You should make yourself an account if you want to do serious editing. Oleg Alexandrov 21:42, 10 Mar 2005 (UTC)
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)
- I'm with linas on this one: (a) this is a good and useful article; (b) I also found the reference to "half-planes" confusing. After reading the discussion here, I think I've figured out what it means, but it's clear that "upper half-plane" as used on this page does not mean the same as upper half-plane. There must surely be a better way to describe this construction, although at the moment I can't think of a way to clarify the page without getting very verbose... Jowa fan (talk) 02:57, 28 December 2009 (UTC)
- A graph is an abstract structure of vertices and pairs of vertices (edges), devoid of any geometric or visual interpretation. A diagram is any visual representation of any kind of information. So neither of these is specific enough to describe what's going on here. A graph drawing is a visual representation of a graph, and that's closer. In this context, the visual part is less important than the fact that it's a graph embedded within a 2-manifold, so graph embedding might be more apropos. But we already have the graph embedding link in the next paragraph, and I think for the lede the phrase graph drawing conveys the flavor of the title (which also refers in French to a kind of drawing). —David Eppstein (talk) 22:27, 10 October 2010 (UTC)