Talk:Riemann sphere

message: Take a look at the short paragraph below:
(Another reader notes that afaik this article has the stereographic projection wrong - the sphere and the plane should not have the same origin. The sphere rests on the plane and the sphere point is defined as the 'other' intersection of the line with the sphere. 62.49.9.94
:Stereographic projection can be performed onto many different planes. Choosing the plane to have the same origin as the sphere is particularly convenient for the Riemann sphere, because it makes the unit circles line up, etc. Joshua R. Davis 15:47, 3 September 2007 (UTC)
thread_title: This article needs lots of work

message: Elroch made a lot of good edits to this article. It is now more mathematically correct, but it is hard to understand for somebody not knowing math however. I believe the geometric viewpoint, which, if not entirely accurate, was helpful in illustrating what is going on. Wonder what you think. Oleg Alexandrov (talk) 15:09, 17 April 2006 (UTC)
replies: text: Thank you for your kind comment, Oleg. However, I now realise I have been unnecessarily narrow in my viewpoint. The thing that has shifted my viewpoint is finally realising (as Hipparchus did a little earlier :-) ) that the stereographic projection, although of course not preserving distances, is conformal. This gives the geometrical viewpoint a fully justified role (with all non-trivial holomorphic maps, including Möbius transformations) being conformal on the geometrical sphere). I wish this had sunk in before I made my second "improvement" to the introduction. Oh well, at least this isn't on paper. Of course this justifies increasing the prominence of the geometrical viewpoint again, which should help to address your suggestion. Elroch 19:11, 17 April 2006 (UTC)

text: I have now returned the geometrical viewpoint to the introduction, and added a little more detail. Hopefully this will mean the net change has been positive, and anything that is still less than ideal will be flagged or fixed by those more perceptive than me. Elroch 19:48, 17 April 2006 (UTC)
replies: text:  Thanks, your edits improved the article. Oleg Alexandrov (talk) 03:51, 18 April 2006 (UTC)
thread_title: Recent changes

message: This article (like the Stereographic projection article) prefers the projection onto a plane tangent at a pole over the projection onto a plane through the equator. It even asserts that the former is more popular. This contradicts my experience. Among the texts in front of me, the equatorial version is used books by Rudin, Bredon, Thorpe, Oprea, and Brown/Churchill; the pole-tangent version is used in Do Carmo; Spivak uses both versions.
More importantly, this article contradicts itself. Its claims about the transition maps being z = 1 / w, w = i corresponding to z = -i, etc. are all assuming the equatorial version. (In the equatorial version, the unit circle in \mathbb{C} is sent to the equator; in the pole-tangent version, the circle of radius 2 is sent there, so the unit circles from the two complex planes don't match up.)
In short, while the pole-tangent version of stereographic projection has uses in differential geometry, the equatorial version seems unequivocally better suited to complex analysis, which is where this article comes in. So unless there are objections I'm going to start making these changes. Joshua R. Davis 19:55, 25 April 2007 (UTC)
replies: text: As with many articles I suspect the contradictions have crept in from various editors using different conventions and not being careful as to whether their edits were consistent or not. I agree that the equatorial version is much better suited to this article (despite whatever assertions I may have made earlier on this page) -- Fropuff 03:25, 26 April 2007 (UTC)

text: Actually, now that I look back at my earliest version of this page, I realize that the mistake lies partially with me. The equations I used were for an equatorial projection but I asserted that they were onto a polar-tangent plane. Clearly, I was confused. All due apologies for accusations of edit creep and editor sloppiness. I'll just go and flog myself. -- Fropuff 03:55, 26 April 2007 (UTC)
thread_title:  Stereo proj: equatorial vs. polar-tangent 

message: I appreciate that mathematical experts may be writing this article from their own minds, without referring to source material.  I've done the same in articles where I have expertise.  This is Wikipedia though, and at least a few citations are needed.  At a minimum, cite a few books or papers that describe a Riemann sphere.  Davidwr 15:10, 3 May 2007 (UTC)
replies: text: Yes, this is a serious problem with the article. I am finishing up some revisions that do cite references. Perhaps they will appear later today. Joshua R. Davis 16:03, 3 May 2007 (UTC)
thread_title:  Need citations 

message: I cannot deduce from the formulaes in the article how to transform a point P=x+i·y of the Gaussian plane onto the x,y,z Cartesian coordinates (or longitude,latitude spherical coordinateS) of the Riemann sphere. :-/ Is it that complicated? --RokerHRO (talk) 09:50, 18 May 2010 (UTC)
replies: text: The section "As a sphere" contains the formula ζ = (x + i y) / (1 - z). So if we write ζ = X + i Y (like your P = x + i y), we have X = x / (1 - z) and Y = y / (1 - z). This is the map from the sphere to the plane; you just have to invert it to get the map from the plane to the sphere. There are more details in the Stereographic projection article. Mgnbar (talk) 19:32, 18 May 2010 (UTC)
thread_title:  Where is an explicit projection formula? 

message: ∞ + ∞ = ∞ is stated at the section Arithmetic operations. I think this operation must be moved to the undefined operations (together with ∞ - ∞ and 0·∞). You can find a counterexample at page 30 here. I'm not an expert, so I'm afraid I may not be correct and I didn't change it. So please, if someone agrees, change it.--Ssola (talk) 17:57, 7 March 2012 (UTC)
replies: text: I'll change it (as I have no annswers).--Ssola (talk) 14:33, 19 March 2012 (UTC)
replies: text:  For the title of this section: there is no "in general" situation for addition with ∞, because ∞ is not a real number, so "in general" is a bit confusing.  You are correct that ∞ + ∞ is undefined on the extended complex plane, however (as your source clearly indicates). TricksterWolf (talk) 03:18, 6 February 2014 (UTC)
thread_title:  ∞ + ∞ = ∞ is false (in general) 

message: This article is to dispell the obvious mistake made in the Riemann sphere definition which states that "∞" infinity is near to very large numbers and that zero "0" is near to very small numbers. This is false for two reasons:
1) nothing and infinity are unlimited concepts and are therefore by their very nature completely incompatible mathematically and conceptually with limited numbers.
2) The concept that zero is close to small numbers and infinity is close to large numbers is based on the assumants inability to conceive of something infinite outside of space and time resulting in the assumants mind counting up or down forever caught in a limited process that given enough time would never reach its infinite goal for the simple fact that it is always limited.
Conclusion this theory is logically refuted as false.  — Preceding unsigned comment added by 193.200.145.253 (talk) 10:52, 23 July 2013 (UTC)
replies: text: When mathematicians use the Riemann sphere, they do not make broad claims about the philosophical nature of nothingness and infinity. Rather, they are talking about a specific mathematical space, with very precisely defined properties. Nothing about the Riemann sphere is controversial in mainstream mathematics. On the other hand, if you can find reliable sources to support your argument, then perhaps we can incorporate that material into the article. Mgnbar (talk) 18:06, 23 July 2013 (UTC)
thread_title:  Riemann Sphere disproven 

message: Why does this article say infinity plus infinity is undefined?? It has always made sense to define it as infinity. Infinity plus negative infinity is undefined though. Georgia guy (talk) 22:00, 16 January 2014 (UTC)
replies: text: Really more of a question for the refdesk, but I suppose it might come up again here, so OK.  The ∞ of the Riemann sphere is unsigned (negative infinity equals infinity).  Look at the figure of the sphere near the top of the article — you can approach ∞ from any direction. --Trovatore (talk) 23:02, 16 January 2014 (UTC)
replies: text: Ssola provided this counterexample already.  See chapter 3 to learn more about why these particular definitions are chosen for this model.  (You must realize math is a tool for selecting certain parameters and seeing what comes up; not for discovering the One True Algebra.  There are lots of One True Algebras.  They're like reproductions of the One Ring: cheap and ubiquitous.)  TricksterWolf (talk) 03:38, 6 February 2014 (UTC)
thread_title:  Infinity plus infinity 

message: It is claimed in the Rational functions section that "the set of complex rational functions ... form all possible holomorphic functions from the Riemann sphere to itself".  As far as I know, e^x is not a rational function, yet it is holomorphic.  So this claim appears to be false.  The only way out I can see is if we consider e^x to be a "polynomial" on the grounds that it has a power series, but this seems to be a stretch. Am I mistaken?  Luqui (talk) 05:19, 8 July 2018 (UTC)
replies: text: Hi. I'm not an expert, but I think that e^x is not a counterexample. To make it a counterexample, you must extend e^x from the complex plane to the Riemann sphere, by defining its value at \infty. Let y = 1 / x, so your function is e^{1 / y}. Now try to define the value of the function at y = 0. I think that you will run into problems. Mgnbar (talk) 11:26, 8 July 2018 (UTC)

text: Another tack: You're going to have a problem defining e^\infty to make e^x continuous (let alone holomorphic). Because in the real numbers \lim_{x \to -\infty} e^x = 0 and \lim_{x \to \infty} e^x = \infty, but in the Riemann sphere -\infty and \infty show up as the same point. Mgnbar (talk) 13:31, 8 July 2018 (UTC)
thread_title:  All holomorphic functions are rational?