Talk:Euclidean space

Today, a well-meaning editor added a one-sentence paragraph to the lede, stating that Euclidean space is a metric space. That's true, but it's just one of many such statements that could be added. I propose that the paragraph be either deleted or expanded --- perhaps to something like, "Euclidean space is an important archetype for many kinds of spaces in mathematics, including manifolds, topological vector spaces, metric spaces, ...". Mgnbar (talk) 23:29, 12 February 2023 (UTC)

This probably does not need to be in the lead section, but it could be useful to add to a new section near the bottom of the page. Hilbert space is linked from the "see also" section, but that and other structures generalizing Euclidean space (e.g. Lp spaces) could profitably fill out at least one top-level section near the bottom. There is a little bit of relevant material in Euclidean space § Metric structure. Ideally any discussion would aim to be lay-accessible, leaving interested readers to click through for more advanced coverage.

While we are at it, the section Euclidean space § Isometries is far too general and abstract and filled with jargon. This section should focus on isometries of Euclidean space specifically, describe them concretely and in lay-accessible language with concrete examples and link to more advanced (abstract) articles for anyone who wants a more expert-level discussion discussion. Readers would be better served by e.g. material about crystallography or inverse kinematics or about the changing conception of Euclidean space as the focus shifted from studying physical forces toward rigid motions for their own sake (concretely studied by Olinde Rodrigues in 1840 decades before the Erlangen program), rather than nitpicky formalities.

The same is to some extent true of this article in general. As many concepts/sections as possible should be pitched to be legible to e.g. high school students or laypeople. It should include more specific details about concrete geometrical figures (or at least obvious links to other articles where those are discussed) and less reliance on lofty abstraction and jargon.

It’s wild to me that this article does not discuss the concepts of spheres or conics/quadrics (or any other curves or surfaces) or simplexes (or any other polyhedra), only mentions the word "triangle" in the context of the triangle inequality and does not mention the word "trigonometry", does not describe what a glide reflection or screw motion is, has no discussion of differential geometry, only mentions 'analytic geometry' per se in a throwaway aside, barely mentions other coordinate systems (and what little it does is inaccessibly jargony/abstract), makes no mention of the geometry of oriented planes, etc. etc. The fundamental notions that make a "Euclidean space" are lines (and higher-dimensional flat subspaces), parallelism, angles and perpendicularity, and distance and spheres. Just as planar Euclidean geometry substantially focuses on lines and line segments, angles, triangles and circles (and then also conics, algebraic curves, ...), higher-dimensional Euclidean geometry substantially focuses on those same planar objects and higher-dimensional analogs like tetrahedrons, spheres, and quadrics.

We could in theory lean more on Euclidean plane for some of this, except that article has most of the same problems. –jacobolus (t) 01:45, 13 February 2023 (UTC)

You raise good points. If I recall history correctly, this article sat at a particular, heavily contested equilibrium for many years. Then, over the past year or two, User:D.Lazard and a couple of other editors broke the equilibrium and essentially re-wrote it (mostly for the better, in my opinion). My point is that the current text is, in a sense, new/immature and ripe for continued polishing/expansion. Mgnbar (talk) 02:39, 13 February 2023 (UTC)

This should need further discussion. However, in my opinion, most topics that jacobolus would add to this article would be better placed (if they are not there) in Euclidean geometry, as this article is about the definition of the space of Euclidean geometry, not about that that can be done with it, which is Euclidean geometry. D.Lazard (talk) 09:22, 13 February 2023 (UTC)

But the title is Euclidean space, not one modern formal definition of Euclidean space. ;-)

Joking aside, I think this focus poorly serves many/most readers. Readers are not generally coming here because they want to find out the formal/technical details and logical ordering of the basic definition(s) Marcel Berger chose to use in his (very fine) textbook, which was convenient for his particular purposes of setting up a framework for the rest of the book, in a logically consistent order, intended for a 2-semester undergraduate course. Readers of Berger's book per se can just directly refer to the relevant definitions in context. Instead, many readers who come here are looking for a more accessible and/or broader view: what is the concept about, why does it matter, what implications does it have (Berger spends 2 volumes on this subject, only a few pages of which we are describing here), what is its mathematical context, how has the idea changed over time, etc. Or when it comes to formal definitions per se: why was this one chosen, which other definitions are equivalent, how does it relate to definitions of other structures, and so on. For instance, this article currently makes no mention of Cayley–Klein metrics, which could provide an alternate of Euclidean space. It also is entirely focused on the geometry of points in Euclidean space, ignoring Laguerre's dual geometry of oriented planes, which is equally much a fundamental part of "Euclidean space".

Even if we accepted the current structure this article is very hard for non-expert readers. For example the concepts of "inner product" and "affine space" are first mentioned as:

The standard way to mathematically define a Euclidean space, as carried out in the remainder of this article, is as a set of points on which a real vector space acts, the space of translations which is equipped with an inner product. The action of translations makes the space an affine space, and this allows defining lines, planes, subspaces, dimension, and parallelism. The inner product allows defining distance and angles.

This is mostly incoherent to someone without significant math background; we are more or less assuming someone has studied 2+ years of undergraduate level pure math courses before trying to approach this article.

I agree that some such material as I mentioned in my previous comment could also conceivably fit in an article titled 'Euclidean geometry'. But elaboration about such details mostly does not fit particularly well at the article Euclidean geometry as it is currently structured. That article is organized around meta-discussion, starting with a description of The Elements, then proceeding to discuss axiom systems, a (currently very incomplete and somewhat idiosyncratic) historical summary, an idiosyncratic smattering of sentences and pictures about applications, and concrete details mainly provided as examples rather than as the primary topic. It would also be great to dramatically improve the article solid geometry, which is currently a wreck. But even if we assumed that those articles were rewritten / dramatically expanded, the scope of this article should be broad enough to include a summary. –jacobolus (t) 20:51, 13 February 2023 (UTC)