Talk:Möbius plane
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new version of the article on Möbius plane[edit]
The current version gives not the right impression and information about Möbius planes. At least one essential mistake is contained: there exists no complex Möbius plane M(C) over the field of complex numbers, because the field of complex Numbers is algebraically closed. The important examples of miquelian Möbius planes are not mentioned and introduced. If a reader wants to get just an idea of a Möbius plane he should read the article on Benz planes.
I suggest to replace the current article by the version in my sandbox. Ag2gaeh (talk) 09:07, 26 March 2013 (UTC)
- Could you provide the link to the article in your sandbox so that other editors could have a look at it?--Kmhkmh (talk) 12:42, 26 March 2013 (UTC)
- The link to my sandbox: http://wiki.alquds.edu/?query=User:Ag2gaeh/sandbox Ag2gaeh (talk) 10:17, 27 March 2013 (UTC)
- Since the copyright issue has been resolved, I think replacing the current article by your sandbox version is a good idea. Personally however I'd would suggest a few modifications, in particular I like the lead of the current article somewhat better than the lead in your sandbox. Simply because the lead (ideally) should already provide an intuitive notion and/or summary of the most important aspects. That is the intuitive/"anschauliche" notion of the classic Möbius plane and the role of incidence relations should appear in the lead.--Kmhkmh (talk) 14:05, 27 March 2013 (UTC)
- The link to my sandbox: http://wiki.alquds.edu/?query=User:Ag2gaeh/sandbox Ag2gaeh (talk) 10:17, 27 March 2013 (UTC)
Nonmathematical phrases?[edit]
In Möbius plane § Classical real Möbius plane contains the adjectival phrases "rather homogeneous" and "highly transitive". Does the first word in either add any mathematical meaning? —Quondum 00:56, 2 August 2015 (UTC)
Equivalence with the complex projective line[edit]
This article does not go as far as to say that the Möbius plane and the complex projective line are the same thing. As far as I can tell, they are the same thing (correct me if I'm wrong): it seems to me that the structures are isomorphic. Should the article not be clearer on this? A more compelling equivalence (for illustrative purposes) is the night sky. This is possibly of more interest to physicists, but may be worth a mention: the transformations of the positions of the stars under boosts and rotations of an observer are those of the complex projective line (or Riemann sphere, according to Roger Penrose in The Road to Reality). Would mention of this association with the real world not make it more understandable? —Quondum
- The real Möbius plane and the complex projective line are different incidence structures. The Möbius plane deals with points and circles, the projective line consits of one line and its points. The common thing are the descriptions of their point sets and their permutations: the Möbius transformations and the inversions .--Ag2gaeh (talk) 07:32, 2 August 2015 (UTC)
- I can see that what I am saying is getting confused by the labelling of incidences, so let's omit P1(C). Let's stay with the celestial sphere – it fits the description of this article with cycles as plane sections of a sphere: cycles are preserved under the transformations. It would therefore be reasonable to say that the geometry of the celestial sphere is the Möbius plane, unless I've got something very wrong; you do not seem to have contradicted me on this point. Mention of a real-world example of the Möbius plane seems to me to be appropriate. —Quondum 13:33, 2 August 2015 (UTC)
- I think the idea of the celestial sphere is to describe far distant objects by spherical coordinates. Plane sections (circles) do not play a special role. The objects on the sky usually move not on circles. But the hint to the article Stereographic projection is given. The more important property for practical use of the stereographic projection is that it conserves angles.--Ag2gaeh (talk) 07:01, 3 August 2015 (UTC)
- I think you misunderstand me. The points in the celestial sphere are the directions from which light rays arrive. I don't mean the positions of stars, or how they move in time, but rather how the apparent direction of arrival of a light ray changes as I, as an observer, rotate and change my relative velocity. If I rotate (as the Earth does), the apparent direction of every light ray rotates, preserving apparent angles on the celestial sphere. If I accelerate in one direction (as the Earth does in its orbit), the apparent direction of arrival of light rays crowds forwards, but in such a way that apparent angles (what you are referring to as angles) are preserved. Circles on the sphere (plane sections) are preserved by these transformations. In short, the points of this sphere transform conformally: these are exactly the transformations of the Möbius plane, with the exception of the reflections. This is a result of the theory of special relativity. —Quondum 13:19, 3 August 2015 (UTC)
- I think the idea of the celestial sphere is to describe far distant objects by spherical coordinates. Plane sections (circles) do not play a special role. The objects on the sky usually move not on circles. But the hint to the article Stereographic projection is given. The more important property for practical use of the stereographic projection is that it conserves angles.--Ag2gaeh (talk) 07:01, 3 August 2015 (UTC)
- I can see that what I am saying is getting confused by the labelling of incidences, so let's omit P1(C). Let's stay with the celestial sphere – it fits the description of this article with cycles as plane sections of a sphere: cycles are preserved under the transformations. It would therefore be reasonable to say that the geometry of the celestial sphere is the Möbius plane, unless I've got something very wrong; you do not seem to have contradicted me on this point. Mention of a real-world example of the Möbius plane seems to me to be appropriate. —Quondum 13:33, 2 August 2015 (UTC)