Talk:Projective plane

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

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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)

Contents 1 Symmetry group of Fano plane 2 affine planes 3 latin squares 4 axiomatic affine space/steiner system 5 Missing definition of ordinary geometric projective planes??? 6 Missing definition of affine plane

[edit] 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 [1] of online finite geometry notes at Ghent University. Cullinane 13:29, 9 August 2005 (UTC)

[edit] 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?"

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)