Talk:Affine plane

Untitled
OK. Perhaps I was a bit sloppy with my recently reverted edit, but there is something about this page that I am finding not quite right. It is jarring my mathematical sensibility ... if that makes any sense to anyone. From my perspective, all three uses are talking about the same concept, so I am looking for the right language to convey the distinctions that others are making without destroying the basic unity. I could use some other insights. Bill Cherowitzo (talk) 23:37, 12 December 2011 (UTC)
 * Per Disambiguation, the goal here should be to briefly list all of the articles people are likely to intend to link to when they write Affine plane and to make it possible for people who get sent here to find the article they're looking for. So although there are certainly ways of describing the articles listed here in a unified way, that's not really the goal of this page. —David Eppstein (talk) 00:55, 13 December 2011 (UTC)

I totally agree, but here is the issue I'm wrestling with. If the uses of the topic are very distinct, writing a DAB page is very straightforward. A difficulty arises when there are only subtle differences in the uses of the term. In this case, to guide the reader to the correct article one would need to be a bit more proactive than is typical, since you want to direct the reader to a particular slant on the subject or possibly to the appropriate level of article. It is not likely that a reader coming to this type of DAB page will have a clear enough idea about what they are after to navigate through the shades of meaning unaided. When I referred to a unity, I really meant an internal(to me) guiding principle to help provide that aid in a coherent manner.

I don't, at present, have a good grasp on what that organizing princple should be, but I can be more specific about what irks me about this page. An affine plane is essentially a plane which abstractly shares the concept of parallelism with the Euclidean plane. Looking at the current page I don't even see a hint of this. The description of affine plane (incidence geometry) would be the appropriate place to see this, but only the minor condition is mentioned. I also object to the use of abstract whenever axiomatics are mentioned (I have seen this POV in several articles.) The Euclidean plane, while clearly the appropriate example that all readers can relate to, and the "mother" of all geometric abstractions, has too many other properties to be considered the archetypical affine plane. Perhaps a better description would be that it is a metric affine plane. There is a clear reference to C2, which, since there isn't an appropriate article, gets a link to Cartesian coordinates. That is just sending the reader on a wild goose chase. There is also a statement about affine being defined to distinguish these geometries from something that they are not (their projective completions) which I find awkward. I think I should stop here. Bill Cherowitzo (talk) 05:36, 13 December 2011 (UTC)
 * I agree with . IMO there is only one notion of affine plane. Thus a dab page is a nonsense here. I have rewritten the article as a stub, with links to every pages that was listed as disambiguation items. D.Lazard (talk) 18:59, 8 February 2014 (UTC)
 * Ok, but if it's an article rather than a dab, it needs references. Right now it has none. —David Eppstein (talk) 20:26, 8 February 2014 (UTC)

Merger proposal
I propose to merge Affine plane (incidence geometry) in this article. In fact, both articles are devoted to the same object which may be defined by two equivalent ways, at least over a field or a division ring (through incidence geometry and through linear algebra). This is stated, with a reference, in Affine geometry. As both articles are very short, there is no reason for keeping them distinct. D.Lazard (talk) 07:21, 14 June 2015 (UTC)
 * Non-Desarguesian planes cannot be coordinatized algebraically, as I understand it. So incidence-based definition is more general, and (if the articles are to be merged) needs to be kept primary. —David Eppstein (talk) 07:29, 14 June 2015 (UTC)
 * I agree that non-Desarguesian planes cannot be coordinatized algebraically, that is the reason of "at least". (Note that Affine plane (incidence geometry) talks of affine planes "over a field" without any definition.) However, "non-Desarguesian plane" is a highly technical notion, and the incidence-based definition seems, nowadays, rarely used. Thus I would suggest to give first the linear algebra-based definition, then the incidence definition and the equivalence, when it makes sense, and finally the non-Desarguesian example. D.Lazard (talk) 08:10, 14 June 2015 (UTC)
 * , are you saying that the most general category should always be primary topic of a page? If so, I disagree; that would suggest that no math article should contain a Generalizations section.  The topic of non-Desarguesian planes is not specific to affine planes, and thus to make a big deal of it in articles on affine planes is out of place, even though it should be mentioned.  I think D.Lazard's approach looks good, and I support a merge .  —Quondum 15:13, 14 June 2015 (UTC)
 * To make clear what I am suggesting, let me suggest an absurd parallel situation: suppose someone suggested merging the article on rings into the article on fields, with the motivation that most of the algebra studied e.g. by undergraduates doesn't really need the full generality of a ring, and that rings are a highly technical notion that doesn't need a separate encyclopedia article. You would squawk, right? Ring theory has developed to become a separate subject with its own depth that can't really be treated properly as an afterthought to fields. My feeling is that the combinatorial definition of affine planes bears a similar relation to the algebraic definition. —David Eppstein (talk) 19:50, 28 June 2015 (UTC)
 * Understood, but I think the fact that the synthetic approach happens to generate a superset of the algebraic approach should not be treated as being a generalization. I'm taking a neutral position on the merge proposal.  Perhaps we should wait as Bill proposes below.  —Quondum 02:46, 29 June 2015 (UTC)
 * I agree with and see a need to keep these articles separate, at least for the moment. Affine planes are an important type of incidence geometry (in their full generality) and have several applications in (block) design theory, graph theory, matroid theory, etc. Having this topic subordinated, as  is suggesting, to the special case makes these connections difficult to see. I am currently in the process of expanding our articles dealing with incidence geometry (see Incidence structure, Incidence geometry and I'm currently working on Incidence (geometry)) and I haven't gotten around to this one yet, but I was planning on getting to it soon. Of the three articles that I have mentioned, only the last is devoted to working over fields exclusively, and you can see by comparison that it is a horse of an entirely different color. I would suggest that we postpone this merger discussion until after I get a chance to develop this page so that we can see what we are talking about more clearly. (By the way, I've only been concerned with the content of the individual pages and there are issues (problems?) with the overall structure of these related pages that I haven't addressed. Other eyes and suggestions are welcome!) Bill Cherowitzo  (talk) 17:25, 14 June 2015 (UTC)
 * Closing merge given weak consensus to keep separate for now (also note that the discussion is stale). Klbrain (talk) 22:41, 3 October 2017 (UTC)