Talk:Klein quartic

message: :Title added. —Nils von Barth (nbarth) (talk) 00:52, 17 April 2010 (UTC)
replies: text: Certainly the claim about seven regular heptagons is incorrect as stated, since the affine curve is not compact. Katzmik 12:26, 1 August 2007 (UTC)
replies: text: You're right, there's confusion here. It seems to me that: the article focuses on the affine curve; my note about the congruence subgroup correctly applies to the affine curve; the claim about covering by regular hexagons applies to the compact curve. Is this correct? Tesseran 02:33, 2 August 2007 (UTC)

text: A reasonably complete discussion of an affine curve would include also the number of punctures of the complete hyperbolic Riemann surface of finite area.  Also, I am not sure about the uniqueness assertion in this context.  At any rate, certainly the most interesting case is the compact surface.  All the pictures on J. Baez's site are certainly of the compact surface.  I don't know what the focus is of the current version of the article.  I find it rather poorly written.  Eventually I would like to rewrite it when I find time, unless somebody else gets to it first.  As far as the arithmetic background for the compact surface, it is described in detail by Elkies in his contribution to the "Eightfold" volume (with one small error that he corrected in another paper of his). Katzmik 07:23, 2 August 2007 (UTC)
thread_title:  Affine version 

message: The affine curve can indeed be represented by the congruence mod 7 in the modular group.  This information can also be found in Elkies' papers. Katzmik 07:24, 8 August 2007 (UTC)
thread_title: Affine case

message: Is it possible to use Klein's original illustration (a tessellated regular hyperbolic 14-gon with instructions for gluing) or the redrawing found in the MSRI volume? I don't know whether either of these would violate copyright. It would also be nice to show Helaman Ferguson's sculpture "The Eightfold Way." Ishboyfay (talk) 17:25, 20 September 2008 (UTC)
replies: text: I've substituted a high-res scan of the cover of "The Eightfold Way" (the book) for the low-res Amazon file that was there.  I've checked with the publisher that it's OK.  Regarding Klein's illustration, it's not under copyright anymore, so anyone can make a scan.  I can also provide a Postscript file of the figure in the book, which is my redrawing of Klein's illustration.  levy@msri.org Silvio Levy (talk) 23:31, 13 June 2011 (UTC)
thread_title: Illustrations

message: The article states:
"The Klein quartic can be viewed as an algebraic curve over the complex numbers C, defined by the following quartic equation in homogeneous coordinates:  x^3 y  +  y^3 z  +  z^3 x  =  0."
This is *misleading*, because the locus of that equation in the complex projective plane is the definition of the Klein quartic, not merely a way that it "can be viewed".
Later in the article, the Klein quartic is described as being tilable by 24 congruent regular hyperbolic heptagons.  This statement is false, because the geometry of the Klein quartic does not have constant negative curvature.
Rather, what is tilable with 24 congruent regular hyperbolic heptagons is a surface that is *conformally equivalent* to the Klein quartic.
This is an important distinction, since it is a common mathematical error to confuse the two concepts.Daqu (talk) 04:59, 17 November 2009 (UTC)
thread_title:  The Klein quartic has just one valid definition 

message: The text uses a definite article to describe an affine version of the compact quartic.  Is an affine curve unique in any sense?  Tkuvho (talk) 07:47, 7 June 2010 (UTC)
thread_title:  THE affine quartic? 

message: I'm also uncomfortable with the terminology affine quartic. If you start from the usual projective equation x3y+ y3z+ z3x of the quartic and try to restrict it to an affine chart of P2C (example: z=1) then you will remove no more than 4 points. However H2/Gamma(7) has 24 punctures. Or maybe affine means that there is a (natural) local chart system on the punctured quartic for which the coordinate changes belong to the group group of affine transformations az+b of C? This should be investigated. Citations needed? Arnaud Chéritat (talk) 20:41, 25 February 2014 (UTC)
thread_title:  The ''affine'' quartic? 

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