a) or c). I think these articles should start with a), then get to c). (Wikipedia should be easier to read than Bourbaki.) MotherFunctor 08:02, 23 April 2007 (UTC)
As long as some people insist that R is a vector space, others that it's a field, and yet others that the vector space R and the field R are the same thing, this argument is going to keep going round in circles. R, and hence R^2 etc., can be many things. By way of perspective on this discussion (meaning something that "only mathematicians can understand", as opposed to what's appropriate for the article itself), here's a partial list of what R^2 can be. (i) A set. (ii) The cartesian square of the set R. (iii) A topological space. (iv) The (weakest topology on the) square of the topological space R. (v) A vector space over the field R. (vi) The square of the vector space R and hence a real Hilbert space. (vii) An associative algebra over the field R, namely the Clifford algebra R1,0. (viii) The square of R as an associative algebra over the field R, and a coordinatization of R1,0 called the double field (oriented 45 degrees from the hyperbolic number coordinatization, as can be seen from 1 = (1,1), e = j = (1,-1)).
Each of these eight incarnations resides in a distinct category. Four of these categories, associated with the odd-numbered kinds above, are Set (sets and functions), Top (topological spaces and continuous functions), VctR (vector spaces over the reals and their linear transformations), and AlgR (associative algebras over the reals and their homomorphisms). The other four are monoidal categories, namely the same four categories equipped with cartesian product as the monoid (they are in fact cartesian categories in the sense that the monoid is categorical product in all four cases, but not necessarily cartesian closed categories because we are ignoring whatever closed structure there might or might not be).
Only in (i) is R^2 isomorphic to R (in (ii) none of those isomorphisms, a.k.a. bijections, respect product, in (iii) R^2 suitably topologized is not homeomorphic to R because the connectivity resulting from removing a point is different). What makes the even-numbered kinds different from the odd-numbered is access to the two projections from R^2 to R, allowing one to identify the points of R^2 with coordinate pairs (x,y). This is what is meant by a coordinate space, and points up that "coordinate space" by itself is not a sufficient designation to prevent talking at cross purposes.
It seems pretty clear that the intended meaning of "coordinate space" here is (vi), R^2 as the square of the vector space R. This furnishes R^2 with the basis {(1,0),(0,1)}, thereby defining dot product as the canonical choice of inner product for R^2.
Euclidean space with an origin and a scale is intermediate between (v) and (vi). (v) is too weak for Euclidean space because there are no notions of orthogonality, angle, or distance. (vi) on the other hand overshoots the mark because although a choice of basis gives canonical notions of orthogonality, angle, and length, it is not basis-independent - geometry as conceived by Euclid is invariant under similarity.
In the finite-dimensional case it is clear that a Euclidean space with an origin and a scale should be identical to what is meant by a real inner product space, which for what it's worth is in turn the same thing as a real Hilbert space. At the least the article says this, but it buries it in so much lead-up as to have dissipated all the force of that definition - no one trying to understand what a "Euclidean space" is will cotton on that as the key property as the article is currently written.
Regarding "infinite-dimensional Euclidean space", absent any standard notion of this it seems the best that can be done here is to point to such standard notions as Hilbert space and topological vector spaces. If I had any say in the definition of "Euclidean space" I'd restrict it to finite dimensional spaces.
"Pure" or origin-free and scale-invariant Euclidean space, in agreement with Euclid's conception of geometry as being invariant under similarity including translation and dilatation, seems to be a Wikipedia research project to judge by Talk:Affine space. --Vaughan Pratt 07:44, 9 September 2007 (UTC)
Vaughan, I agree with your math (except that I would substitute rotation where you use similarity). But someone trying to understand what a "Euclidean space" is will not gain any insight from the explanation that it's a real Hilbert space, right? Although WP is not a textbook, we should try to make it useful for learning. Balancing correctness with accessibility is difficult, as you can tell by all of the intro attempts below.
I'm fine with restricting this article to finite dimensions. Other than that, your viewpoint seems to match the Intuitive Overview section pretty well. Is your complaint that the Intro section is poor or too long? If so, welcome to the club. Joshua R. Davis 19:10, 9 September 2007 (UTC)
I wasn't complaining about the article so much as about the discussion. It's not clear what to change in the article as long as the discussion is at cross purposes.
Regarding rotation vs. similarity, I said "geometry as conceived by Euclid is invariant under similarity." If you disagree, what theorem of Euclid is not invariant under similarity? My understanding of Euclid's geometry is that it is not Euclidean geometry but similarity geometry. Unlike Euclidean geometry, similarity geometry has no concept of origin or unit length, but it can tell when two line segments are orthogonal or have the same length, unlike affine geometry. --Vaughan Pratt 06:58, 11 September 2007 (UTC)
I assumed (incorrectly) that you meant Euclidean geometry when you said Euclid's geometry. I have only a passing familiarity with the latter and hence no valuable opinion on it. Cheers, Joshua R. Davis 20:33, 11 September 2007 (UTC)

[edit] Anon's merge proposal

See Talk:Coordinate space for why I disagree. Melchoir 19:06, 25 January 2006 (UTC)

[edit] Euclidean Vector Space not equal to Euclidean Space

Please understand this. They are not the same.

[edit] Isomorphic, so equal?

Of course you oughtn't combine them. I would go further. The isomorphism between Euclidean space and R^n is nontrivial. It rests on geometric theory. And the fact that Euclidean space is equipped with a Euclidean norm ought to be at the top of the article.--VKokielov 14:50, 6 June 2006 (UTC)

I agree with your points. I added Euclidean norm to the intro. I'm not sure what part of the article you're referring to with the equality/isomorphism bit, nor what change you have in mind. Could you clarify? -lethe talk + 14:55, 6 June 2006 (UTC)
Sure. We ought to highlight the structure which the link with geometry imposes on Rn, and explain how analytic geometry arises from the correspondence. --VKokielov 14:59, 6 June 2006 (UTC)
I'm still not sure what problem you're trying to correct, but I don't think the new section looks very good. -lethe talk + 15:18, 6 June 2006 (UTC)
A Euclidean Space need not have any notion of coordinates, so why is the article based on this?
I've reverted VKokielov's edits, since I think the original version is better. Paul August ☎ 17:33, 6 June 2006 (UTC)

[edit] Obnoxious article

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)
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)
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)
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)

I don't know math words well. Would I be correct in saying, "Euclidean space is a geometrical term used to describe a mathematical space, such as a line or a plane, that follows the laws of Euclidean geometry. One of the theories of Euclidean geometry is the theory that any two given parallel lines are the same distance from each other at all points"? --Raijinili 08:03, 2 September 2006 (UTC)

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)
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)

[edit] What??!?

What the hell is this article on about? Please explain it in plain English for us idiots.

[edit] intro

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)

[edit] Revision of 10 Sep 2006

Hello all. I just rewrote the intro, added a section on intuition, and fiddled with the rest as well. In doing this I reworded a lot of writing that was already good, and removed some other good stuff entirely, in my attempt to make it all coherent. I apologize for stepping on any toes.

I think that we can remove the "needs context" tag. Comments? Joshua Davis 16:07, 10 September 2006 (UTC)

I have removed the "needs context" tag and restored the original introductory paragraph, including a request to keep it accessable to the non-mathematician. I think ideally, this article would work its way up to absolute rigor in a series of steps or cycles, so that a reader could stop when they were overloaded rather than skipping it entirely. PAR 14:55, 11 September 2006 (UTC)
Gentleness and accessibility are important, and my version is weak on these. In my opinion, the current version is also weak, as follows. The average reader most wants to know:
  • What is a Euclidean space?
  • What is it good for?
Thus I think the introduction needs to have a definition (not necessarily rigorous) along the lines of "A Euclidean space is a ...", and it needs to mention some "applications". In addition, the intro must explicitly mention the term n-space, since that redirects here (although from the tags maybe that term should have a disambig page that points here instead). I also think that
  • mentioning non-flat spaces should wait for later (and in retrospect my second intro paragraph should too), and
  • mentioning that this is "that geometry stuff you probably learned in secondary school" is a powerful, personal way to give context. (Let me know if this is USA-centric.)
Although no one asked for it, here is another stab at an intro...
Plane Euclidean geometry, invented by Euclid around 300 BCE, describes the geometry of an idealized two-dimensional plane. The fundamental concepts of this geometry are distance and angle; in terms of them, higher-level notions such as congruence (equivalence between shapes) can be defined. Over the millenia the theory has been tremendously successful in solving practical spatial problems, and for that reason it has become a standard part of school mathematics curricula throughout the world.
However, the universe that we inhabit appears to be three-dimensional, and so it is natural and useful to attempt a generalization of plane Euclidean geometry to some kind of three-dimensional "Euclidean space". Indeed, many scientific problems are most conveniently stated in more than three dimensions, so even higher-dimensional "Euclidean spaces" are desirable.
It turns out that one can recast Euclid's fundamental concepts in more abstract language, with the advantage that they then generalize naturally to higher dimensions. The result is n-dimensional Euclidean space (n-space, for short — often denoted En), a space with properties of distance and angle analogous to those of the familiar Euclidean plane. This kind of space serves as the standard geometric background for most problems in science, engineering, economics, and other quantitative disciplines. It is also the prototype for still more general concepts of space in mathematics.
(By the way, I may be out of communication for the next couple of days.) Joshua Davis 20:57, 11 September 2006 (UTC)
Can a Euclidean space be correctly described as having the property that it extends infinitely in all directions? The representation of another type of geometry is represented by the sphere, which I don't think extends infinitely in all directions. Also, can the Euclidean space be described as the most intuitive of the spaces? Or would that be too non-NPOV? --Raijinili 03:34, 12 September 2006 (UTC)
I think you are getting at compactness. The sphere is compact, while Euclidean space is not. But this is not really the defining characteristic of Euclidean space; that would be something like flatness. Joshua Davis 12:56, 12 September 2006 (UTC)
I've reverted to Joshua Davis' version since I think it is better than what was there before. In particular the first sentence is much better:
In mathematics, specifically in geometry, a Euclidean space is a mathematical space that follows the axioms of Euclidean geometry.
The previous one:
The three dimensional space that we live in is, for most practical purposes, a three dimensional Euclidean space and the properties of objects in this space are described by solid geometry.
sounds more like the begining of an article about physical space rather than the abstract mathematical concept. Paul August ☎ 04:01, 12 September 2006 (UTC)
  • To a mathematician, the statement makes sense. To anyone else, Its about one step away from saying "A Euclidean space is a space which is Euclidean." Its a really bad sentence from this standpoint. Also, physical space is essentially the same as the abstract Euclidean space when you actually deal with it (as opposed to thinking about it), and for humans up to about 100 years ago, it was entirely Euclidean. Why not use this fact to give an intuitive introduction?
other problems:
  • "The Euclidean plane E2, which is two-dimensional, was originally studied by Euclid around 300 BCE; today it is a standard part of school mathematics curricula throughout the world." The Euclidean plane is not a standard part of school mathematics curricula throughout the world. The study of plane Euclidean geometry and solid Euclidean geometry is.
  • "The key concepts of plane geometry, such as distance and angle, generalize naturally to n-dimensional Euclidean space, denoted En and often called simply n-space". This sentence assumes one knows what "n-dimensional" means. A non-mathematician probably will not.
etc. etc. etc.
I have reverted to the other introduction. I am not saying that it is the best, but please, the whole point of this is to get an introduction which gives a quick intuitive understanding of what a Euclidean space is and get rid of the "insufficient context" complaint. Very simply, its the space we live in as organisms. Our minds and bodies and senses react to space as if it were Euclidean. Only when we reach high school and begin to study what amounted to higher mathematics a few hundred years ago do we begin to understand that there are (what used to be) very esoteric experiments that indicate that perhaps we do not live in a Euclidean space. Also, we might imagine what it would be like to be two dimensional beings living on the surface of a sphere, etc. but the bottom line is that we are Euclidean animals. Why can't we use this fact to introduce Euclidean geometry? Somebody who is not a mathematician comes to this page, they don't want to hear about how Euclidean space is a mathematical construction in which the concepts of blah blah blah, they want to know WHAT IS IT? Its simple - its the space we live in, practically, or the space of a flat piece of paper we draw on.
Then you start to explain the aspects of the space that are important, in simple ways.
Then you build up to the mathematical definition. PAR 04:42, 12 September 2006 (UTC)
PAR, I think you are confusing physics and mathematics. This article is about the mathematical concept of Euclidean space, it is not an article about physics. Like all mathematical notices, Euclidean space is an abstraction, so leaving aside the apparent fact that the physical space we live is not correctly modeled by Euclidean geometry, it is incorrect to say that Euclidean space is the space we live in, since you can't live in an abstraction. Granted it was motivated by a desire to model the geometry of physical space as perceived by our senses (incorrectly as it turned out). So it might be correct to say that physical space is WHY IT IS, but not WHAT IT IS. Paul August ☎ 05:52, 12 September 2006 (UTC)
Please do not revert the intro back to my version. I agree with PAR's criticism that the first sentence is nearly vacuous. Most of his other objections I have mitigated in the newer version. I think it's better than the standing version. In addition to the criticisms I wrote above, which PAR has not addressed, I strongly agree with Paul August, that the standing version confuses physics with math. Two things that the public doesn't understand about math are
  • math involves precise definitions and proven theorems, and
  • mathematics is not physically real; it's just a language used to model reality, with varying success.
I think that the current introduction perpetuates these misconceptions. In particular, when ones says that E^3 is "the space we live in", exactly how much do you mean? Do you mean that the speed of light in it is 186,000 miles per second? Do you mean that it's mostly empty of matter? No, none of that. You just mean that there are three degrees of freedom at every point, and that distance and angle are just so.
Paul and PAR, I wonder what you think of my second version, posted above. It's very gentle, introducing n dimensions only after 2 and 3. It also hints at what we really want, which is the role of the inner product in defining distance and angle. It gives historical context and mentions usefulness. The complaint that it doesn't include 3D in "standard worldwide curriculum" can be addressed easily. Of course, feel free to amend it; I don't intend to own the article. Joshua Davis 05:37, 12 September 2006 (UTC)

I agree the second version is good, perhaps better than the present one, except I really think that for a "newcomer" the point should be immediately made that we live our everyday lives in a Euclidean space. Ok, strictly speaking, that's not true. We live our everyday lives in a space in which geometrical relationships are very accurately modelled by the abstract mathematical system known as a "3-dimensional Euclidean space". At this point the newcomer has to read this sentence two or three times, then wonder what does "modelled by" mean?, what does "abstract mathematical system" mean? and then finally goes away or maybe asks on the talk page why this is such an obnoxious article. And I think rightly so. For the sake of communication with a newcomer, can we leave the distinction between the physical space and the abstract system until later? Joshua said that "Two things that the public doesn't understand about math are ..." and that's exactly why the introductory sentence should not assume this understanding.

The present introductory sentence has the advantage of immediately orienting the newcomer to the subject matter by drawing on everyday experience and referencing the topics of plane and solid geometry which everybody studies in high school. Now we can begin the process of refining concepts, particularly the importance of angles, distance, etc. PAR 17:51, 12 September 2006 (UTC)

I don't see anything wrong with saying that a Euclidean space is a space that follows Euclidean geometry. You can give an intuitive explanation, too, but if they want to really understand it, they need to know what a mathematical space is and what Euclidean geometry is. The introduction should simply consist of that statement (with the wikilinks), some descriptions/properties, and an analogy or two. Anything else is a problem for the Euclidean geometry and Space (mathematics) pages. --Raijinili 22:27, 12 September 2006 (UTC)
Raijinili is right - there is nothing wrong with a first sentence that is slightly vacuous, as long as some idea of what Euclidean geometry is is given immediately afterwards. Even the Britannica article referred to earlier as helpful begins in this way (although it wrongly restricts the defn to 2 or 3 dimensions). It is also vital to make the distinction between physical space and the abstraction - ignoring it is as bad as assuming it. How about:
In mathematics, particularly geometry, a Euclidean space is a space that follows Euclidean geometry, with axioms involving the fundamental concepts of distance and angles which were originally intended as statements about physical reality. Beginning with Euclid around 300 BCE, this geometry has been a useful model for spatial problems and to this day geometry in two and three-dimensional Euclidean spaces is a standard part of school mathematics curricula.
The properties of these familiar spaces have been naturally generalised to an arbitrary dimension. An n-dimensional Euclidean space, often called n-space, is equivalent as a metric space to the n-dimensional real coordinate space or Cartesian space, that is, the set of ordered n-tuples of real numbers, with distance defined by the Euclidean metric. Higher dimensional spaces are useful in many quantitative disciplines.
In modern mathematics, Euclidean spaces form the protoypes for other, more complicated geometric objects, such as manifolds, which embrace both Euclidean and non-Euclidean geometry. From this point of view, the essential property of Euclidean space is that it is flat — that is, not curved. Modern physics, specifically the theory of relativity, demonstrates that our universe is not truly Euclidean. Although this is significant in theory and even in some practical problems, such as global positioning and airplane navigation, a Euclidean model can still be used to solve many other practical problems with sufficient precision.
There is room for additions, and the wording isn't wonderful yet, but I think we need to cover at least this material, particularly the redirect terms and flatness, in a way that starts with what a Euclidean space is, not what Euclidean geometry or the physical world is. JPD (talk) 10:02, 13 September 2006 (UTC)
We begin to get an intuitive understanding of Euclidean geometry and Euclidean 3-space from the moment we are born. An intuitive (kinesthetic) understanding of Euclidean 3-space is about fully developed in a 12 year old child. Why is it that you want to write an introduction in which a high-school student is told "you will not be given an intellectual explanation of Euclidean space until you have intellectually mastered the concepts of Euclidean geometry, mathematical space, distance, and angle. Why can't we relax and let them know its not a secret, that they already have a good grasp of what a Euclidean 3-space is, they just don't know it.
Here's an introduction to an article on a "chair"
"A chair is particular type of mechanical support system constructed of material having the proper modulus of elasticity and coefficient of compressibility to prevent the downward acceleration of the supported object in a gravitational field or, equivalently, an accelerated reference frame."
Why can't we begin by saying "A chair is something a person sits down upon"? Oh, because thats not strictly correct. Not all people are able to sit down on a chair and besides, a non-person (e.g. Pan troglodytes) can also sit down upon a chair. Also we have to keep a strict separation between the use of the chair and its theory of construction. God forbid we confuse the two!
Ok, I'm being a bit sarcastic, but this illustrates the feeling I get when I read an introduction which ignores the fact that Euclidean 3-space is right here and now, its what you are living in! PAR 15:23, 13 September 2006 (UTC)
We may well naturally develop a kinesthetic understanding of the real world. However, the real world is not Euclidean 3-space and Euclidean 3-space is most definitely not the real world. We don't bother talking about things like Euclidean space until we are considering the abstraction. If you can reword the first sentence or two so that they more clearly bring to mind the high-school/intuitive understanding of the real world as the origin of the Euclidean model, then do so. However, don't change the whole emphasis of them so that they are no longer mainly talking about what Euclidean space is. Making an article accessible does not mean writing in textbook style. JPD (talk) 15:55, 13 September 2006 (UTC)
Ok, then can we leave the first sentence more or less as it is? I am not sure what you mean when you say the real world is not Euclidean 3-space. Do you mean that Euclidean 3-space is a mathematical abstraction as opposed to reality, or do you mean that Euclidean 3-space is a poor description of the real world? When I say the real world is a Euclidean 3-space, I mean that if we identify certain physical measurements with certain mathematical operators, the real world is practically isomorphic to Euclidean 3-space for every "kinesthetic" situation and for most other situations. PAR 16:36, 13 September 2006 (UTC)
When I say the real world is not Euclidean 3-space I am referring to the fact that Euclidean space is not a perfect model. More importantly, in saying that Euclidean 3-space is not the real world, I mean that Euclidean 3-space is not a physical reality like a chair is. It is an abstraction and nothing more, so that kinesthesia is not relevant. I really don't like how the article starts at the moment, not because the first sentence is inaccurate, but because it is talking about the space we live in, instead of Euclidean space. The first few sentences of my version were intended to convey the connection between the intuitive understanding of the real word and what Euclidean space is, but I readily admit that they probably don't do it very well. Would you be happier with something like: "In mathematics, particularly geometry, a Euclidean space is a space with properties that generalise the basic properties of plane geometry and solid geometry. These properties (the axioms of Euclidean geometry) originated as statements about the space we live in, made by Euclid around 300 BCE." ? JPD (talk) 17:11, 13 September 2006 (UTC)
A chair is not a deep concept. A Euclidean space, however, is a term in math, that doesn't mean anything outside of it. It's not the same thing. Besides, a chair is an object that is used for the purposes of resting the buttocks, and anything else goes on the disambiguation page. >_>. --Raijinili 04:12, 14 September 2006 (UTC)

[edit] Intro redux

I would like to offer some contextual remarks to all parties trying to arrive at a satisfactory intro.

This may be one of the more difficult introductions to do well in all of our mathematical articles. I will explain why, but please bear that in mind as editing and debate progresses.

Mathematical introductions are frequently difficult because of a tension between precision and intuition. One approach to writing begins with a formal definition and follows with exploration. Another begins with context and motivation, then moves to formalities. We see the tension between these approaches here.

Every technical Wikipedia article faces a second challenge, which is not only to introduce the topic to a broad general audience, but also to provide substance for a specialist audience. This issue is addressed somewhat by the Mathematics Manual of Style, which suggests “It is a good idea to also have an informal introduction to the topic, without rigor, suitable for a high school student or a first-year undergraduate, as appropriate.”

This article presents a still greater challenge, because so many people "know" something about it, at different levels of expertise. Where not many people would have the confidence to edit the article on “Čech cohomology”, say, millions have a passing familiarity with Euclidean geometry, which means dozens (at least) of potential editors.

Toes will be stepped on, maybe yours. It will be a miracle if everyone is equally satisfied with the outcome. And today's hard-fought consensus will be obliterated in a few months, by editors who have no sensitivity, no awareness, no interest in past debates. Really good editors will walk away, shaking their heads sadly, leaving the field to the really persistent editors.

One practical observation: It is virtually impossible for a single opening sentence to meet all the goals of all the editors. Be prepared to use a paragraph or two instead. The first sentence should inform, engage, and set the stage as much as possible, but it cannot carry the weight of a full article. Use it wisely.

Try never to lie. If no phrasing can be found that makes a statement both exact and accessible, use a qualification like “informally” or “is like” to appease the demand for rigor. For example, rather than make a false statement by saying “a flat piece of paper is a two dimensional Euclidean space”, make a true informal statement by saying “a two dimensional Euclidean space is like a flat piece of paper.” That one extra word and slight rephrasing can make a huge difference in acceptability!

Non-mathematicians need to understand that the history of mathematics includes an early phase of informality followed by a later phase of rigor, and all modern mathematicians have drilled into them the reasons why rigor is essential. Mathematicians need to understand that rigor without insight and intuition is useless.

Real world concerns prevent me from engaging in this effort today, but I hope these observations can facilitate the process for others. I offer as a beacon of hope the “manifold” article, which despite a difficult past has become a point of pride for Wikipedia. --KSmrqT 08:04, 14 September 2006 (UTC)

Thank you for the comments, KSmrq; all of this is worth keeping in mind.
There are several very specific problems with the current version, which PAR has not addressed. In an attempt to make this article conform to extremely basic standards, I have made the following extemely mild alterations to the intro:
  • It now has a non-rigorous "definition" of Euclidean n-space, accomplished by simply putting those words in bold. The posters calling this article "obnoxious" seem to hold this as their main complaint.
  • It explicitly mentions the term "n-space", since that redirects here.
  • Just by putting in a paragraph break, it emphasizes that what makes Euclidean space different from other kinds of three-dimensional manifolds is flatness. (The PAR opening sentence is certainly too vague to detect this distinction.) I don't think this is necessarily the right approach to the concept, but if we choose to go with it, then we need to explain flatness later in the article. I have also corrected the oversight that relativity without gravity is, in fact, flat.
I've also moved the paragraph on manifolds to its own section, unified with other manifold stuff. This edit addresses the most egregious of my complaints. Now for the more subtle ones:
  • As Paul August pointed out above, the opening lines are about (pseudo-)physics, not math.
  • If the inner product is the key idea to understand in the generalization, then the intro should at least allude to this fact. In my version I attempt this by discussing "distance" and "angle", which are simple, understandable proxies for the inner product. The current version doesn't do this at all. It just says "abstract", which if anything is counter to PAR's goal of not scaring the reader. Joshua Davis 22:33, 14 September 2006 (UTC)
The first sentence reads: "The three dimensional space that we live in is, for most practical purposes, a three dimensional Euclidean space". What is wrong with this sentence? Yes, it is physics, but it most emphatically NOT pseudo-physics (i.e. crackpot craziness). It doesn't say that special and general relativity is wrong, it doesn't say that we live in an absolutely Euclidean space, it says for most practical purposes which is correct AND informative. If we map certain mathematical operators of the abstract Euclidean space to certain physical operations, then the space we live in is, with respect to these operations, a Euclidean space, for most practical purposes. This is what mathematical physics does. It notes that certain physical situations are, under the proper definitions, identical to certain abstract mathematical systems. The "obnoxiousness" of the original article, as I understand it, was just the refusal to introduce any kind of intuitive assistance to a newcomer to orient themselves as the the meaning and importance of a Euclidean space. This desire to remain absolutely detached from the real world from the beginning is counterproductive. The original first sentence read: "In mathematics, n-dimensional Euclidean 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." I actually like that sentence. To a mathematician, it is a neat, concise summary of Euclidean space. But to anyone else it is obnoxious, and I am sure that this is how this article earned that epithet.
Please, when it comes to the introduction, edit away. I just became stubborn about new edits because it seemed to me that no one really "got" what the problem was. The introduction kept drifting back towards the original impenetrable obnoxiousness in search of mathematical precision and purity. As KSmrq said "It is a good idea to also have an informal introduction to the topic, without rigor, suitable for a high school student or a first-year undergraduate, as appropriate" PAR 15:00, 15 September 2006 (UTC)

The first sentence is fair enough by itself, just not as the opening sentence to this article. It is definitely a good idea to have an informal introduction to the topic, but it should still start with at least a sentence about what the subject of the article is. This is appropriate for a Wikipedia article, whatever our usual approach to mathematical writing is. It doesn't need to be the "obnoxious" technical precise definition, though, and as KSmrq says, we shoudl expect to need more than one sentence to fit in everything that people would like to see at the start. Does anyone have any constructive comments concerning my suggestions above? I agree that talking about distance and angle, is a good alternative to inner product, and point out that all the terms I had in bold in my first suggestion redirect here, and so should feature in the intro. JPD (talk) 18:20, 15 September 2006 (UTC)

I'm sorry that I forgot to acknowledge it. Among the mathy versions, it's probably best. It mentions distance/angle and in the second paragraph mentions how distance gets generalized. I would rearrange the first paragraph a bit for wording (but I guess it's a matter of taste):
In mathematics, particularly geometry, a Euclidean space is a space with concepts of distance and angle that follow the axioms of Euclidean geometry. These axioms were originally developed by Euclid around 300 BCE as a model for physical two- and three-dimensional space. Indeed, Euclid's geometry has been tremendously useful in describing and solving spatial problems, and to this day geometry in two and three-dimensional Euclidean spaces is a standard part of school mathematics curricula.
PAR asks again what is wrong with the current first sentence. I have been trying to figure out what I don't like about it. How about this: It describes any three-dimensional space, not just Euclidean 3-space. So I don't like that it tries to "define" Euclidean space as the space we live in; I would prefer that it merely attempt an analogy between the two. So here is a revision that I prefer:
The physical space in which we live appears to be three-dimensional, and a good model for this three-dimensional space is solid geometry. An idealized plane surface in this space, akin to a flat sheet of paper, is modeled by plane geometry. Both are examples of Euclidean geometry, developed by Euclid around 300 BCE.
The fundamental concepts in these geometries are distance and angle; in terms of them, higher-level notions such as congruence (equivalence of shapes) may be defined. It turns out that distance and angle can be abstractly reformulated, in the guise of an inner product, so that they make sense in any dimension. The resulting construct is n-dimensional Euclidean space — n-space, for short — a version of Euclidean geometry in n dimensions.
(If anyone wonders why I keep bringing in translation/rotation/congruence, it's because I'm trying to allude to the Erlanger Programm.)
PAR, I am not angry with you, and I do not mean to flame you, but may I politely request that you offer revisions of your own, or accept one of the offered revisions? Right now I feel that JPD, I, and others are putting all kinds of work into this, because the article sorely needs it, but you veto our attempts without offering your own improvements. Respectfully, Joshua Davis 19:32, 15 September 2006 (UTC)
No problem. To echo a comment made above, I don't intend to "own" the article. How about a chronological introduction:
Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called "plane Euclidean geometry", which is the study of the relationships between angles and distances of objects drawn on a flat two-dimensional surface. This was soon extended to three dimensions to describe similar relationships for solid three-dimensional objects. Ultimately, all of the results of Euclid were encoded into an abstract mathematical space known as a two or three dimensional Euclidean Space. These mathematical spaces may be easily extended to apply to any dimensionality, and such a space is called an n-dimensional Euclidean space or an n-space. This article is concerned with the mathematical structure of such a space.
It still needs work. I have not addressed the legitimate concerns of everybody, but you get the general idea. PAR 20:05, 15 September 2006 (UTC)
As a matter of style, I still prefer that the first sentence is not simply about Euclid did, but describes (vaguely) what a Euclidean space actually is. Apart from that, Joshua's rewordings seem reasonable, although Euclid possibly did think of his model as more than a model. I'm not too keen on PAR's verion after "Ultimately, ...", though. Axioms are encoded in the idea of Euclidean space, not Euclid's results, and these can be generalised to higher dimensions. Such a space is a mathematical structure, so it seems redundant to say that the article is about it's mathematical structure. JPD (talk) 11:03, 16 September 2006 (UTC)
The chronological development has advantages.
  • It is (or can be made) strictly correct.
  • It does not blur the distinction between the physical space and the abstract space.
  • It (should) build up from simple knowledge (angles, distance, flat surface, plane geometry, solid geometry) to an explanation of Euclidean space.
What I am trying to say after "Ultimately..." is how the mathematical abstraction known as Euclidean space was developed from the axioms, results, theorems, whatever, of Euclid. If it is not correct as it stands, then please, fix it. I've tried to address your concerns below:
Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called "plane Euclidean geometry", which is the study of the relationships between angles and distances of objects drawn on a flat two-dimensional surface. This was soon extended to three dimensions to describe similar relationships for solid three-dimensional objects. All of the axioms of Euclid were then encoded into an abstract mathematical space known as a two or three dimensional Euclidean Space. These mathematical spaces may be easily extended to apply to any dimensionality, and such a space is called an n-dimensional Euclidean space or an n-space. This article is concerned with such spaces.

Sorry, I thought I had replied to this already, but obviously not. The chronological approach has many advantages, and I don't have too many problems with this version as a paragraph about Euclidean spaces. However, I think it departs from the normal style of a first paragraph in a Wikipedia article unnecessarily. This is a reference work, not a text book. It doesn't hurt to start with a sentence that actually mentions "Euclidean space" and gives a rough idea what sort of thing it is. JPD (talk) 08:46, 20 September 2006 (UTC)

[edit] A point to keep in mind

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)

Also, use more headers. This is a terrible, terrible mess. --Raijinili 08:58, 16 September 2006 (UTC)

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)
What about "intuitive" explanation having its own section? --149.4.211.70 14:38, 21 September 2006 (UTC)

Regarding the 4th sentence of the introduction, I would question the utlity of the phrase "encoded into an abstract mathematical space". The word "encoded" generally implies that information has been condensed into some difficult-to-read form. Is that true for the axioms of Euclid? I do not know enough about Euclidean space to make definitive changes to the article, so I'll leave that to someone else. But perhaps something like:

"The concrete axioms of Euclid have been taken together to describe an abstract mathematical concept which is a two- or three-dimensional space, called a Euclidean space."

I've tried to keep that as simple as possible since I think the first paragraph ought to be highly accessible to all kinds of people.

Perhaps this might work well if we try to keep in mind writing for a younger audience, such as elementary school children, at least at the beginning of the article. More sophisticated readers can easily scan through the article to reach the part that interests them. And wouldn't it be great if elementary school children did read the article and get something out of it? Cheers. -Paul 198.81.125.18 19:33, 11 September 2007 (UTC)

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)

[edit] Orders on the Euclidean space

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)

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)
Perhaps we need a separate page on Rn then?--Patrick 21:55, 20 May 2007 (UTC)
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)

[edit] R^0?

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)

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)