# Talk:Euclidean space/Archive 1

This is an archive of past discussions. Do not edit the contents of this page. If you wish to start a new discussion or revive an old one, please do so on the current talk page. |

Archive 1 | Archive 2 |

## 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
**R**^{1}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 *:
**R**xV |--> 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**R**^{0}is (here) defined through a special definition of n-tuples, which starts out from the 0-tuple {}, for every set S: S^{0}=**R**^{0}= {{}}. The set**R**^{0}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**R**^{1,0}. (viii) The square of**R**as an associative algebra over the field**R**, and a coordinatization of**R**^{1,0}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),**Vct**_{R}(vector spaces over the reals and their linear transformations), and**Alg**_{R}(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.

- Vaughan, I agree with your math (except that I would substitute rotation where you use similarity). But someone

- 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)

## Anon's merge proposal

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

## Euclidean Vector Space not equal to Euclidean Space

Please understand this. They are not the same.

## 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 , 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)

- I'm still not sure what problem you're trying to correct, but I don't think the new section looks very good. -lethe

- Sure. We ought to highlight the structure which the link with geometry imposes on , and explain how analytic geometry arises from the correspondence. --VKokielov 14:59, 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)

## 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)

## What??!?

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

## 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)

## 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...

- 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:

- 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.

- 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

- 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**(, for short — often denoted*n*-space**E**^{n}), 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.

- 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

- (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.

- In mathematics, specifically in geometry, a

- 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.

- The three dimensional space that we live in is, for most practical purposes, a three dimensional

- 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)

- 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

- 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*.

- 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

- 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)

- Paul and PAR, I wonder what you think of my second version, posted above. It's very gentle, introducing

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, is equivalent as a metric space to the*n*-space*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.

- In mathematics, particularly geometry, a
- 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)

- 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:

- 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!

- 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.

- 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)

- 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

- 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)

## 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. --KSmrq^{T} 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.

- It now has a non-rigorous "definition" of Euclidean
- 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.

- 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

- 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.

- In mathematics, particularly geometry, a

- 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**—, for short — a version of Euclidean geometry in*n*-space*n*dimensions.

- 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

- (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.

- Around 300 BC, the Greek mathematician

- 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.

- Around 300 BC, the Greek mathematician

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)

### 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)

Eucledian geometry exist only by approximation and does not exist sub- and super perseptaul. PW Botha 2009 09 21

## 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. --KSmrq^{T} 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 -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
**R**^{n}then?--Patrick 21:55, 20 May 2007 (UTC)

- Perhaps we need a separate page on

- I don't think so, given that most things about
**R**^{n}as a set are probably covered in general articles about Cartesian products and so on. Why should an article about Euclidean spaces, or**R**^{n}, 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**R**^{n}, 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)

- I don't think so, given that most things about

## 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**R**^{n}and sometimes called**real coordinate space**."

Shouldn't this be:

- "For any
__positive__integer*n*..."

Or do we need to explain what **R**^{0} is? - dcljr (talk) 15:22, 4 June 2008 (UTC)

- A mathematician would certainly acknowledge
**R**^{0}= {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)

## improper lead writing style

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)

- 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)

The Wikipedia:Lead section Opening paragraph [1] manual of style clearly states that "The article **should begin with a short declarative sentence, answering two questions for the nonspecialist reader: "What (or who) is the subject**?" and "Why is this subject notable?"... Please try to fix this, or I feel obliged to try to do it myself (although I am *not* a mathematitician, so I really prefer that someone more familiar with math would do that!) Kaarel (talk) 22:26, 15 June 2009 (UTC)

See also the "Writing better articles" [2] Kaarel (talk) 23:04, 15 June 2009 (UTC)

## ariplane navigation?

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)

- Search Google for GPS and relativity together and you get many hits --- for example, [3] or [4]. 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)

## More concise definition

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)

- 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)

## Orientation in euclidean space

The article says "Rotations of Euclidean space are then defined as orientation-preserving linear transformations T that preserve angles and lengths:"

but it doesn't mention orientability earlier. Does one have to asume that euclidean space is an *oriented* vector space with the dot product? Or orientation is not necesary? — Preceding unsigned comment added by 186.22.56.37 (talk) 05:45, 8 January 2012 (UTC)

- The concept of "orientation-preserving linear transformation" belongs to vector spaces, not Euclidean spaces. Basically, "orientation-preserving" means "positive-determinant". The article links to Orientation (vector space) at that point. I think that it's enough. Mgnbar (talk) 14:28, 8 January 2012 (UTC)

- I don't agree with Mgnbar. The article Orientation (vector space) says "The orientation on a real vector space is the arbitrary choice of which ordered bases are "positively" oriented and which are "negatively" oriented", and "A vector space with an orientation is called an oriented vector space, while one without a choice of orientation is called unoriented", so not every vector space is oriented. As euclidean spaces are vector spaces, it is important to know if they are oriented or not, or if it's the same whether it's oriented or not. And if it is, what exactly the "arbitrary" choice of positive bases. It is not clear this way. — Preceding unsigned comment added by 186.58.69.141 (talk) 06:47, 15 January 2012 (UTC)

- Unless I'm mistaken, an orientation on a vector space is not required, to define which linear transformations are orientation-preserving. For example, the identity map on the vector space is inherently orientation-preserving, right? More generally, choosing one orientation on a vector space, or its opposite, leads to the same definition of which linear transformations are orientation-preserving. For either choice of orientation, the orientation-preserving linear transformations are simply the positive-determinant linear transformations. Am I right? If so, and if we added a note in the article to that effect, would that be enough to clarify the article? Mgnbar (talk) 14:29, 15 January 2012 (UTC)

- With the disclaimer that I'm not an expert, it would certainly appear to me that Mgnbar's logic is correct. — Quondum
^{☏}✎ 17:40, 15 January 2012 (UTC)

- With the disclaimer that I'm not an expert, it would certainly appear to me that Mgnbar's logic is correct. — Quondum

- Mgnbar two things aren't clear to me in your reasoning. First, you say that "the orientation-preserving linear transformations are simply the positive-determinant linear transformations", but what if the vector space is infinite dimensional? What is the determinant of a linear transformation (I asume you mean an automorphism on the vector space) if the space is infinite dimensional?. Second, the orientation article says "The orientation on a real vector space is the arbitrary choice of which ordered bases are "positively" oriented and which are "negatively" oriented", so if chosing an orientation is completly arbitrary, I don't think that one can define orientation-preserving maps without first establishing an orientation for the vector space in question. Using determinants wouldn't necesary agree with the orientation (ie. a map with negative determinant but doesn't change orientation). And third, I still doubt if the orientation is part of the definition of the euclidean space or not (it is common to use the right hand rule in 3 dimensions in physics in eclidean 3-space), but the article doesn't mention that there has to be an orientation on euclidean space. — Preceding unsigned comment added by 186.58.22.241 (talk) 22:18, 17 January 2012 (UTC)

- I hadn't considered the infinite-dimensional case. That's a good point. But I think that infinite-dimensional inner product spaces are not usually called "Euclidean spaces". You might want to check out Hilbert space, too. The rest of my remarks apply to finite-dimensional inner product spaces. I agree that three-dimensional Euclidean space is commonly oriented "according to the right-hand rule". Also, two-dimensional Euclidean space is commonly oriented "counterclockwise" in mathematics. These are the popular choices of orientations on two of our most popular Euclidean spaces, but they are not essential to the concept of Euclidean space.
- As I argued above, one can define "orientation-preserving linear transformation" without any orientation being chosen on the vector space in question. Let me make this more explicit. Suppose that you and I are discussing a particular vector space, whose bases are partitioned into two sets A and B. You view bases A as positively oriented and bases B as negatively oriented, while I prefer to call bases B positively oriented and bases A negatively oriented. For you, the orientation-preserving linear transformations are the ones that take A to A and B to B. For me, the orientation-preserving linear transformations are the ones that take B to B and A to A. See? We have the same notion of orientation-preserving linear transformation, even though we disagreed on the orientation. Mgnbar (talk) 22:57, 17 January 2012 (UTC)
- I have been thinking about it and now I realize I don't see any difference between a finite dimensional Hilbert Space over and a non-oriented Euclidean Space. Are they equivalent? — Preceding unsigned comment added by 186.22.56.37 (talk) 20:14, 28 January 2012 (UTC)

- A finite-dimensional real Hilbert space (FDRHS) is the same as a finite-dimensional real inner product space (FDRIPS), because any finite-dimensional space is going to be complete with respect to the induced norm. In turn, a FDRIPS is
*essentially*the same thing as a Euclidean space, I think, but the word "essentially" hides at least two details. First, as this article mentions, a Euclidean space is not itself an FDRIPS, but rather an affine space on which the FDRIPS acts. Second, Euclidean space is usually defined not just with any inner product, but specifically with the dot product. However, it seems to me that any inner product can be made to look like the dot product by changing basis (this follows from the spectral theorem). So then it's just an issue of how coordinates are chosen; any FDRIPS is isomorphic to Euclidean space of the same dimension. But maybe this isomorphism isn't canonical, and maybe that's important...I don't know.

- A finite-dimensional real Hilbert space (FDRHS) is the same as a finite-dimensional real inner product space (FDRIPS), because any finite-dimensional space is going to be complete with respect to the induced norm. In turn, a FDRIPS is

- By the way, this talk page is not a help desk for math questions, but Wikipedia:Reference desk/Mathematics is. Mgnbar (talk) 22:27, 28 January 2012 (UTC)

## Today's lead change

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)

- "
**Euclidean space**is the Euclidean plane and three-dimensional space of plane geometry" ?! Gibberish! —Tamfang (talk) 19:54, 11 October 2012 (UTC)

- 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)

- 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)

- 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)

- This article has always been (at least in my memory) about the modern mathematical concept called

- 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)

## Making two good articles instead of one poor

After my recent work on the article pseudo-Euclidean space I found that this article, arguably a more important one, does not inform the reader better about modern understanding of Euclidean geometry. Worse, it presents in a confusing way the distinction between ℝ^{n}, the Euclidean space proper, and a vector space with an Euclidean structure, which cannot be ignored in pure mathematics. So, I propose the following plan:

- Real coordinate space (edit | talk | history | links | watch | logs) goes away as a separate article, together with redirects real n-space, R-n, R^n, Standard topology, and Usual topology (see [5] for full list of redirects).
- Only those facts about topology are retained here, which are relevant to Euclidean structure. The bulk of content should be expelled to real coordinate space.
- The article Euclidean space become more preoccupied with geometry, and less preoccupied with coordinates.
- The Euclidean space – vector space distinction has to be formulated in terms of point–vector distinction: the article should refer to any element of a vector space as a vector, and to any element of Euclidean space as a point. So, point − point = vector, point + vector = point, etc.
- This 8-years-old talk page is moved to talk: Euclidean space/Archive 1, with links both from talk: Euclidean space and talk: Real coordinate space.
- Real coordinate space is expanded to mention Cartesian product, matrix (mathematics), and other things directly relevant to the ℝ
^{n}formalism, but only tangentially – to Euclidean spaces. There will be some overlap with vector (mathematics and physics).

Comments? Incnis Mrsi (talk) 13:40, 14 April 2013 (UTC)

- Oops… vector (mathematics and physics) is not an article. Even better: there is no thing to overlap with. Incnis Mrsi (talk) 14:00, 14 April 2013 (UTC)
- What you say is reasonable, but please be aware of the protracted battles that have taken place over this article. To oversimplify, on one side you have mathematicians, who prioritize precision and logical clarity. On the other side, you have non-mathematicians, who feel that the mathematicians make things too technical too quickly, at the expense of the reader's motivation, intuition, etc.
- So it is difficult to comment on your plan without seeing the specifics. The core of the current article is the inner product and how it leads to notions of distance and angle. Would you remove this material from the article? Would you instead focus on an axiomatic treatment of Euclidean space, along the lines of Euclid? Adding axiomatic material would be fine with me, but I think that removing the inner product would be a mistake. Mgnbar (talk) 21:40, 14 April 2013 (UTC)
- I pointed to the article pseudo-Euclidean space as a suggestion how the future article Euclidean space should look like. No, I do not propose to focus on axiomatic treatment of Euclidean space, along the lines of Euclid, because Euclidean geometry exists. Of course, I do not propose to remove the inner product. The problem is that real coordinate space redirects here, which causes the article to consider some Euclidean-unspecific stuff which makes the concept of Euclidean structure unclear. I think that all things about ℝ
^{n}which do not require this structure should go to real coordinate space – it can be linked from Euclidean, pseudo-Euclidean articles, as well as from many others. Also, I think that point–vector distinction should be explained in the article affine space and be used uniformly here, in pseudo-Euclidean space, and elsewhere. One should not explain in details in each article that an (affine) triangle is three independent points, which correspond to three vectors with one linear dependence, and it is not three*independent vectors*what the triangle is. Incnis Mrsi (talk) 05:15, 15 April 2013 (UTC)

- I pointed to the article pseudo-Euclidean space as a suggestion how the future article Euclidean space should look like. No, I do not propose to focus on axiomatic treatment of Euclidean space, along the lines of Euclid, because Euclidean geometry exists. Of course, I do not propose to remove the inner product. The problem is that real coordinate space redirects here, which causes the article to consider some Euclidean-unspecific stuff which makes the concept of Euclidean structure unclear. I think that all things about ℝ
- I just noticed that the article says not especially much about the topological structure. But an important point which is missed is that this structure does not depend on a particular metric (i.e. Euclidean structure). That's why “real coordinate space” should be separate. BTW, when I re-examined the section about it, I found it nonsensical. Perhaps it is the result of some years-old compromise, but one should not explain ℝ
^{n}starting from the concept a basis. It is a complete rubbish. Contrary, a basis in a linear space is a useless thing until one understood what ℝ^{n}(or*F*^{ n}, for an arbitrary field) means. Generally, any mathematical article should distinguish structures (metric space, vector space, metric tensor, affine space, and so on) and presentations (ℝ^{n}, other coordinate systems). Incnis Mrsi (talk) 07:26, 15 April 2013 (UTC)- Also, I just found an “interesting” article coordinate space, which IMHO is completely wrong. Incnis Mrsi (talk) 07:51, 15 April 2013 (UTC)
- I have no specific objection to your plan, and I do agree that the vector space/affine space distinction could be treated more carefully. But let's dig into it a little more. One can talk about
- Cartesian products in set theory
**R**^{n}as a vector space in linear algebra**R**^{n}as a topological space (with topology induced by various norms) in topology**R**^{n}with an inner product in geometry.

- Fundamentally, there could be an article on each of these four topics. The question is: How much should each article summarize from the previous articles? At one extreme, there is no recap/overlap/summary, so that the reader has to read all four articles, to understand Euclidean space. Although Wikipedia is not a textbook, that approach seems downright hostile to learning. I think that Euclidean space attempts to briefly summarize the earlier articles, while focusing on its own material (the inner product).
- Maybe the best way to proceed is: We (you) write a Real coordinate space article. Once that is in decent condition, we can see how to reshape Euclidean space? Mgnbar (talk) 18:22, 15 April 2013 (UTC)
- Agreed, and I’ll try to reuse some content from coordinate space, but what should happen eventually to that latter article? Incnis Mrsi (talk) 18:45, 15 April 2013 (UTC)
- I have not heard the term "coordinate space" used widely in mathematics. Is it a useful concept, in itself? In its greatest generality, it seems to be nothing more or less than Cartesian product. In its greatest specificity, it seems to be about
**R**^{n}or maybe**C**^{n}. So I can't tell whether it deserves an article separate from Real coordinate space. Certainly the safest thing to do is just write Real coordinate space separately, and later decide whether to merge, delete, etc. - Although you seem to be a highly experienced editor, in no need of my advice, my advice is to generate content first, and then clean up inter-article organization later. :) Mgnbar (talk) 22:07, 15 April 2013 (UTC)

- I have not heard the term "coordinate space" used widely in mathematics. Is it a useful concept, in itself? In its greatest generality, it seems to be nothing more or less than Cartesian product. In its greatest specificity, it seems to be about

- Agreed, and I’ll try to reuse some content from coordinate space, but what should happen eventually to that latter article? Incnis Mrsi (talk) 18:45, 15 April 2013 (UTC)

- I have no specific objection to your plan, and I do agree that the vector space/affine space distinction could be treated more carefully. But let's dig into it a little more. One can talk about

- Also, I just found an “interesting” article coordinate space, which IMHO is completely wrong. Incnis Mrsi (talk) 07:51, 15 April 2013 (UTC)

I saved my version of real coordinate space. Feel free to edit, and let’s start its talk page for ℝ^{n}-specific issues. Incnis Mrsi (talk) 10:01, 16 April 2013 (UTC)

## Assessment comment

The comment(s) below were originally left at Talk:Euclidean space/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

Needs pictures; plus more on history, context and applications. Tompw 17:06, 5 October 2006 (UTC) |

Substituted at 06:04, 24 July 2016 (UTC)

## Related discussions

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)

## Non-Cartesian coordinates

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)

## Wrong redirection from 2-norm.

Must be redirected to Norm (mathematics). Jumpow (talk) 19:04, 7 August 2013 (UTC)Jumpow

- Done. An alternative could have been Lp-space, but Norm (mathematics) seems more straightforward. —
*Quondum*20:00, 7 August 2013 (UTC)

**R** or ℝ?

WP:MOSMATH #Common sets of numbers prefers **R** over ℝ, and the majority of mathematical articles which use HTML formatting follow this. NevilleDNZ, is there some specific reason to ignore the MoS? The U+01D53C 𝔼 MATHEMATICAL DOUBLE-STRUCK CAPITAL E is an astral character and is missing from many fonts (including those used for {{math}} 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)

- BTW, de-italicizing of the variable n 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)
- 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) - 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)- 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)

- While I do not disagree with the above comments, I would like to note that WP:MOSMATH #Common sets of numbers might not be

- Without bothering to find a policy specifically addressing it, I strongly agree that

## Use of a different syntax in the markup

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)

## Generalization

The article makes the statement

*Any two distinct points lie on exactly one line. Any line and a point outside it lie on exactly one plane. These properties are studied by affine geometry, which is more general that Euclidean one, and can be generalized to higher dimensions.*

This suggests that this is where these properties "belong", whereas these properties live naturally under the more general umbrella of projective geometry, of which affine geometry can be considered a restriction. Should this not be a preferred description? —*Quondum* 08:57, 11 February 2014 (UTC)

- You are partially right and partially wrong. Yes, the projective geometry is “the most natural” one to consider the incidence of points, lines, and planes. But it lies farther from the Euclidean geometry than affine geometry. My text suggests (although does not say explicitly) that you can easily made an affine space from a Euclidean space, just by adding all transformations that preserve lines to the group of motions. You can’t make a projective space from a Euclidean space so easily. Affine geometry is about
*the same shapes in the same space*, only their classification slightly differs (for example, “circle” and “right triangle” go away). Projective geometry is not about the same shapes: even the definition of a triangle becomes~~problematical (you can’t distinguish its interior and exterior angles, for instance)~~. Incnis Mrsi (talk) 11:30, 11 February 2014 (UTC)- Actually, a good definition of a (non-degenerate) triangle requires some supplemental structure beyond just three vertices: namely, one must choose one of four quadrants in a vertex, that entails such choices for other two vertices. It becomes
*tricky*, similarly to the spherical case. Incnis Mrsi (talk) 19:22, 11 February 2014 (UTC)

- Actually, a good definition of a (non-degenerate) triangle requires some supplemental structure beyond just three vertices: namely, one must choose one of four quadrants in a vertex, that entails such choices for other two vertices. It becomes
- The particular properties mentioned do not include "shapes". The normal concept of angle disappears in affine geometry. And producing a projective geometry from either a Euclidean or affine geometry is hardly more difficult: it involves adding the (very natural) plane at infinity (3d case, or hyperplane at infinity more generally), and then adding all the transformations that preserve lines. And yes, it is "further", but this is the nature of generalization. My point is that projective geometry is precisely the level of generalization at the concept makes sense, and restricting the geometry (by fixing/stabilizing or removing sets of points, thus restricting the symmetry group), allows the flats to be retained essentially unmodified. The statement, as it stands, invites one to make an incorrect inference about which level of generality the concepts of line, plane etc. belong. In projective geometry as with affine geometry, the circle generalizes to a conic section, although in the affine case conic sections will partition into three sets according to how many of its points intersect the hyperplane at infinity. And though I realize that the projective plane is not orientable, I am having difficulty in seeing what you mean about "interior" and "exterior": a line does not divide a plane into two, but a triangle does still seem to have an interior and an exterior. —
*Quondum*17:58, 11 February 2014 (UTC)- An Euclidean/affine triangle is uniquely defined with three its points. A projective triangle is not (see above). Euclidean and affine definitions of a triangle are compatible. The projective one is not. At last, when you get a Euclidean space and forget all its structure but points, lines, planes, and their incidence, which geometry do you obtain? Incnis Mrsi (talk) 19:22, 11 February 2014 (UTC)
- I understand that three non-collinear points in elliptic and projective geometries define three lines that divide the plane into four triangles, not one of which has status over the the others as defining the "inside": one must choose. Yet three line segments joining three points do distinguish an "inside". You are seeking esoteric differences. In the context of generalizations, starting with the study of properties of flats as stated, these apply equally to Euclidean, hyperbolic, elliptic, affine, de Sitter and projective geometries. There is nothing in the statement that relates it to Euclidean geometry, other than the fact of the context of which article it is in. I could equally validly put the statement
*Any two distinct points lie on exactly one line. Any line and a point outside it lie on exactly one plane. These properties are studied by projective geometry, which is more general than an elliptic one, and can be generalized to higher dimensions.*

- into an article Elliptic space, since projective geometry is in a sense to elliptic geometry what affine geometry is to Euclidean geometry. And a reader seeing the two will be confused. Also, as to "which geometry do you obtain", this is a loaded question. You can remove points, forget things, add things, whatever, and end up with various different things. You might as well ask: if you exit the door, turn right, walk two blocks, turn left and then enter the second door, where will you be? I pointed out that a minor difference to your prescription makes the difference between ending up at affine or projective geometries. In fact, if you start with a Euclidean geometry but consider it to include the points at infinity and which can still be regarded as Euclidean geometry in a true sense, and apply your question
*without alteration*, which geometry do you obtain? Tweaking your questions to arrive at the answer you would prefer is called rationalization (or wordplay , depending on your frame of mind). —*Quondum*03:30, 12 February 2014 (UTC)- From the most recent edits, I see the intended direction. I think that it is a good idea to draw attention to the similarities and differences between Euclidean and affine geometries. I will look at choice of words, but will leave the meaning unchanged. —
*Quondum*21:01, 13 February 2014 (UTC)

- From the most recent edits, I see the intended direction. I think that it is a good idea to draw attention to the similarities and differences between Euclidean and affine geometries. I will look at choice of words, but will leave the meaning unchanged. —

- I understand that three non-collinear points in elliptic and projective geometries define three lines that divide the plane into four triangles, not one of which has status over the the others as defining the "inside": one must choose. Yet three line segments joining three points do distinguish an "inside". You are seeking esoteric differences. In the context of generalizations, starting with the study of properties of flats as stated, these apply equally to Euclidean, hyperbolic, elliptic, affine, de Sitter and projective geometries. There is nothing in the statement that relates it to Euclidean geometry, other than the fact of the context of which article it is in. I could equally validly put the statement

- An Euclidean/affine triangle is uniquely defined with three its points. A projective triangle is not (see above). Euclidean and affine definitions of a triangle are compatible. The projective one is not. At last, when you get a Euclidean space and forget all its structure but points, lines, planes, and their incidence, which geometry do you obtain? Incnis Mrsi (talk) 19:22, 11 February 2014 (UTC)

## Velocities, projectivizations, and non-Euclidean geometries

### Velocities became hyperbolic?

This edit assumes that lightspeed and tachyonic velocities are not "velocities", though my own edit was based on a misconception. Is there a way in which we can correctly describe the full space of velocities (other than as the Lorentz group)? —*Quondum* 22:54, 15 February 2014 (UTC)

- My vote would be to strike that whole paragraph. A typical reader won't be able to get anything out of it, and it's straying far into the territory of other articles that would treat it much better. Mgnbar (talk) 00:40, 16 February 2014 (UTC)
- Which paragraph? Incnis Mrsi (talk) 07:12, 16 February 2014 (UTC)
- The paragraph that was edited in the diff that Quondrum cited. That is, the one paragraph in the article that contains both of the words "hyperbolic" and "velocities". Mgnbar (talk) 08:41, 16 February 2014 (UTC)

- Yes, just remove the paragraph. It's going way off topic talking about velocities, while the links are just confusing: click on them and nothing seems to happen; even looking at the target (by hovering and looking at the tooltip) it takes a few seconds to work out what it's jumping to.--JohnBlackburne
^{words}_{deeds}17:40, 16 February 2014 (UTC)

- Which paragraph? Incnis Mrsi (talk) 07:12, 16 February 2014 (UTC)

- Light velocities are ideal points (I was wrong when sent the reader to the nearby Riemannian geometry instead of hyperbolic geometry directly, that should explain both flavours of points). Why should I care about mysterious
*tachyonic velocities*? Incnis Mrsi (talk) 07:12, 16 February 2014 (UTC)- We have two problems here in my view:
- There is nothing mysterious about faster-than-light velocities; perhaps I confused the matter by referring to these as "tachyonic".
~~Even though these do not occur as classical boosts of classical real-world particles, they are legitimate elements of the Lorentz group.~~Thus, the "space of velocities" includes them.~~And they cannot be excluded from quantum-mechanical calculations of QFT, as I understand it. If you want to make such an abstract connection, it must be mathematically valid.~~ - The treatment of even sub-light velocities as a hyperbolic geometry is too non-obvious to be handled only via this link. To regard the space of velocities as a geometry, it must be clear how velocities map to the geometry: to points on the hyperboloid, or to transforms? It is relatively easy to map velocities onto points of a union of the hyperbolic (for sublight), conformal (for lightspeed, the "ideal points") and de Sitter geometries for supra-light,
~~but that distracts from understanding of what the geometric transforms, lines, planes etc. (in short, the bulk of the geometric properties) correspond to~~.

- There is nothing mysterious about faster-than-light velocities; perhaps I confused the matter by referring to these as "tachyonic".
~~On my second point here, the mapping that would really of interest, that of a homomorphism from some "subgroup of boosts" to the transforms of the geometry, does not exist. This can be seen as the boosts, even in a single reference frame, do not form a group. (The velocity-addition formula shows that they are not associative and they are actually not even closed under composition, whereas the transformations of hyperbolic geometry are both).~~Even if my argument turns out to be technically wrong, the point that this link is too obscure~~even if it were valid~~would still remain. —*Quondum*16:09, 16 February 2014 (UTC)- You think about
*relative*velocities, whereas I think about*absolute*velocities, that are lines in a spacetime (not necessarily in Minkowski, and nobody mandated they must be*all*lines), usually up to translational equivalence (although in the de Sitter it isn’t an option). There is no group**structure**on absolute velocities, I already said something about it. There is a group*action*on them, do you understand the difference? Your “homomorphism from some "subgroup of boosts" to the transforms of the geometry” is the projectivisation of the (1/2, 1/2) (a.k.a. vector) representation of the Lorentz group! There is no*some subgroup*. You can consider the entire**RP**^{3}of all directions in the Minkowski space and the action on it, but I consider only projectivisation of the cone of the future. Both actions are faithful, but “my” space is consistently metrizable whereas your one is not. I could comment on your arguments about QFT, but it would be a deep off-topic. To me, this quasi-geometrical stuff is next to trivial and I wonder that local frequenters experience such problems with it. Incnis Mrsi (talk) 17:11, 16 February 2014 (UTC)- My 2c: Either beef out in the article or scrap the sentence. Suggestions if you don't scrap it: Define "space of velocities" =
*V*( = subset of**RP**^{3})? probably, yes; you can also view it as a subset of spacetime upon identification of spacetime with its tangent space) properly in the article. At the very least, point out that it applies to*physical*velocities, attainable from the rest frame through a (1/2, 1/2) Lorentz transformation*or*the velocity of light in any spatial direction. The reader would also wonder what is meant by hyperbolic here. He'll wonder, is*V*geometrically a hyperboloid, is it (as a manifold in its own right) naturally endowed with some structure (a Riemannian metric?) justifying the terminology? The reader will want to know. The link is malfunctioning (or I'm too stupid to figure it out, probably the latter;) YohanN7 (talk) 19:00, 16 February 2014 (UTC)- This is a #-link. It pulls you slightly upwards, where examples of Riemannian manifolds are described. But it becomes too complicated, I see. One should describe this stuff in articles like velocity first, and only after that one may refer to these facts from here. Incnis Mrsi (talk) 19:39, 16 February 2014 (UTC)

- [@Quondum: Faster-than-light velocities are not legitimate "elements" of the Lorentz group, surely, you didn't mean to write that?. Then I don't know what you mean by not excluding super-luminal (is that a word) velocities from QFT calculations. You may think of the path integral or its field theoretical generalization, but nothing that really exists is moving faster than the speed of light here either, it's just a popularized description of what's really going on.] YohanN7 (talk) 19:00, 16 February 2014 (UTC)
- I'm evidently getting confused by non-connected components of the Lorentz group in that statement. However, there is nothing wrong with a world-line with a tangent that lies outside the light cone, for example the spot of a scanning laser beam on a wall at a large distance; this gives a 4-velocity belonging to a distinct space (4-velocities that square to –1 instead of +1). One needs to allow different normalizations of the tangent vector to a world line to allow for photons, which I'm sure you'll agree are physical. That super-luminal velocities are non-physical is not relevant; they exist in the projective space that 4-velocities belong to. This projective space is separated into three components: the hyperbolic (sublight velocities), the conformal (lightspeed) and deSitter (super-luminal). The Lorentz group acts on all of these, but every orbit is contained within one of the three components. That all aside I've been getting confused between 4-velocities and boosts. One needs to separate them, and then the space of 4-velocities within the light cone (which should be called 4-velocities for clarity, not simply velocities) have a hyperbolic geometry, which under the Erlangen program transform under the Lorentz group.
- Okay, now that I've retracted much of what I've said, the question would be whether it has value to describe an example of a hyperbolic space formed by (subluminal) 4-velocities under the action of the Lorentz transform as a real-world geometry. This would not belong in the section in which it was, but perhaps under curved geometries. Its local metric is also definite, not indefinite. A further real-world and easily pictured example is the conformal geometry of the heavens as seen by observers travelling at different velocities. —
*Quondum*21:50, 16 February 2014 (UTC)

- My 2c: Either beef out in the article or scrap the sentence. Suggestions if you don't scrap it: Define "space of velocities" =

- You think about

- We have two problems here in my view:

### Projectivization of 4-vectors

It is an off-topic, but the discussion already happened. One can embed the Minkowski space into **RP**^{4}, but this way gives botched, geometrically abhorrent compactifications of pseudo-Euclidean spaces (possibly, I 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 **RP**^{3}), 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 **RP**^{3}. 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)

- 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)

### Motions of elliptic spaces

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)

- Doesn’t the current text say that Euclidean rotations are direct motions of the (
*n*− 1)-sphere? Apparently it doesn’t, but it is a mishap. Similar statements are already scattered over Wikipedia, e.g. there and there (I 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)- 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)- 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 I 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)- 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)- "The hyperbolic space of its ideal mumbo-jumbo-plane". I start to understand your thinking pattern… that’s why you make two steps where I 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 I 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, I have to sleep now. Incnis Mrsi (talk) 20:37, 17 February 2014 (UTC)
- 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)

- 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"? —

- "The hyperbolic space of its ideal mumbo-jumbo-plane". I start to understand your thinking pattern… that’s why you make two steps where I 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 I 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, I have to sleep now. Incnis Mrsi (talk) 20:37, 17 February 2014 (UTC)

- 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? —

- What do you mean under

- 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. —

## Euclidean space

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)

- Removal 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)

## Wrong, wrong, wrong

The last paragraph of the introduction reads as follows:

"*From the modern viewpoint, there is essentially only one Euclidean space of each dimension. With Cartesian coordinates it is modelled by the real coordinate space (R ^{n}) of the same dimension. In one dimension, this is the real line; in two dimensions, it is the Cartesian plane; and in higher dimensions it is a coordinate space with three or more real number coordinates. Mathematicians denote the n-dimensional Euclidean space by E^{n} if they wish to emphasize its Euclidean nature, but R^{n} is used as well since the latter is assumed to have the standard Euclidean structure, and these two structures are not always distinguished. Euclidean spaces have finite dimension.[2]*"

1. "From the modern viewpoint, there is essentially only one Euclidean space of each dimension." is utterly meaningless. For it was left unstated **what** kind of object is there with "essentially only one Euclidean space of each dimension" — and what does "essentially" mean???

For: R^{n} can be thought of, in increasing complexity, as a) a topological space, b) a differential manifold, c) a Riemannian (metric) manifold.

2. R^{n} is not assumed to have a "Euclidean structure", which in mathematics is called a "Euclidean metric", unless that is explicitly stated.

Rather, R^{n} stands for the topological space that is the Cartesian product of n copies of the set of real numbers.

3. E^{n} is an old-fashioned notation for R^{n}, nothing more, nothing less.

But R^{n} is often considered to have a differential structure. In this case, for n ≠ 4, R^{n} is always diffeomorphic to the standard differential structure, given by the Cartesian product mentioned in 2. above. But for n = 4, there are a very large (infinite) number of inequivalent differential structures on (the topological space) R^{n}.Daqu (talk) 17:59, 9 September 2015 (UTC)

- First, in case you haven't been reading this talk page for multiple years, please understand that there is a lot of contention between mathematicians and non-mathematicians, particularly about the introduction. What we have currently is an imperfect compromise.

- Yes, "essentially" is a weasel word. The problem is that we cannot begin with a rigorous definition, because it does not appeal to non-mathematicians. For them, it may be more useful to approximate the truth by first laying out a hierarchy (line, plane, 3-space, 4-space, etc.) and then going back to fill in the details.
- You are right that R^n is not always assumed to have Euclidean structure. Nor is it always assumed to have a topology. Sometimes it is a vector space or just a set.
- I am not convinced that E^n is an antiquated synonym for R^n. Honestly, I'd want a reliable source for this assertion.

- Regarding your last point, I'm rusty on low-dimensional topology, but isn't there a "standard" differentiable structure for R^4 that fits into the pattern R^1, R^2, R^3, ..., R^5, R^6?

- In summary, the article is far from perfect, but bettering it is difficult. I invite you to draft a new introduction here, on the talk page. Try to satisfy non-mathematicians who "just want to know what Euclidean space is." Also satisfy mathematicians who regard E^n as a vector space. Also mathematicians who insist that E^n is not a vector space at all, but only an affine space (and this distinction must appear in the introduction or else it is wrong). Also mathematicians who use R^n to denote a vector space, as in linear algebra, without any topology or geometry. You will also have to contend with amateur physicists who think that we live in E^3. And so on. Mgnbar (talk) 20:31, 9 September 2015 (UTC)

## Weak Definition

"Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces"

This definition is weak and almost incomprehensible to a newcomer to the subject. A good definition encompasses all possibilities, so the use of the term "certain other spaces" evades the subject - it could mean anything - the definition should either describe those certain other spaces or define the term in a different way which encompasses 2D 3D and the other spaces in a single phrase. Also defining Euclidean space in terms of Euclidean plane is almost tautological, the definition should not include the term Euclidean twice but should describe what a Euclidean plane is. — Preceding unsigned comment added by 2A02:C7F:C405:700:6CB9:5150:B5C:7739 (talk) 08:21, 21 June 2018 (UTC)

- The text you are quoting is not intended to be the definition. The definition comes later in the article. The intro is just a warm-up intended to give readers an intuitive idea of what Euclidean space is. Many math articles are structured in this way, because starting with a formal definition is too scary and incomprehensible to newcomers. Mgnbar (talk) 11:04, 21 June 2018 (UTC)

- I came here to say this. I disagree with Mgnbar that such an "intuitive idea" of what Euclidean space is is constructive. In my opinion, the first sentence should be an informal summary of the formal definition, similar to the Vector spaces article. In the end, the formal definition is what makes up an Euclidean space, and what distinguishes it from other concepts of math. I guess it could be useful to mention in a small sentence that Euclidean spaces are what is usually taught as 2- and 3d spaces in elementary school in order to make the connection for students, but this intuitive approach is not how math is practiced, and thus not how this article should be structured. Sgyger (talk) 17:24, 9 March 2019 (UTC)

- Your points are reasonable, but the devil is in the details. So please craft a new introduction and post it here for us to discuss. While you craft your text, you might want to consult Wikipedia:Manual of Style/Mathematics and the archives of this talk page. The latter are contentious. Mgnbar (talk) 18:25, 9 March 2019 (UTC)
- (edit conflict) You suggest to give in the first sentence an "informal summary of the formal definition", but you do not give any idea of the formal definition that you have in mind. In fact, there are two formal definitions that have been proved to be equivalent, but none is given in this article. One of these definitions (Hilbert's axioms), that of synthetic geometry appears essentially in Euclid's
*Elements*, but has not been formalized before 1899. It cannot be summarized in a single sentence, and is sufficiently technical for not be given in this article. The other definition through vector spaces can be short, but is still more technical (*A Euclidean space is an affine space over the reals, whose associated vector space is a finite dimensional inner product space*). This definition deserves to be expanded and*explained*in section "Technical definition", which is still empty. - You say "this intuitive approach is not how math is practiced". I do not know how you practice geometry, but the intuitive representation of a Euclidean space has been used by all geometers since more than 2000 years, and, still now, I do not know anybody who can
*understand*a non-trivial proof of geometry without using an intuitive representation of the Euclidean space, for example by drawing a figure. However it is essential, for a good practice of mathematics, to well distinguish between*understanding*and*verifying*a proof. Both are fundamental. - I do not pretend that the lead cannot be improved, but this is certainly not by removing intuitive explanations. D.Lazard (talk) 19:13, 9 March 2019 (UTC)
- I have 2 phds and a BS in computer science and I can barely understand what the introduction is trying to say. It's absolutely awful, though this is a consistent problem with many math and science articles on wikipedia. In my humble opinion the introduction needs to be drastically simplified and related to something that most people are more familiar perhaps the cartesian system. Regardless of what is chosen, a common point of reference that people can grab onto, whatever that is, would be huge improvement. Zuchinni one (talk) 20:13, 18 April 2019 (UTC)
- This is an example of a math article that draws both math-expert and math-novice editors. And they have trouble agreeing on the right level of precision and abstraction. So what you see here is a flawed compromise arising from many pages of argumentation. (See the archives of this talk page.) Feel free to propose new text for us to argue about. :) Mgnbar (talk) 21:24, 18 April 2019 (UTC)

- I have 2 phds and a BS in computer science and I can barely understand what the introduction is trying to say. It's absolutely awful, though this is a consistent problem with many math and science articles on wikipedia. In my humble opinion the introduction needs to be drastically simplified and related to something that most people are more familiar perhaps the cartesian system. Regardless of what is chosen, a common point of reference that people can grab onto, whatever that is, would be huge improvement. Zuchinni one (talk) 20:13, 18 April 2019 (UTC)

## Definition of Euclidean spaces

I have reverted three edits by three different editors. These edits are motivated by the fact that I began to rewrite this article, but never finished the work. This results that the technical definition is still lacking, and that next sections need to be updated or rewritten for being understandable. Another point was confusing: it was not clear that the subsections preceding "Technical definition" were about the definition. Have thus renamed them "History of the definition" and "Motivation of the modern definition".

I have also added a one line definition, and tagged it with {{expand section}}. In fact this definition is much too technical for most readers. It must be expanded for detailing the different concept that are used, and relating this to the preceding subsection.

I have tagged the next section with {{update}} because it seems to refer to an old state of the article, and, in particular, its first sentence is, at least, confusing.

In fact, I have introduced the section "Definition" and its subsection "History" by this edit, where I have also moved the section "Motivation" into a subsection of "Definition". I intended to continue to upgrade this article, but some other tasks came on the top of "to do" pile. Reading the present state of the article, I am convinced that I have to move this article to the top of my "to do" pile.

For the record, my edit of the article had two motivations: A definition is needed in the article, and was completely lacking in the previous version. Worse, it is often said, here, and in many other Wikipedia articles, that a Euclidean space *is* a vector space. This is definitively wrong, but for fixing this error, one need a correct definition here, and this implies to completely rewrite this article. D.Lazard (talk) 21:26, 20 August 2019 (UTC)

- Your motivations are good, and I'm excited to see what you do. I confess that I...gave up on the article a few years ago after some unpleasant battles.
- You are technically correct that Euclidean space is not a vector space. I am eager to see how you maintain this level of rigor while making the treatment understandable to readers and reliably sourced. Mgnbar (talk) 21:54, 20 August 2019 (UTC)

If D.Lazard strives to reform the definition in a gradual fashion, then let him do it in Euclidean_space/sandbox or wherever instead of pushing intermediate stuff online. The edit [6] added such ignominy as “n-tuples of real numbers equipped with the dot product” – hey, you don’t need to specially “equip” ℝ^{n} with anything which is expressed with a simple formula. Contrary, D.Lazard discarded a genuinely useful paragraph on Euclidean vectors vs points. **Both** points and vectors are commonly called Euclidean, which was pretty clear from pre-D. revisions. Why does the user try to improve namely this article? Incnis Mrsi (talk) 15:02, 23 August 2019 (UTC)

- The previous version asserted that a Euclidean space is a vector space. This is definitively wrong, although an inner product space of finite dimension is equivalent to a Euclidean space with a specific point chosen. I have fixed this in the section "Motivation...". I agree that the next sections become nonsensical, but they were already so (at least the first one) before my first edit. D.Lazard (talk) 15:15, 23 August 2019 (UTC)

D.Lazard disrupted connectivity of Wikipedia by removal of all links to Euclidean vectors, which are (by the way) extensively discussed at vector space. Instead D.Lazard set up a redirect serving, essentially, an own inline WP:content fork of the “Euclidean vector” article. Moreover, a bizarre link *real vector space* popped up instead (which is astonishing, given the past D.Lazard’s wise reaction to another content fork). What’s special one should learn about namely ℝ-vectors? Incnis Mrsi (talk) 20:13, 26 August 2019 (UTC)

- This post is hard to accept because to its WP:Personal attack tone, and the presentation as disruption of some details that can be easily fixed by some edits here or in other articles (the latter ones have already been fixed). Nevertheless, it raises some points that deserve to be fixed. Here are the details:
- Links to Euclidean vectors: I do not remember to have removed these links. In any case, as the definition of Euclidean vectors requires a definition of Euclidean spaces, such links cannot be used for defining Euclidean spaces. On the other hand, "Euclidean vector" is another name for "translation vector" and "free vector", and this must be added (with link(s)) to the article. I'll do it soon.
- Redirect Euclidean vector space: This redirect existed with Euclidean space as target. I have simply added an anchor for redirecting to the section where the term is defined. Unfortunately, I miscapitalized the anchor name. This is now fixed.
- Content fork with Euclidean vector. The only things that I have added is a proper definition of Euclidean spaces. Reading Euclidean vector again, I do not see any content fork.
- Link to Real vector space: This has nothing bizarre; this is a standard term for specifying a vector space over the reals, and the redirect existed already. However the definition was difficult to find in the target article. Thus I have edited if for giving it in the lead.

- D.Lazard (talk) 09:42, 27 August 2019 (UTC)
- My bellicose comment was prompted by necessity to watch multiple edits in nearly real time, instead of comparing the historical article against a largely complete piece
*in a sandbox,*as already stated above. But now it’s IMHO already a moot point. No, “translation vector” is not synonymous with*Euclidean*vectors – the point–vector distinction is not specific to the Euclidean case. It’s merely an element of the vector space acting on a space of points. Only in*this local*context “translation” refers to the action by Euclidean vectors. Incnis Mrsi (talk) 13:11, 27 August 2019 (UTC)

- My bellicose comment was prompted by necessity to watch multiple edits in nearly real time, instead of comparing the historical article against a largely complete piece

- D.Lazard: I haven't been examining every one of your edits, figuring that they were stepping stones on the path to a major rewrite. But now that you've made many edits and declared at least one section "complete", let me please raise this issue: It's hard for a reader to tell that your version is supported by reliable sources rather than original research. (The old version of the article also had this problem.) Do you plan to cite sources? Mgnbar (talk) 12:00, 29 August 2019 (UTC)
- Everything that I have added must (and much more) must be in Berger, Marcel (1987),
*Geometry I*, Berlin: Springer, ISBN 3-540-11658-3. This book must be cited as a source, but also as "Further reading". Unfortunately, I have not it under hand, and I am unable to provide inline citations to specific pages. Maybe, it would be worth to add in the lead a sentence likethe definition and properties of Euclidean spaces presented in this article are based on Marcel Berger's book

I strongly recommend this marvelous book to everybody interested in classical geometry (and less classical aspects such as tesselation and M. C. Escher's drawings). D.Lazard (talk) 13:12, 29 August 2019 (UTC)*Geometry*.^{[1]}

- Everything that I have added must (and much more) must be in Berger, Marcel (1987),

- Great. It would be nice to have multiple sources, for several reasons. The article's existence rests on the notability of Euclidean space as a concept. The article implicitly asserts that there is consensus in the math community about the definition. Also it would be nice to avoid merely reproducing that one source. Mgnbar (talk) 13:37, 29 August 2019 (UTC)
- A problem here is that most textbooks avoid to define properly Euclidean spaces because this is too technical for their audience, and advanced textbooks, specially in physics and mechanics suppose that Euclidean space is known. Berger's book is an exception, because it issued from a course for future professors of mathematics. Its first title was
*Géométrie élémentaire approfondie*(that is*Advanced elementary geometry*). I am generally not good for sourcing. Nevertheless, other possible sources are Artin's Geometric Algebra and some Coxeter books (I have never read the latters). In any case, Artin's book must be cited, as being the main source for the equivalence between the axiomatic and the algebraic definitions. - About consensus: There is clearly a consensus that there is no specific origin in a Euclidean space. There is also a clear consensus to prefer, in modern mathematics, the algebraic definition to the axiomatic one, because it makes easier to use the power of linear algebra. The definition that I have introduced in the article is (except possibly for details) the only existing one that combines these two consensus.
- IMO, the lead must be rewritten for giving this kind of explanations. But before that, the remainder of the article must be updated from the section on angles on. D.Lazard (talk) 16:31, 29 August 2019 (UTC)

- A problem here is that most textbooks avoid to define properly Euclidean spaces because this is too technical for their audience, and advanced textbooks, specially in physics and mechanics suppose that Euclidean space is known. Berger's book is an exception, because it issued from a course for future professors of mathematics. Its first title was

- Great. It would be nice to have multiple sources, for several reasons. The article's existence rests on the notability of Euclidean space as a concept. The article implicitly asserts that there is consensus in the math community about the definition. Also it would be nice to avoid merely reproducing that one source. Mgnbar (talk) 13:37, 29 August 2019 (UTC)

- [Edit conflict] On my bookshelf I've found one source that approximately agrees with yours:
- Boothby,
*An Introduction to Differentiable Manifolds and Riemannian Geometry*. Pages 2-6 have a lot of commentary, which I will attempt to summarize. They define an*n*-dimensional real vector space to be "Euclidean" if it is equipped with a positive-definite inner product. Then**R**^{n}is a Euclidean vector space with "a built-in orthonormal basis and inner product". If we choose coordinates*x*,*y*on the Euclidean plane**E**^{2}, then there is a mapping**E**^{2}->**R**^{2}given by*p*-> (*x*(*p*),*y*(*p*)). Then, "with qualifications" (namely keeping our arbitrary choice of coordinates in mind) we can identify**E**^{2}with**R**^{2}.

- Boothby,
- Meanwhile, I have also found several sources that essentially define Euclidean space as
**R**^{n}equipped with the dot product:- Artin,
*Geometric Algebra*. Page 178 says, "We specialize*k*to the field**R**of real numbers. Let*V*be an*n*-dimensional vector space over**R**with an orthogonal geometry based on the quadratic form*x*_{1}^{2}+*x*_{2}^{2}+ ... +*x*_{n}^{2}. Then we say that*V*is a euclidean space." - Bredon,
*Topology and Geometry*. Page 1 opens with, "We are all familiar with the notion of distance in euclidean*n*-space: If**x**and**y**are points in**R**^{n}, then dist(**x**,**y**) = ..." (What follows is the definition in terms of the dot product.) - Fleming,
*Functions of Several Variables*. Pages 2-3 define "euclidean*n*-space", denoted**E**^{n}, essentially as**R**^{n}equipped with the dot product. Page 11 even defines the dual space of**E**^{n}as the functions**E**^{n}->**R**. Page 28-34 even discusses non-Euclidean norms on**E**^{n}! - Kasriel,
*Undergraduate Topology*. The index says that "Euclidean space" is covered on pages 58-80. That's Chapter 2 of the book, whose title is "Structure of**R**and**R**^{n}". As far as I can tell, the word "Euclidean" appears only briefly, in defining the "Euclidean metric" on**R**^{n}in terms of the dot product. - Kelley,
*General Topology*. In the context of Cartesian products, page 31 defines "Euclidean*n*-space" as the set of all real-valued functions on {0, 1, ...,*n*- 1}. Page 60 defines the "Euclidean plane" as the set of all pairs of real numbers, equipped with what is essentially the product topology. Page 89 combines the two, defining "Euclidean*n*-space" as the functions {0, 1, ...,*n*- 1} ->**R**with the product topology. - Rudin,
*Real and Complex Analysis*. Page 34 says, "...all vector spaces appearing in this book will be complex, with one notable exception: the euclidean spaces**R**^{k}are vector spaces of the real field." - Spivak,
*A Comprehensive Introduction to Differential Geometry*, Volume I. Page 1 defines "Euclidean*n*-space**R**^{n}" with its "usual metric" given by the dot product. - Stein and Weiss,
*Introduction to Fourier Analysis on Euclidean Spaces*. Page 1 lets**E**_{n}denote "*n*-dimensional (real) Euclidean space". It says, "We consistently write*x*= (*x*_{1},*x*_{2}, ...,*x*_{n}) ... for the elements of**E**_{n}", defines the standard inner product, and proceeds to measure theory and function spaces. - Warner,
*Foundations of Differentiable Manifolds and Lie Groups*. Defines**R**^{d}to be "the*d*-dimensional Euclidean space".

- Artin,
- Admittedly, there are a few books on my shelf that I haven't checked. And my bookshelf may not be perfectly representative of the math community. But it seems that the article should acknowledge that many authors view Euclidean space as simply
**R**^{n}with the dot product and its resulting norm, metric, topology, etc. Mgnbar (talk) 17:09, 29 August 2019 (UTC) - In response to your most recent post: I agree that there is a problem of level. Authors of advanced texts (e.g. most of the ones that I mentioned above) probably regard the distinction between
**E**^{n}and**R**^{n}to be irrelevant to their goals or even pedantic. Your Berger source seems admirably careful. I'm not sure how to reconcile that opinion of mine with our responsibility to summarize the literature without giving undue weight to minority views. Mgnbar (talk) 17:33, 29 August 2019 (UTC)- Kelley seemingly describes the real coordinate space ignoring this proper term. The function stuff clearly pertains there, not here. Incnis Mrsi (talk) 18:02, 29 August 2019 (UTC)
- It's interesting that Real coordinate space has no inline citations, and has only two references at the end. One of them is Kelley, who uses the term "Euclidean
*n*-space" rather than "real coordinate space", as far as I can tell. The other is Munkres, which I don't have here. Does he use the term "real coordinate space"? Mgnbar (talk) 21:47, 29 August 2019 (UTC)- It turns out that I asked you the same question in 2013 here. :) Mgnbar (talk) 21:57, 29 August 2019 (UTC)
- Do English-speaking writers specially define the real coordinate space? Not sure, but German:
*reellen Koordinatenraum*seemingly is in use for decades. Incnis Mrsi (talk) 05:28, 30 August 2019 (UTC)- German Wikipedia's Koordinatenraum is slightly better referenced --- but only slightly. Mgnbar (talk) 11:19, 30 August 2019 (UTC)

- Do English-speaking writers specially define the real coordinate space? Not sure, but German:

- It turns out that I asked you the same question in 2013 here. :) Mgnbar (talk) 21:57, 29 August 2019 (UTC)

- It's interesting that Real coordinate space has no inline citations, and has only two references at the end. One of them is Kelley, who uses the term "Euclidean

- Kelley seemingly describes the real coordinate space ignoring this proper term. The function stuff clearly pertains there, not here. Incnis Mrsi (talk) 18:02, 29 August 2019 (UTC)

- [Edit conflict] On my bookshelf I've found one source that approximately agrees with yours:

For discussing the definition of Euclidean spaces, and the sources that must be referred to, we must first establish which are the strong consensuses. It seems that all sources agree that

- A Euclidean vector space is a finite dimensional inner product space over the reals
- A Euclidean vector space is a Euclidean space
- equipped with the dot product is a Euclidean vector space and thus a Euclidean space.
- There is "essentially" one Euclidean space in each dimension. That is, the choice of a Cartesian frame (origin + orthonormal basis) defines (whichever definition is chosen) an isomorphism (affine transformation that is an isometry) of Euclidean spaces, which allows identifying any Euclidean space with

None of the above references (except Berger's) are devoted to the study of Euclidean spaces (that is Euclidean geometry). Because of the last item, it is thus natural that they use the weakest definition that is sufficient for their need. However, restricting the definition of Euclidean spaces to that of Euclidean vector spaces is not convenient in some contexts, specially in pure geometry. For example, the definition must include that a flat is a Euclidean space. Also, what should be an isometry of a Euclidean vector space? A linear map, or an affine map, including translations?

Being an encyclopedia, WP must cover all contexts. The only available definition of an Euclidean space that covers all context is that I began to give and is inspired by Berger (I have finished the part of the definition relative to Affine structure, but not the one related to the metric structure, which still needs much work. Nevertheless, I agree that the relationship with more restricted definitions must been improved. This is in this spirit that I have added a reference to flats and renamed "Typical examples" into "Protoypical examples". Nevertheless — Preceding unsigned comment added by D.Lazard (talk • contribs) 14:04, 30 August 2019 (UTC)

- Your comments about the needs of pure geometry make sense. However, I suspect that most use of the term "Euclidean space" by recent mathematicians happens outside pure geometry.
- I don't know terminology around Euclidean vector spaces, because I haven't seen that term used much (and I'm not convinced that it's an important concept). I would guess that
*isomorphisms*of a Euclidean vector space do not include translations, because they do not preserve the 0. I don't know what*isometries*should be. Clearly isometries of Euclidean space should include translations. Flats in a Euclidean vector space might be forced to pass through the origin, while flats in a Euclidean space obviously aren't. Ugh. - Your argument about covering all contexts worries me. It seems to logically imply that we should always focus the most general version of a definition. But Wikipedia policy is that we should be summarizing the secondary literature. That principle suggests that we should focus on
**R**^{n}and relegate the no-preferred-point-hence-isometries-include-translations view to a section. - I don't plan to try to stop you. You are definitely improving the article. :) I'm just anticipating the battles that will happen 1-5 years from now, when one editor complains that the treatment is too abstract, another complains that it doesn't reflect how mathematicians use the term, and they're both right. Mgnbar (talk) 18:25, 30 August 2019 (UTC)
- Following up with one more reference...
- Munkres,
*Topology*(first edition). Page 37 (the only one mentioned in the index) defines "Euclidean*m*-space" as the set**R**^{m}. On page 115,**R**^{n}is given the product topology. Pages 124-126 develop the "Euclidean metric" on**R**^{n}.

- Munkres,
- By the way, I cannot find any mention of any version of the phrase "real coordinate space". Mgnbar (talk) 21:27, 5 September 2019 (UTC)
- I agree that the phrase "real coordinate space" is meaningless and weakly (at least) sourced. But we have a problem that goes far behind this article: We must have an article on
**R**^{n}. Presently, this article is named Real coordinate space. It results from this discussion that it should be renamed. At first glance, I had no idea for a better name. But looking on the existing redirects, I find Real n-space, which seems convenient. Thus, I suggest to replace (here) the direct link by this redirect, and requesting a move real coordinate space -> Real n-space (I have not the rights for making the move myself). D.Lazard (talk) 07:45, 6 September 2019 (UTC) - P.S. I have modified the first occurrence of the phrases. Other modifications are needed, but, as these are minor edits, I prefer to wait that the rewrite of the article will be finished. Be free to make these edits immediately, if you prefer to not wait. D.Lazard (talk) 07:55, 6 September 2019 (UTC)

- I agree that the phrase "real coordinate space" is meaningless and weakly (at least) sourced. But we have a problem that goes far behind this article: We must have an article on

- Following up with one more reference...

References

**^**Berger, Marcel (1987),*Geometry I*, Berlin: Springer, ISBN 3-540-11658-3

## Rewriting the article

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.

**Done**Not removed, but moved and completely rewritten. D.Lazard (talk) 08:24, 15 September 2019 (UTC) - IMO the section "Applications" must be removed. Euclidean spaces are so widely used in mathematics and physics, that it is impossible to list the applications, and even to decide which are the most important.
**Done**not removed but renamed and rewritten. D.Lazard (talk) 09:37, 14 September 2019 (UTC) - The section on topology must be rewritten and extended for including the fact that Euclidean spaces are complete and locally compact.

**Done** - The section "Alternatives and generalizations" must be rewritten. It must include Klein geometries (that I have removed from the section on isometries), and non-Euclidean geometries. It must clarify the relationship between Euclidean spaces and manifolds. That is, section "Curved spaces" must be renamed "Manifolds" and rewritten. On the other hand, sections "Hilbert spaces" and "Indefinite quadratic forms" do not deserve more than a mention in "See also section".
**Done**, but not exactly as described above, and renamed "Other geometric spaces". D.Lazard (talk) 15:00, 18 September 2019 (UTC) - Section "See also" must be updated.
**Done**D.Lazard (talk) 15:00, 18 September 2019 (UTC) - References must be completed, and must include Berger's book.
**Done**with the removal of a link to stack exchange. D.Lazard (talk) 14:28, 11 September 2019 (UTC) - A section must be added on the equivalence between the definition of this article and axiomatic definitions of synthetic geometry.
**Done**D.Lazard (talk) 14:30, 10 September 2019 (UTC)

If you disagree with some of the above points, or if I have forgot something, please discuss here before I proceed. D.Lazard (talk) 13:09, 7 September 2019 (UTC)

- 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)- In the previous list, I forgot "updating the lead". This is done. D.Lazard (talk) 18:12, 8 September 2019 (UTC)
- I have finished what I am able to do for improving this article (ping Brirush). D.Lazard (talk) 15:00, 18 September 2019 (UTC)

- In the previous list, I forgot "updating the lead". This is done. D.Lazard (talk) 18:12, 8 September 2019 (UTC)

## Fundamental?

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 **Euclidean space** is the fundamental space of geometry.*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)

- 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)
- 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)

## Prototypical examples

Shouldn't in this section be written: an origin?Madyno (talk) 20:33, 26 August 2020 (UTC)

**Fixed**D.Lazard (talk) 07:37, 27 August 2020 (UTC)

## Associated vector space

There is no formal definition of the associated vector space. Madyno (talk) 20:40, 26 August 2020 (UTC)

- The associated vector space of an affine space is formally defined in Affine space. D.Lazard (talk) 07:41, 27 August 2020 (UTC)

The term is mentioned, not defined. Madyno (talk) 21:11, 27 August 2020 (UTC)

- Now, it is defined. D.Lazard (talk) 10:36, 28 August 2020 (UTC)

## Sentence Structure

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)