Talk:Projective plane

Java app
Have written a java app to find projective planes. Has reached order [16 lines * 17 points] so far.

Degenerate examples
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)

Symmetry group of Fano plane
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)

affine planes
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)

I meant the combinatorial definition (thus without use of any linear space over any field) : three noncollinear points at least, all different two points are on exactly one line, for each point not on a given line there is exactly one parallel line through it. So should i just write "combinatorial affine plane?"


 * 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)

latin squares
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)

axiomatic affine space/steiner system
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)

I added a reference to Axiomatic projective space in the Finite geometry article. No doubt the reference would also be useful in other articles, but I don't know which ones.

Cullinane 20:06, 15 February 2006 (UTC)

Missing definition of ordinary geometric projective planes???
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)


 * 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)


 * 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)


 * 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)


 * 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)


 * 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)


 * 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)

Missing definition of affine plane
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)

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

Linear algebra definition
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)

Topological embeddings
The new subsection "Topological embeddings" needs some work, at least in its terminology. Embedding is the wrong word to use here; the real projective plane cannot be embedded into Euclidean 3-space. And some of the other examples destroy the topology in other ways; e.g. the hemisphere is not homeomorphic to the projective plane, so stereographically projecting it does not yield a projective plane. Mgnbar (talk) 14:24, 28 December 2010 (UTC)
 * Please follow up the references given before you make such uninformed claims. For example the hemisphere as it stands is not homeomorphic to the projective plane, but if pairs of opposing points around the rim are identified as single projective points then it becomes a homeomorph. Weeks (The shape of space, CRC, 2002, p 59) explains it like this; "The projective plane is a surface that is locally like a sphere, but has different global topology. It's made by gluing together the opposite points on the rim of a hemisphere." and gives a nice illustration. You might also care to consider why Klein thought the stereographic projection is also a homeomorph. &mdash; Cheers, Steelpillow (Talk) 15:02, 28 December 2010 (UTC)
 * Hi, SteelPillow. Just so we're clear, in my post above I did not say that the section should be thrown out; I was trying to say that the terminology should be clarified. The statements you've made in your post are correct, but they do not refute my argument. In fact, your quote about how "The projective plane is a surface that is locally like a sphere, but has different global topology." supports my argument: The projective plane, the sphere, the half-closed hemisphere, and the plane all have different topologies, and this is why the word embedding is inappropriate. In the cases of Boy's surface and the Roman surface, the projective plane is immersed in 3-space. The fact that the real projective plane cannot be embedded in 3-space is well-known; see the first paragraph of Real projective plane, or Massey: A Basic Course in Algebraic Topology, p. 6, or Bredon, Topology and Geometry, p. 353, for example. Mgnbar (talk) 16:56, 28 December 2010 (UTC)
 * "Just so we're clear" indeed. This could take a while. You misrepresent Week's quote. He says that the projective plane and the sphere are not homeomorphic (they have different global topology). He does not make this claim for the projective plane and the hemisphere as you suggest he does, in fact quite the reverse - he explains how the two topologies are equivalent - his "gluing together" is just layman's language for "identifying". You also quote the Real projective plane article out of context. It remarks that the plane cannot be embedded in 3-space without intersecting itself. I never said that it could. Another mistaken assertion is that the Roman surface is an embedding. It has Whitney umbrellas (or similar features?), therefore it cannot be an embedding. Historically it was widely conjectured that no smooth embedding existed. Boy was set the task of finding a proof - only to discover the counterexample that bears his name. You may confirm all this for yourself from the references I note in the article. If Massey and Bredon truly disagree with Ap&eacute;ry and Brehm then we have an interesting time ahead of us. &mdash; Cheers, Steelpillow (Talk) 17:43, 28 December 2010 (UTC)
 * Whoops! My apologies. I got embedding and immersion the wrong way round. I'm always doing that. sigh. &mdash; Cheers, Steelpillow (Talk) 18:19, 28 December 2010 (UTC)
 * No problem; we all make mistakes. I like your recent changes, but we do still have some work to do. The Boy surface is an immersion but not an embedding, because it is not injective. I don't know much about the Roman surface, but Roman surface seems to explain in some detail that it's not even an immersion. We should also mention the cross cap, which seems to be an immersion. (By the way, Cross-cap seems to have some errors.) The map from the sphere to the projective plane is very valuable, but it is not an immersion, because it's not injective; it's a quotient map, a covering map, and a submersion. The map from the hemisphere to the projective plane is a quotient map. I vote that we retitle this subsection to something like "Relations to other topological spaces". Regards, Mgnbar (talk) 21:21, 28 December 2010 (UTC)
 * Yes, I now see that you are right that none of these are embeddings. We need a general terms for dropping one manifold inside another space (to allow either injection or surjection). Offhand I cannot think of a suitable one - I believe have seen "projection" used in this context but am not sure if this is a correct choice. Perhaps "examples" is adequate - I'll try it. We need to be clear that these are projective planes in their own right. For example note that Weeks is clear that he is constructing "the projective plane" from his hemisphere, not just an equivalent surface. To a topologist all these forms are the same object, indistinguishable from each other save by external properties. In fact I think it might help if we promote this section a level, call it "topological definition", and provide a bit more explanation. However I am ignorant of the topological properties of the complex projective plane - which is why I put this section under the real projective plane. What do you think? &mdash; Cheers, Steelpillow (Talk) 10:50, 29 December 2010 (UTC) (Edited &mdash; Cheers, Steelpillow (Talk) 10:52, 29 December 2010 (UTC)]
 * Oops; I made a typo above. The map from the sphere to the real projective plane is an immersion, but not an embedding. Sorry.
 * The functions in question are all different enough that I can't see a common term for all of them beyond the generic "function" or "map". By the way, here's another topological definition: Take a Möbius strip and a disk, both of which have circles as their boundaries, and glue these circles together. The complex projective plane is rather different from the real projective plane --- a complex 2-manifold, and hence an orientable real 4-manifold. It should be described somewhere in this article, but maybe not in this section. Regards, Mgnbar (talk) 13:27, 29 December 2010 (UTC)
 * I think we agree that the current version is now correct. There are many topological constructions, e.g. slice a Klein bottle in half, stitch the sides of a square together in a certain way, etc. I think most of these go beyond the scope of this article (perhaps some of mine have already done so). I agree about the complex projective plane and have started a discussion below. &mdash; Cheers, Steelpillow (Talk) 15:08, 29 December 2010 (UTC)

Restructuring suggestions
IMHO this article only makes sense if it focuses on the common features of, and key differences between, the real and complex projective planes.

Some key properties that I think deserve their own (sub?)sections include:
 * Projective duality as it applies to the plane.
 * Treatment of the line at infinity, especially its projective lack of distinction with respect to any other line.

I don't know much about the complex one (e.g. how can it be orientable when, as I understand it, the real projective 4-space is not?), so I can't judge how much of the present content applies to both or only to the real one. For example does the bit on duality in the real projective plane apply to both kinds?

For the detailed treatment of each we have separate articles for the Real projective plane and Complex projective plane, but I think that separate (sub?)sections on each are called for here, to provide a home for their key differences.

Can anybody restructure/clarify the relevance of the present content for each type, either below here or just edit the article?

&mdash; Cheers, Steelpillow (Talk) 15:08, 29 December 2010 (UTC)


 * I fully agree that the real and complex projective planes deserve their own articles, and that this article should provide just a brief treatment and link to them. However, projective planes can be defined over any field (and even more generally than that), so these are not the only two examples. For example, the three-trait variation of the game Set amounts to finding points in a projective plane over the field of three elements. So I guess I vote that we move most of your new topological stuff to Real projective plane and then clean up Projective plane as you say, to discuss mainly those features common to all projective planes, such as the "key properties" that you mention above. By the way, Projective plane could easily become redundant with Projective space as well. Deciding how many articles there should be, and exactly how material should be divided among them, is not easy. Regards, Mgnbar (talk) 16:55, 29 December 2010 (UTC)


 * Looks like a plan. I would expect that the projective plane has enough meat to warrant an article independent of projective space, we'll see. I'll have to leave most of the work to those more knowledgeable than me. Would it make sense to move wholesale the section on visualising the real projective plane into the article on the real projective plane? &mdash; Cheers, Steelpillow (Talk) 12:22, 30 December 2010 (UTC)


 * Really I think that the article should be rewritten from scratch along these lines:
 * first definition: using linear algebra over division ring
 * common properties
 * mention real and complex cases, and link to those articles, but don't go into depth on them
 * projective transformations
 * second definition: combinatorial
 * properties, theorems, etc.
 * planes of small order, etc.
 * I'll start reorganizing and then working on the first definition. I don't have the knowledge or references to handle the second definition. Mgnbar (talk) 16:28, 31 December 2010 (UTC)


 * This makes sense if the properties of projective planes over fields or division rings are used to motivate the abstract (combinatorial) definition.
 * The real and complex planes really do need separate treatments. Most of their uses are not general to projective planes but are specific to planes that are, at the least, manifolds. Zaslav (talk) 08:12, 27 March 2011 (UTC)

Linear-algebraic definition
The lede is much improved, thanks; but I'm struggling with the opening line of the first proper section: "The projective plane is the set of lines through the origin in 3-dimensional space, and a line in the projective plane arises from a plane through the origin in 3-dimensional space. This idea can be made precise as follows." I think the first clause is simply wrong as it stands, but I'm not sure. Certainly it is unclear. --ColinFine (talk) 11:32, 1 January 2011 (UTC)


 * It has the right idea. I guess it might be more accurate to say that the set of lines through a point can be defined as a projective plane: enclose the point by a sphere, and the point-pairs of intersection with those lines make it easy enough to see that the sphere is a projective plane with respect to these point-pairs (antipodal points on the sphere are identified as a single projective point). Likewise a plane through the original point meets the sphere in a projective line (great circle) on the sphere. There is no need to draw the sphere in order to treat the lines and planes through the original point as a projective 2-space. Taking the original point as the origin makes life easier. Not sure how to express this rigorously in terms of linear algebra, so I won't touch the article myself. &mdash; Cheers, Steelpillow (Talk) 13:23, 1 January 2011 (UTC)


 * ColinFine: I know that I have yet to add references (I'll try to do so today) but the idea you're discussing is quite standard in mathematics. For example, from Baez: The Octonions: "If K is any field, there is an n-dimensional projective space called KPn where the points are lines through the origin in Kn+1, the lines are planes through the origin in Kn+1, and the relation of ‘lying on’ is inclusion." Is your objection that we're tying the projective plane to a particular construction or representation? We could fix that by adding "In one view...".


 * SteelPillow: I agree that any point could in 3-space be used to define the projective plane. This is analogous to how mathematicians often talk about "the sphere" as being defined by x^2 + y^2 + z^2 = 1, when the details of radius and center don't matter to the math that they're doing. For the projective plane, introducing an alternative "center" wouldn't help anything; you'd get an equivalent definition with more complicated notation, I think. Regards, Mgnbar (talk) 15:23, 1 January 2011 (UTC)
 * Yes, that's why I said taking it as the origin makes life easier. One could equivalently take any spheroid in place of the sphere, probably any quadric surface if one chose an appropriate reference point, but that would get even more complicated. &mdash; Cheers, Steelpillow (Talk) 22:15, 1 January 2011 (UTC)
 * Agreed. Regards, Mgnbar (talk) 22:19, 1 January 2011 (UTC)
 * ColinFine: In case my post above came across as dismissive, what I should have said is: The treatment given is correct, but if it's unclear or unhelpful, then please elaborate, so that we can improve it. Mgnbar (talk) 17:39, 1 January 2011 (UTC)

Yes, the usual definition of this type simply takes all lines through the origin. You gain nothing by generality. The proper way to say it, I think, is not to say "the plane" is the set of lines, but that the points of the plane are the lines through the origin. You also have to say what the projective lines are; they are the sets of coplanar lines. I think this change is necessary because some interested readers are not going to understand the definition without its being more concrete. (I have a reader in mind.)


 * The current text is common in math textbooks, because mathematicians are accustomed to conflating an object with its underlying set when necessary. But your new text ("the points of the plane are the lines...") is more precise and not overly wordy, so please put it in.

On the other hand, if you're discussing the concept of projection from an arbitrary point in an affine or projective plane, then you would take the lines through your point. I suggest that the former is best for a definition and the latter for discussing properties of, and constructions in, affine and projective spaces. Zaslav (talk) 08:07, 27 March 2011 (UTC)


 * Yes, the former should be used in the definition, because that is standard in textbooks, and the latter has its place elsewhere.


 * I'm glad that you're making edits to the combinatorial definition, which I edited very little in my recent reorganization. For one thing, you could do a great service by adding some references. Mgnbar (talk) 13:22, 27 March 2011 (UTC)

Definitions
There is a fundamental problem with this page and it has to do with the meaning of a definition. There are some here who view projective planes as whatever is obtained by the standard field plane construction (I am using the term field here in the sense that the French sometimes do, it need not be commutative) which is called here the "Linear Algebra Definition" (should be vector space construction, but that's probably nit-picking). The "combinatorial" (and I wonder where that term came from) definition is then introduced as a generalization (or alternative) of the first. This is wrong on several levels. First of all, the concepts are different, so one could never refer to them as being alternatives. The axiomatic definition encompasses the Linear Algebra construction so it should take precedence. A construction should never be taken as a definition unless you don't know what the object you are constructing really is. In mathematics you will find that there are good definitions and bad definitions, but hopefully the bad ones get weeded out over time. The function of a "good" definition is to provide a way to decide whether or not an object is in the class of things being defined. The problem with using a construction as a definition is that if an object is obtained in a different way you need a theorem to say that the object is isomorphic to something obtained by the construction, rather than simply verifying that the definition is satisfied. While it is true that you see objects referred to as a "generalized" this, or a "generalized" that, when this is properly done, the term "generalized" is part of the name (as in Generalized quadrangle) and has its own specific definition. Advocates of the "Linear Algebra definition" would want to refer to projective planes defined axiomatically as "generalized projective planes". Frankly, this is just not done by anyone. A quick look at any text with Projective Plane(s) in the title or more generally Projective Geometry, will show that only definition is the axiomatic definition. I believe that a reorganization of this page is in order. Wcherowi (talk) 18:31, 26 August 2011 (UTC)


 * You raise some good points, but they do not convince me that the article should be substantially changed. Please consider these counterpoints:
 * When one is trying to learn a difficult concept, it is helpful to see examples before an abstract definition. (I have seen educational studies to support this idea, although I confess that I cannot cite them.) Wikipedia is not a textbook, but it still tries to make concepts accessible. The former definition/construction of projective plane, insofar as it is an example of the latter one, is valuable for this reason.
 * You assert that books entitled Projective Plane (etc.) all use the axiomatic definition. I counter that every topology or algebraic geometry book that I have ever seen uses the linear algebraic definition. So maybe this is a disagreement between sub-disciplines of math. Indeed, from my experience (which I admit is not universal) it seems that projective planes over (A) the real numbers and (B) algebraically closed (commutative) fields account for the vast majority of discussion of projective planes in the modern math literature. So it makes sense to have them explicitly constructed in this article.
 * You also raise a number of semantic points. I agree with you that the latter definition is not really an "alternative" to the former. You also say that the latter definition "encompasses" the former, but is not a "generalization" of it; this makes no sense to me, but maybe it's not important. As far as "linear algebraic definition" vs. "vector space definition" goes, I am indifferent.
 * Would you be satisfied if the first definition/construction, rather than being called "Linear algebraic definition", were simply renamed "Construction over a division ring"? And, by the way, have you seen Projective space? Mgnbar (talk) 19:57, 26 August 2011 (UTC)


 * Thanks for responding so quickly. I think that I should clarify some of the points that I made.
 * Examples are extremely important and this article is in need of more, I am not advocating the removal of any examples. The difficulty that I am trying to point to is calling an example a definition.
 * It is good practice, when writing a textbook or other educational exposition, to proceed an abstract definition with good examples ... provided the reader is familiar with at least some of the examples. If the examples are as novel as the definition, the whole point of doing this is lost.
 * I object to the confounding of abstract and axiomatic. The axiomatic approach is familiar from high school geometry. The axioms for a projective plane are very simple statements and there are only three of them - far less abstract than the concept of an equivalence relation on 3-dimensional vectors defined over a division ring.
 * Recall that the title of the article is Projective Plane, not the "Topological Aspects of Some Projective Planes" or the "Algebraic Geometry view of Projective Planes". I would believe that books with the same title as the page are probably more closely aligned with the intended meaning of the term.
 * There is a reason that you see only the classical construction in Algebraic Geometry books. There are, with very few exceptions, two assumptions made about the geometries that are dealt with, namely, the geometric dimension is at least three and the division ring does not have characteristic two. These assumptions are made so that the theory works out well without exceptions. If either assumption is violated then one has to deal with special behaviours that do not fit into the general framework. Of course, it is sometimes necessary to talk about planes, so when that occurs the planes are restricted to be those of the classical construction and generally over division rings of characteristic zero, so that the general framework does not have to be modified.
 * The reason for the above restriction to dimension three or higher is that it is fairly easy to prove that Desargues' Theorem holds universally there and so the projective space does come from the classical construction, but this is not true in dimension two. In other words, only in geometric dimension two can you get projective geometries which do not come from the classical construction. Such projective planes can not be embedded in projective space and so are not part of that theory. Some will consider these planes to be aborations, but those of us who work in this area celebrate the richness of the variety of these planes. If it wasn't for these examples there wouldn't be much of a reason to have this article, it could just be a section of the projective space article. Wcherowi (talk) 03:32, 27 August 2011 (UTC)


 * Hi again, Wcherowi. I appreciate that you've put some thought into this. Regarding your points here:
 * From my experience in teaching students (which may contradict yours or others'), constructions are always more concrete than non-constructive axiomatic definitions. If you present them with the latter, they ask, "But what is it?" They like triples of numbers, and they love pictures. So I am still of the opinion that having the construction before before the axiomatic definition is useful, or that we should somehow tell the reader to read Real projective plane before this article.
 * The fact that this article is called "Projective plane" does not convince me that books of that title should take precedence, as if it is somehow their turf, or that topological and algebro-geometric properties of projective planes should not be treated. The "intended meaning of the term" in Wikipedia is, to be honest, in constant flux, because it's decided upon by you and me. However, we should be guided by least surprise. Imagine a reader who comes to this page. What is she looking for? Probably RP2 or CP2, really. So we should quickly point out the articles for those two cases. We should also give the construction over an arbitrary division ring, because it generalizes those two cases. We should also give the axiomatic definition, of course.
 * I understand that projective spaces of two dimensions are qualitatively different from those of higher dimension. That is a good reason to have this separate article, as you say. But there are other reasons to have this separate article. People learning a topic find it helpful to learn a low-dimensional version first. Also, low-dimensional examples are disproportionately important; look at how important CP1 is.
 * In case you are not aware (your editing history suggests that you've started editing only recently, but this could be misleading), the mathematics articles on Wikipedia are under constant complaint from readers as being "too abstract" and "written for mathematicians only". For me, this is a key consideration.
 * What changes to the article do you want, exactly? I have proposed renaming the first definition as a mere "construction". Would that satisfy you? What if we moved the construction after the definition? Would that satisfy you? Or do you still feel that a rewriting of the article is warranted? I am not convinced of that. Regards. Mgnbar (talk) 10:26, 27 August 2011 (UTC)

At risk of lowering the tone of this discussion, I am one of those "constant complainers" who seek everyday explanations. The lead introduces the linear algebraic and axiomatic definitions. Good. There follows a section discussing the linear algebraic view. But there then follows a section on the "combinatorial" definition. Are we to take this as another name for the axiomatic? Further, to me the linear algebraic definition arises as a consequence of the geometric observation that a sphere with antipodal points identified is a projective plane (we draw lines joining antipodal pairs). Meanwhile the geometric construction arises as a consequence of the axiomatic treatment (plus some definitions of "point", "line, "plane" and "incidence"). And is not algebra supposedly based on axioms too (or have those been successfully supplanted by set theory, as the latter part of the 20th century set out to do)? It seem to me that these relationships and their explanation should guide the construction of this article. &mdash; Cheers, Steelpillow (Talk) 12:12, 27 August 2011 (UTC)


 * Hi, Steelpillow. It's fine with me to rename "combinatorial" to "axiomatic". I disagree that the linear algebraic definition follows from the geometric definition; either one can be made to follow from the other. (The geometric definition of RP2 arises from the algebraic one by picking two points on each line, and then modding out. Very simple.) In modern geometry, it is extremely common to define spaces by "constructions" involving "data". Look at manifold or orbifold or scheme, for example. I say all of this while acknowledging that we should probably change the article to make the axiomatic definition "the definition" and the linear algebraic definition "the crucial class of examples". Mgnbar (talk) 12:31, 27 August 2011 (UTC)
 * Thanks for the prompt reply. I am not saying that the geometric takes absolute primacy over the algebraic, merely that as a simple-minded soul who loves pictures, that's the way I see things. &mdash; Cheers, Steelpillow (Talk) 15:03, 27 August 2011 (UTC)


 * Hi again. I don't think we are really very far apart. My pet peeve is what I perceive as an inappropriate use of the term "definition" when "construction" is meant. That aside I don't really see major revision of the page, merely a reorganization (actually along the lines that Mgnbar suggested) that would remove a pov bias (as I see it). So here is what I am proposing -

I hope this gives a better idea of what I had in mind. Wcherowi (talk) 17:59, 27 August 2011 (UTC)
 * 1) Introduction - essentially ok. I would modify the second paragraph a bit and put a stronger emphasis on the link to the Real Projective Plane.
 * 2) Definition - the axiomatic definition, it need not be labelled as axiomatic.
 * 3) Some Examples
 * 4) RP2 - Given as the extended Euclidean plane (i.e., add points to destroy parallels)
 * 5) PG(2,q) - Maybe just a specific one like PG(2,3) with a hint about other finite fields.
 * 6) Moulton plane - An example of an infinite plane which is easy to describe and is non-Desarguesian.
 * 7) Vector Space Construction
 * 8) Classical examples (section 1.1)
 * 9) Topological properties and anything else that needs to be mentioned about the classical examples.
 * 10) Finite Examples - including the Fano plane.
 * 11) Affine Planes
 * 12) Definition and Examples
 * 13) Construction of Projective Planes from Affine Planes (essentially section 2.3 done more generally and a tie in with the very first example. Also includes section 1.2)
 * 14) Generalized Coordinates (PTR's, etc.  essentially section 2.2)
 * 15) Degenerate Planes (section 2.5, but redone to clarify what the seven types are)
 * 16) Collineations (Includes section 1.3 but goes beyond this to include fixed-point configurations)
 * 17) Lenz-Barlotti Classification Scheme of Projective Planes (which is based on the types of degenerate planes)
 * 18) Finite Projective Planes (essentially section 2.4)
 * 19) Projective Planes in Higher Dimensional Projective Spaces (sections 1.4 and 2.6 combined)


 * Now that I have a better idea of what you're suggesting, I like it. Please go ahead. By the way, you might want to read Wikipedia talk:WikiProject Mathematics, where a discussion of the "example before definition" idea seems to be underway. And Steelpillow: I thrive on pictures too. :) Mgnbar (talk) 12:13, 28 August 2011 (UTC)

Major overhaul
I've just finished (or at least – come to a reasonable stopping point of) an overhaul of this page. I've reorganized it and added new material without (I hope) treading too heavily on what was already on the page. Please look it over and give me your reactions, suggestions, etc. Thanks. Bill Cherowitzo (talk) 03:42, 2 October 2011 (UTC)


 * In that nobody speaks, I'd say go ahead! One technical name for a major rewrite that changes structure but not content is a (major) refactor, that might be a good term to use in some of your edit summaries.


 * But one word of caution... the algebraic usage is equally important to the usage in naive projective geometry.


 * Agree that the current article would be improved by discussing the naive approach first, but I think your current draft doesn't strike a balance yet. We might need to consult an algebraist, I did study projective geometry back in the 1980s but that was a while ago! Andrewa (talk) 14:21, 18 October 2011 (UTC)

Having bounced around for the past week or so I am now prepared to go ahead with my major refactor of this page (thanks for the terminology Andrewa). I am being a little timid here, but there is a subtle POV issue in the background and I want to make sure that I get it (NPOV) right. For the past 60-70 years there has been a split in the projective geometry research community. Camp A, most notably algebraic geometers but others as well, believes that projective planes are just the 2-dimensional versions of projective spaces and tends to ignore anything that does not scale up to higher dimensions or puts too much of a burden (in terms of exceptions and alternate treatments) on the algebraic underpinnings. Camp B, I'll call them "foundationists", consider the primary objects of interest to Camp A to be well studied and known topics and are more interested in the projective planes which can not be imbedded in higher dimensional spaces. From the Camp B viewpoint, the field planes (as Camp A sees them) are a very important but special class of examples. As I see the current article, it is written from the Camp A viewpoint with an offhandedly perfunctory nod to Camp B. I believe that this is not the way to write an encyclopeadic article and so I rewrote the article to reflect the more inclusive worldview of Camp B. This did not involve changing a lot of content but it did require some modifications and a lot of reorganization. A comparison of the current article and my refactorization will look like I have totally rewritten the page &mdash; but this is really not the case as a closer examination will show. Have I given Camp A equal treatment? - No. Have I given the Camp A point of view? - No. Have I provided enough of an emphasis on the Camp A special case? - that's debateable, and I probably haven't &mdash; but what I have done is to open up the structure of the article so that this material can be easily added by any editor. I would urge anyone who is interested in this topic to take a look at User:Wcherowi/Projective planes and let me know if I have stepped over any boundaries that I shouldn't have. Thanks. Bill Cherowitzo (talk) 16:16, 19 October 2011 (UTC)

IP comment on section Projective planes in higher dimensional projective spaces
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)


 * 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)


 * 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)


 * 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)

I doubt that the "Definition" suffices to avoid all degenerate planes.
The textbook ''Lars Kadison und Matthias T. Kromann: Projective Geometry and Modern Algebra. Birkhäuser, Boston/Basel/Berlin 1996, ISBN 3-7643-3900-4'' 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)

I saw now that my example Y does not fulfill 2. and is wrong. But an inline-citation for the axioms would be nice! :-) (Or did I overlook something?) A little soubt remains... --KlioKlein (talk) 15:26, 8 August 2013 (UTC)


 * 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)

Poor English sprinkled through
Poor English sprinkled through: "...lines keep the points with negative x-coordinates..." Lines are inanimate objects, and they are incapable of possessing or doing anything at all. Furthermore, there were statements about "the points" (in a plane) which means any points whatsoever - all of them - instead of a specified subset of points. You really must write clearly and specifically, and not vaguely and indeterminently - and especially in mathematics. Also, there was the problem of writers making up words that do not exist in the dictionary or common usage. Even if those are "jargon words" in the field, that is BAD, because that is jargon that should be avoided at all costs. This article should be written in Plain English that the general reader can read and understand, even if he cannot understand the mathematics. I will just make one up as a specific example: pseudohypergeometrization. Don't do that. I am not stating that his one was really there, mind you, but I am creating an EXAMPLE.98.67.106.148 (talk) 15:42, 22 September 2013 (UTC)
 * About English: Some of your complaints have been fixed. Other examples are idiomatic to mathematics writing, and hence may not be fixed.
 * About jargon: Mathematics, like any highly technical field, has its own jargon, which does not appear in non-mathematical dictionaries. Part of the function of Wikipedia is to explain and exemplify jargon for readers, so that they can understand other texts that use such jargon.
 * About difficulty: This is a common complaint about mathematics articles on Wikipedia. (See FAQ at the top of the project talk page here.) This Projective plane article is not a bad offender, in comparison to many other articles. It attempts to give visual intuition, historical context, applications, etc. The later parts of the article are more obscure, but only highly motivated readers will attempt those anyway. Fundamentally, this is an article about a moderately advanced mathematical topic, that few non-mathematical laymen ever encounter. So the difficulty seems about right to me. Yes, there is always room for small improvements, but no dramatic change in tone, organization, content, etc. is warranted. Mgnbar (talk) 16:04, 22 September 2013 (UTC)

Vector space sentence correct?
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)


 * 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)