Talk:Dessin d'enfant

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Mathematics rating:	B Class	Low Priority	Field: Geometry
Please update this rating as the article progresses, or if the rating is inaccurate. Click to show/hide comments. Please add to or update the comments to suggest improvements to the article. Geometry guy 21:20, 3 June 2007 (UTC)

[edit] Thanks

Thanks for clarifying the connection between the riemann sphere and Dessin d'enfant; I mean the new paragraph you put in this article today.

You should make yourself an account if you want to do serious editing. Oleg Alexandrov 21:42, 10 Mar 2005 (UTC)

[edit] Glueing to create a Riemann surface

Let me start by saying "Great article! Good job!" before I offer some criticisms.

I know (or thought I knew) how to glue triangles to create Riemann surfaces, but somehow, the description in this article threw me for a loop. I'd like to see that clarified. Please note that the article on Riemann surfaces does not even hint at such a construction. In particular, I'm not sure why one wants to glue together half-spaces, instead of triangles. Yes, the j-invariant tells you how to get from triangles to half-spaces, and v.v. but, to me, "visualizing" glued triangles is easier than visualizing glued half-spaces. I feel like I'm missing some step. The only handy-dandy glueing article I know of on WP is fundamental polygon, which describes glueing to create compact riemann surfaces, as cribbed from a book by Jost of the same name. Compact vs. non-compact tends to be a major distinction; this article doesn't hint how that might enter. (And the WP entries on glueing to form non-compact surfaces are lacking.)

I do have to admit I only skimmed this article; perhaps I lack sufficient background, but it seems I will have to go to other sources to understand the mechanics. linas (talk) 14:18, 26 November 2007 (UTC)

Firstly, the halfplanes being glued are compact, because they include ∞. In fact, they are triangles: triangles having 0, 1, and ∞ as their three vertices. Very big triangles, but still nice compact convex closed triangles. The reason to glue them rather than some smaller triangles is because we're not just creating a Riemann surface, but we also need to simultaneously create a Belyi function from the created surface to the Riemann sphere, and with this particular gluing construction the function is easy to describe (identity within each half-plane). —David Eppstein (talk) 16:16, 26 November 2007 (UTC)