Talk:Projective plane

message: I'm not sure I understand the point being made by the 'degenerate' examples, in detail. Is it just to show what the third axiom rules out?
Charles Matthews 06:54, 26 May 2004 (UTC)
thread_title: Degenerate examples

message: The 168 symmetries of the Fano plane play a significant role in mathematics.  Hence I have added a paragraph linking to a discussion of these symmetries.  That discussion is my own.  For some background, see other discussions of mine on finite geometry cited at Alan Offer's compilation  of online finite geometry notes at Ghent University. Cullinane 13:29, 9 August 2005 (UTC)
thread_title:  Symmetry group of Fano plane 

message: I added this to the properties : the fact that the order n occurs as order of projective plane is equivalent to existence of an affine plane of the same orde.
Why is there no mention at all of affine planes.  In fact, i think you can make an affine plane by removing a random line from a projective plane, and  two such affine planes are isomorphic if and only if there is an automorphism on the projective plane mapping the first chosen line to the second.
:You'd better clarify your definition of 'affine plane'. A combinatorial definition would not be the same as affine plane, qua affine space of dimension 2. Charles Matthews 15:04, 1 February 2006 (UTC)
replies: text: No, IMO.  An affine plane in the sense of this general article is equally general.  Besides, the fact that deleting any one line from a projective plane gives an affine plane is completely general. The fact that adding a line at infinity to an affine plane gives a projective plane is also completely general. Zaslav (talk) 08:16, 27 March 2011 (UTC)
thread_title:  affine planes 

message: I added a reference to mutually orthogonal latin squares.  I find it strange that there was no mention at all.  However, one might argue about the whole new section just for that.  On the other hand it does explain why Euler proved n=6 impossible with his officers problem.
:Sorry, Euler had nothing to do with abstract projective planes; they came later.Zaslav 01:50, 7 November 2006 (UTC)
thread_title:  latin squares 

message: I know it has been pointed out that not everyone thinks the same thing of projective planes and axiomatic projective planes, so this might lead to more confusion.  However, i created an article 
Axiomatic projective space and I was wondering where is the best place to refer to it.  I think it is important as axiomatic projective planes are what makes axiomatic projective spaces hard to classify.
I also added the converse of the steiner system remark.
Evilbu 17:25, 15 February 2006 (UTC)
thread_title:  axiomatic affine space/steiner system 

message: It appears that the article, when it finishes with combinatorial projective planes, immediately launches into how commutativity of the division ring is unnecessary -- and then "Generalizations" -- without ever defining the geometric projective plane at all!  This strikes me as a glaring omission (or does it represent vandalism?).   There needs to be an elementary discussion of what the plain vanilla projective plane of a vector space is before beginning to get into the generalizations thereof.Daqu 21:56, 23 October 2006 (UTC)
replies: text: The reader is supposed to use the link to projective space. There is nothing special about the case of the projective plane associated to a three-dimensional vector space. Perhaps links to the pages on real projective plane, complex projective plane should be more prominent. Charles Matthews 22:04, 23 October 2006 (UTC)
replies: text: Indeed it must be more prominent.  There is nothing informing the reader of what you just wrote, even if you and I know perfectly well that it is true.  (Also, the very fact that there is a term "projective plane" suggests that the best way to explain it may not necessarily be as a special case of a more general concept.)Daqu 22:09, 23 October 2006 (UTC)
replies: text: I fully agree with Daqu. For example, the Surface article is largely redundant with the Manifold article, just so people can learn the concepts in two dimensions before having to deal with n of them. Additionally, in my experience the linear-algebraic definition is by far the most common (but maybe I'm biased.) What's worse, Projective space is also grossly deficient in the linear-algebraic treatment. If I have time, I'll put in some work on these. Joshua Davis 13:32, 26 October 2006 (UTC)

text: Though I actually agree with the sentiment, this is a POV problem ('must be more prominent' does depend on one's orientation within mathematics). Charles Matthews 13:41, 26 October 2006 (UTC)
replies: text: The projective plane, and not higher projective spaces, are of special importance in combinatorics.  It makes natural sense to explicitly discuss the geometric version of this combinatorial object.Daqu 20:20, 14 November 2006 (UTC)

text: I disagree with Charles Matthews (vehemently, actually).  It is not a matter of POV, it is a matter of starting at the beginning.  Currently, the introduction only states that there are two definitions of the projective plane (coming from linear algebra and geometry) but doesn't say what they are.  This article would be vastly improved if someone would replace this with a layman's explanation of what the projective plane is, and shift the technical discussion down to start at the first section.  The fact that I came to Wikipedia to learn about projective planes and came away no wiser is not a demonstration of POV success.  Edmund Blackadder (talk) 04:29, 1 September 2010 (UTC)
thread_title: Missing definition of ordinary geometric projective planes???

message: The article makes several references to affine planes but a) without ever defining them and b) without referring to any Wikipedia or external definition of them, either.  If you're going to use a technical term, there must a definition either here or elsewhere that the reader can find . . . if you have any desire to be understood.Daqu 22:09, 23 October 2006 (UTC)
thread_title: Missing definition of affine plane

message: What does "PG(2,F)" really mean? 192.43.227.18 15:00, 4 October 2007 (UTC)
thread_title:  Notation 

message: This section needs explanation: Let K be a division ring. How does one construct a projective plane? In the text in the article there is no hint about this. And an example may also be helpful.
TomyDuby (talk) 05:47, 14 February 2009 (UTC)
thread_title:  Linear algebra definition 

message: An IP editor added the following comment to the body of the article. I've moved it here:
:I suggest to add a sentence at the start of this section, saying that an alternative definition of a finite projective plane of order n with n > 0 is a collection of n^2 + n + 1 lines and n^2 + n + 1 points such that 1. every line contains n + 1 points, 2. every point is on n + 1 lines, 3. any two distinct lines intersect in exactly one point, and 4. any two distinct points lie on exactly one line. (Perhaps this is slightly easier to grasp than saying that "there are four points such that no line is incident with more than two of them.") 
--JBL (talk) 02:36, 20 July 2012 (UTC)
replies: text: I assume that it is the finite projective plane section that is being discussed. There are some problems with this suggestion. First of all, it is not an alternate definition, it is a theorem which follows from the definition of a projective plane and the assumption of finiteness. The proof of this theorem involves counting, and the result is essentially what is given in the first sentences of the section. Secondly, the suggestion would include the case of n = 1, which gives a triangle. It is almost universally accepted that a triangle is not to be considered a projective plane, but rather a degenerate form of a projective plane. Admittedly, this is a convention, but to do otherwise would require the constant exception of this case in most general statements of theorems concerning projective planes. The IP editor seems to be confused about the meaning of "there are four points ..." in the definition of a projective plane. This condition is specified to eliminate the seven degenerate cases, one of which is the triangle. In those treatments of projective planes which only deal with finite projective planes from a block design point of view, one may see the suggestion given as a definition (even with the casual disregard of the n = 1 case), but this is not acceptable in the geometry community. Bill Cherowitzo  (talk) 18:28, 20 July 2012 (UTC)
replies: text: IP means Internet Protocol, and terms like "IP editor" are nonsense because that obviously doesn't have anything to do with the Internet Protocol. YOU CAN'T DO THAT - which I mean is to create new acronyms and initialisms without explaining what they mean. All such things must be explained upon their first appearance. Furthermore, in Air Force terminology, IP means "initial point", which is a certain place in a bombing run. Don't go around using undefined acronnym, etc., especially when they conflight with well-known ones like IP = Internet Protocol. And then you want to do that on the Internet? Completely ghastly.98.67.106.148 (talk) 15:32, 22 September 2013 (UTC)
replies: text: And you left out Intellectual Property and International Police and about a zillion other possible interpretations in your tirade. In the Wikipedia community, an anonymous user who has not created an account (such as yourself) is identified by their IP address as explained at WP:IP. Such editors are casually referred to as anonymous IP editors, or just IP editors. On talk pages, such as this one, editors talk to other editors about how to improve the associated articles. Jargon such as this is a natural development in such an environment and not the ghastly abomination you seem to think it is. There are also several other conventions that are in use in this community such as: using all caps is called shouting and is frowned upon. One learns about such conventions generally by observation over time. Bill Cherowitzo  (talk) 16:28, 22 September 2013 (UTC)
thread_title:  IP comment on section Projective planes in higher dimensional projective spaces 

message: The textbook Lars Kadison und Matthias T. Kromann: Projective Geometry and Modern Algebra. Birkhäuser, Boston/Basel/Berlin 1996,  has together with the 1st and 2nd instead of the 3rd axiom "There are four points such that no line is incident with more than two of them." in the article now 2 axioms: 
*P3: There exist three noncollinear points and 
*P4: Every line contains at least 3 points.
In my point of view an Y-graph (or Mercedes-Stern without outer circle) (4 points, 3 lines, exactly 2 points on each line, all three lines intersecting in one single point) should fulfill all the present 3 axioms in the article and is rather degenerate. Or abstractly speaking: How the hell should the dual of the third axiom in the article be proven from 1.-3., without making this dual form 3.D an axiom?--KlioKlein (talk) 15:12, 8 August 2013 (UTC)
replies: text: The nondegeneracy axiom stated in the article is equivalent (in the presence of the first two axioms) to the P3 and P4 axioms of Kadison and Kromann. This is a standard homework exercise in a projective geometry course. The overwhelming majority of the references will give the axioms as presented (or equivalents - the usual phrasing of axiom 3 is : There exist four points no three of which are collinear), although there are a couple of notable exceptions (Beutelspacher & Rosenbaum, and Coxeter come to mind immediately). Having 4 axioms instead of 3 is an older tradition and some authors still find it useful for their expositions. I have noticed that Kadison and Kromann tend to use older terminology which is consistent with an algebraic geometry point of view. I could provide the citations and mention this issue if you think it will improve the page (someone else wrote that section and I didn't think that this was important enough to modify it). Bill Cherowitzo  (talk) 20:49, 8 August 2013 (UTC)
thread_title:  I doubt that the "Definition" suffices to avoid all degenerate planes. 

message: In the vector space construction, we're defining planes through the origin in K^3 as (kx + ly) for k,l in K. Then this sentence: 
"This plane contains various lines through the origin which are obtained by fixing either k or l."
Since we're talking K^3, wouldn't fixing k or l just make lines parallel to y or x (resp.), that don't pass through the origin? It seems like it's the ratio of k and l that needs to be fixed to get lines through the origin.
173.25.54.191 (talk) 02:27, 9 April 2014 (UTC)
replies: text: Good catch. I didn't pay much attention to this section and just assumed it was correct, but you are right. As stated you get lines which do not pass through the origin. I'll fix it. Bill Cherowitzo  (talk) 03:29, 9 April 2014 (UTC)
thread_title:  Vector space sentence correct?