Talk:Euclidean space

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One of the 500 most frequently viewed mathematics articles.

Related discussions[edit]

Note about three related (and important) discussions:

Non-Cartesian coordinates[edit]

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)

R or ℝ?[edit]

WP:MOSMATH #Common sets of numbers prefers R over , 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 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)

Wrong redirection from 2-norm.[edit]

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)

Use of a different syntax in the markup[edit]

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)


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)
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 Face-devil-grin.svg, 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)

Velocities, projectivizations, and non-Euclidean geometries[edit]

Velocities became hyperbolic?[edit]

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.--JohnBlackburnewordsdeeds 17:40, 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:
  1. 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.
  2. 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.
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 RP3 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 RP3)? 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)

Projectivization of 4-vectors[edit]

It is an off-topic, but the discussion already happened. One can embed the Minkowski space into RP4, 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 RP3), 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 RP3. 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[edit]

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)

Euclidean space[edit]

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. (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[edit]

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 (Rn) 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 En if they wish to emphasize its Euclidean nature, but Rn 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: Rn can be thought of, in increasing complexity, as a) a topological space, b) a differential manifold, c) a Riemannian (metric) manifold.

2. Rn is not assumed to have a "Euclidean structure", which in mathematics is called a "Euclidean metric", unless that is explicitly stated.

Rather, Rn stands for the topological space that is the Cartesian product of n copies of the set of real numbers.

3. En is an old-fashioned notation for Rn, nothing more, nothing less.

But Rn is often considered to have a differential structure. In this case, for n ≠ 4, Rn 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) Rn.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.
  1. 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.
  2. 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.
  3. 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)