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Talk:Real projective plane

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Continuing the 2013 discussion Talk:Real projective plane/Archive 1 § "Topology is not concerned with flatness, and the real projective plane.."

Then the elliptic plane really should be mentioned, ie. the geometric projective plane as opposed to the topological projective plane. Maybe this distinction should even be written near the top of the article. --2607:FEA8:86DC:B0C0:561:D589:AD84:9BC7 (talk) 15:25, 19 April 2022 (UTC)[reply]

In Cross cap disk You realized six images, maybe drawn in TikZ. For the Figure 1 and 3 You wrote the surface equation, but for Figure 2 there is not a surface equation. Could You write it, please? Thank you so much. Best regards at all. 94.38.53.130 (talk) 14:44, 6 June 2022 (UTC)[reply]

Continuing the 2007 discussion Talk:Real projective plane/Archive 1 § Homogeneous coordinates

A line l in the plane corresponds to some P plane in R^3 (namely span of the union of l and the origin). Though there might be a formal difference (line in the real projective space is a set of points of the real projective space, i.e, a set of the lines through the origin --- whose unionisP), this is their point 178.255.168.233 (talk) 19:31, 31 March 2024 (UTC)[reply]

@178.255.168.233, I am finding your changes today special:diff/1188534080/1216566057 to be pretty confusing; I can't tell what exactly you are trying to say there. Absent a rewrite for clarity, I think they should be reverted. –jacobolus (t) 20:27, 31 March 2024 (UTC)[reply]

@Jacobolus, today I improved the clarity. Originally, I came to learn about the real projective plane in context where it was understood as an extension of the plane. Need to say, the linear structure (of both) is involved. Reading about the Moebius strip right in the introduction did not help me (indeed, on that day, I could not be satisfied by the topological properties of the real projective plane). Actually, the introduction names some of the properties, mostly topological; it does say a word on what the real projective plane actually is (i.e., "space of lines ..."), but not enough for visitors like me. Therefore I added a few words on the relation of the real projective plane and the (ordinary) plane. Today I corrected some typos in improved readability, which I hope can satisfy you. 178.255.168.233 (talk) 20:02, 6 April 2024 (UTC)[reply]

What something "is" is a bit of a philosophical question. There are a number of isomorphic structures any of which can be called the "real projective plane".

Personally I think it is most natural to imagine the real projective plane as the space of orientations of lines in 3-dimensional space. But other interpretations are no more or less inherently valid.

I agree with you that "an example of a compact non-orientable two-dimensional manifold;" is not an adequate definition. –jacobolus (t) 20:51, 6 April 2024 (UTC)[reply]