Talk:Real projective plane

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Contents 1 Informal construction 2 Homogeneous coordinates 3 Embedding in R^3 4 Formal Construction 5 Möbius strip?!

[edit] Informal construction

Can someone clean this section up and perhaps give it a more thorough treatment? For example, what is a circle of "length" 2 * pi? I'm assuming by length, they mean circumference. Vecter (talk) 20:06, 3 January 2008 (UTC)

it's not clear at all what point that section is trying to make. i am going to remove it. Mct mht (talk) 17:48, 4 January 2008 (UTC)

[edit] Homogeneous coordinates

Here:

The set of lines in the plane can be represented using homogeneous coordinates. A line ax+by+c=0 can be represented as (a:b:c).

whyare we speaking about "lines in the plane"? What has it to do with the projective plane?--Pokipsy76 16:06, 21 July 2006 (UTC)

It's probably in there because the real projective plane can also be thought of as the collection of all one-dimensional subspaces of R^(n+1) —Preceding unsigned comment added by 68.184.213.58 (talk) 06:43, 30 November 2007 (UTC)

[edit] Embedding in R^3

The Generalized Jordan Curve Theorem only applies to (co-dimension 1) images of spheres, not to real projective spaces or quotients of spheres, if you like. Is the idea that an embedding of RP^2 in R^3 can be pulled back to a map from S^2 to R^3? I'm unclear about this. 169.233.53.103 16:36, 3 April 2007 (UTC)

The generalized jordan curve theorem says any compact connected boundaryless n-dimensional manifold, when embedded in R^{n+1} seperates it into two components, one bounded, the other unbounded. In fact, you can replace R^{n+1} with any simply-connected manifold. The proof is essentially homological and appears in most intro alg-top texts. I like the proof in the Guillemin and Pollack differential topology textbook. Here is a rough sketch, assuming everything is smooth: let M be the submanifold of R^{n+1}. First of all, M is closed proper subset of R^{n+1}. Let * be a point of R^{n+1}-M. Let x be another point of R^{n+1}-M. Let p be a path, starting at * and ending at x. Let #(p) be the number of times p intersects M, mod 2. This is well-defined for a generic p by transversality/sards theorem, and it does not depend on the choice of p because R^{n+1} is simply connected -- given a 1-parameter family of functions p_t, you can assume it intersects M transversely and then you notice that the preimage of M under that 1-parameter family is a 1-dimensional submanifold in the parameter domain. FYI, I added this version of the GJCT to the Wikipedia page for JCT. Rybu 16:02, 8 July 2007 (UTC)

Oh, I forgot to add one thing: now conzider this Z/2-valued invariant as a function of the point x. Then it is locally constant, takes value 0 near *, and also takes the value 1 at some points (consider it in a neighbourhood of a point of M). That's the proof. Rybu 16:02, 8 July 2007 (UTC)

[edit] Formal Construction

"Consider a sphere, and let the great circles of the sphere be "lines", and let pairs of antipodal points be "points". [...] This is the real projective plane." The omitted portion is not relevant, other than as an argument for the validity of the first statement. As someone not familiar with topology, but who understands what both great circles and antipodal points are, I have no idea how to use great circles as "lines", or how this makes a plane. --68.76.222.122 03:21, 29 April 2007 (UTC)

[edit] Möbius strip?!

I don't get how this is a Möbius strip. The Möbius strip is only attached in one direction. It's an OPEN surface. The Klein bottle is connected on both direction, like the square and arrow diagram shows! Tom Ruen 05:36, 21 July 2007 (UTC)

Okay, I see a Klein bottle is only inverted on one axis and closed in the other, just like the Möbius strip is inverted in one direction but OPEN in the other. So neither exactly represents the projective plane. I guess the Möbius strip is a better representation giving half the connection and imagining the other applied to the open edge. Tom Ruen 18:01, 22 July 2007 (UTC)

(edit conflict) What that paragraph is trying to say is that if you glue the edge of the Möbius strip to itself, thereby closing it, in the right way, you get a projective plane — or at least you would, if you could do that without having to worry about those pesky self-intersections you get in 3D. I've tried to edit the paragraph to make the intended meaning clearer. Any further improvements would be most welcome. —Ilmari Karonen (talk) 18:10, 22 July 2007 (UTC)

Thanks. I got it, just read too quick sometimes. Better wording now - no perfect tricks to make dumb people like me read slower. I'm just wondering about the "fundamental polygon" image. I see ANY even-sided regular polygon can be fundamental, with opposite sides corresponding, or in the limit a circle with center-point reflection defining the correspondence. Unit circle [r,θ]: [1,θ]=[1,θ+180]. Maybe I just like pictures, but I think that would be helpful to see as well. Tom Ruen 18:39, 22 July 2007 (UTC)

Yes, certainly. I originally drew those squares as a series for the fundamental polygon article; as it happens, only one of them is currently used there, and another one seems to have been replaced with a slightly different version. I made them all squares since a square is sufficient to construct both the sphere, the projective plane, the torus and the Klein bottle, and I wanted the constructions to be as directly comparable as possible.

As it happens, the minimal fundamental polygon for the projective plane, in an algebraic sense, is two-sided — but of course a meaningful drawing of such a degenerate polygon is impossible without at least cheating a bit. There is another series of fundamental polygons on Commons that does show two-side polygons — with curved sides — for the sphere and the projective plane.

I do agree that nicer illustrations could probably be drawn for this article. In fact, you've given me some ideas above: perhaps a circular disc with two arrows going around it, or maybe (also?) even a real Möbius strip with similar arrows. In the mean time, if you can think of better illustrations, please do draw some or, if you don't wish to draw them yourself, suggest them here. You could also consider asking at Wikipedia:WikiProject Mathematics/Graphics; it's not a very busy place, but there are some excellent illustrators hanging around there. —Ilmari Karonen (talk) 19:44, 22 July 2007 (UTC)

I can produce images of all four immersions of the real projective plane in R^3 with a single triple point, if that would be useful. I'll post them here to see if they can be worked into the article. Actually, there's not a mention of the fact that there are exactly four ways to immerse it in R^3 (again with a single triple point). Maybe I'll add a section if anyone else thinks it deserves a mention. Apery has a proof, but I pity anyone who actually has to read it. -- A13ean 21:17, 22 July 2007 (UTC)

I suggest moving the 2nd paragraph on the Moebius strip based construction into the "informal construction" section. Perhaps what's in there right now might be combined in. An illustration might help too -- maybe the Sudanese surface with a disk being sewn on to the last edge? Seems like it might be hard to see with the disk being attached to the normal Moebius strip. -- A13ean 21:30, 22 July 2007 (UTC)