Talk:Point (geometry)

2019 Comments on article - imho

I have a number of problems with the article as it stands as of Sept 1, 2019. I came to this article seeking understanding in what seems to me to be the conflicting views of points as locations in space or points as vectors (location vectors?) in space. The article spends an inordinate amount of time detailing the fact that a point is 0-dimensional in several of the more popular types of mathematical spaces. Little else is included. A point has no internal structure, but does have external characteristics - it has relationships to other points and mathematical (geometric) objects. None of the common properties or characteristics are mentioned. A serious deficit, imho. In addition, I offer the following comments as constructive criticism (I hope) of the article. It is true that IF the plane uses Cartesian coordinates that x is conventionally horizontal. What's missed here is the FACT that the editors (authors, actually) have assumed the reader knows that the x,y coordinates are orthogonal and isotropic. That isn't necessary, of course, but is commonly understood as the default. What also isn't mentioned is that while it is true that x & y CAN be numbers, they are values - which might be variables or functions (or more complex stuff) as well as numbers. It is ALSO true that in POLAR COORDINATES the two values are qualitatively different and the angle, Θ, is arguably NOT "infinite" but limited to [0,2π]. I think the discussion of points shouldn't disregard the common grade school coordinate systems. I don't think it is best to state that the jump from 2d to 3d is "easy" nor is it best described as a generalization. I'd say it is an "expansion". Again in talking about 3d the claim is made that an additional number is added. In Spherical coordinates, the additional value is Φ and an angle. In all of these coordinate systems a point is denoted (to use the word the article does to describe the name or label of a point) by values that are relative to an origin (or reference point) as well as 2 or 3 directions (i,j) or (i,j,k) for Cartesian or 1 or 2 angles and a distance for the polar/spherical. The article also implies that the 3rd cartesian dimension, z, is added onto the horizontal x and vertical y plane, while it is just as common for the z direction to be vertical, the y direction to be horizontal, and the x direction to be perpendicular to the paper i.e. claiming z is (generally) "depth" is incorrect in many many cases. The article makes the silly error of claiming Euclid wasn't "definitive"! Unless we count the 2000 years where he was the only authority (in the West). I suggest the authors review the definition of the word. (I don't argue he wasn't wrong or at least incomplete, he was; but he was the very epitome of an authority, hence he WAS definitive!) The article spends, as I already wrote, a lot of e-ink on details about a points dimension. WHY? That is, if you ask the reader to plow thru this mostly irrelevant arcana, you need to justify it, not just start in cold. I'm not a mathematician, but if 1 dot 0 = 0 isn't trivial which the article claims, I don't understand what trivial is. It is trivial, and I'm pretty sure it is used as an example of trivial. This claim (if it's correct, which I doubt) needs further justification and clarification, not just the bald claim. The article has an entire paragraph on the line. Why? This article is about the point. The article has no discussion on the continuity of Euclidean space. The article claims that many constructs contain infinite set of points...ok, so what? If it is not going to provide examples of constructs which are either A) not infinite sets (good luck with that in Euclidean space!) or B) not sets of points, then this seems to me to be not very useful here. If the point is a fundamental concept, don't you think that statement should be fleshed out both with what we CAN do with it (build the line and the plane and objects made from them - line segments, etc. etc.) and what we can't? What can't we do? I don't really know - you can't derive an angle or direction from a point. You can't add points, or subtract them. Can you move them in space? Hmmm. I'd say not, but that's one reason I came here; affine transformations. I'm of the school of thought that to fully describe a concept you need to describe what it is and what it isn't in a meaningful way. I don't think this article does that as well as it could. Two final comments: It took me a while to understand what was meant by L = {(a1,a2,a3)|a1c1+a2c2+...+ancn=d}. Waaay too complicated, and if ai are points (NOT numbers!) shouldn't there be some discussion of what a1*c1 or a1+a2 mean? Is a1*k another point? or a number? if the latter, why does the operation of multiplying a scalar by a point or vice versa produce a scalar? Similarly, why does adding two points together even make sense? (in my view, it doesn't). This is an example of where the article fails to discuss the external relationships (structure) points have in Euclidean space (to start). Finally, it is claimed that the dirac delta function is a generalized function. Well aside from being nearly totally devoid of meaning (to the unanoited) the dirac delta function is NOT a function, it is a distribution. It may be beyond scope to mention that it was used by physicists for a long while before the math was created to handle/justify it. But since it describes the "area" of a point, I'd say a little more should be said, rather than less. Anyway, "generalized function" is not useful unless the reader knows that that is a category of mathematical objects/concepts rather than a type of function. We shouldn't require the reader to pursue a linked topic when we don't have to. I think distribution is a better choice here: The Dirac Delta Function (which is a distribution and not a function) ... That's it.40.142.185.108 (talk) 14:54, 1 September 2019 (UTC)

None of the common properties? As the IP user stated, the article explains that a point is zero-dimensional. The subsequent rant about algebraic operations, third dimension, polar coordinates etc. is weakly relevant bordering off-topical. If the IP expects to see here a chapter from a book on Euclidean geometry, then I have to disappoint the user. Incnis Mrsi (talk) 15:14, 1 September 2019 (UTC)

On the last point, the theory of distributions is quite commonly understood as a generalization of the theory of functions. Hence "generalized functions." Any function defines a distribution (the regular distributions), but they are of different types, so it wouldn't feel right to say distributions are extensions of functions, as it was said that space can be viewed as an extension of the plane. It is common in physics to abuse language and call the Dirac distribution a "function."

None of these issues really bother me because I take a point to be a figure of speech or thought, not a metaphysical element (or "primitive entity"). Perhaps its unsatisfying to describe something that physical space is supposed to be composed of as "pure formalism." From my view, being fictional, the point itself can't be held to blame if it doesn't make much sense when held up to intense scrutiny, and that few people are seriously skeptical that space can be decomposed into points explains the lack of clarity in the article, and in discourse in general.

It seems the ranter is genuinely conflicted about a genuinely perplexing aspect of nature, and this has a long history of influencing people's philosophy in profound ways. I can't think of a better place than this article to include something of the history of debate on the notion of points. Perhaps something about Parminides and Zeno of Elea is in order? Bergson may also be relevant.

-Off Use 2601:647:C900:B6C0:197F:BDF0:7311:C712 (talk) 05:11, 27 January 2023 (UTC)

If the point onest ..

Where ??

176.59.215.189 (talk) 17:48, 9 April 2020 (UTC)

Point in a space

Should there be a about points in a space rather than specifically in geometry? DMH43 (talk) 15:37, 27 December 2023 (UTC)