Talk:Schwarz triangle function

Suggested splits

A lot of this material might be better off on other related pages. There's a section "Hyperboloid and Klein models" that covers the same material as Hyperboloid model and Beltrami–Klein model. Much of "Tessellation by Schwarz triangles," which is way too long, could be pulled into Schwarz triangle or elsewhere.

We also need to deal with the fact that the editor who started this article never actually got to the meat of the article: the section "Conformal mapping of Schwarz triangles" is one sentence long! -Apocheir (talk) 22:36, 7 October 2021 (UTC)

In February 2017 the lead originally said:

In mathematics, the Schwarz triangle function was introduced by H. A. Schwarz as the inverse function of the conformal mapping uniformizing a Schwarz triangle, i.e. a geodesic triangle in the upper half plane with angles which are either 0 or of the form π over a positive integer greater than one. Applying successive hyperbolic reflections in its sides, such a triangle generates a tessellation of the upper half plane (or the unit disk after composition with the Cayley transform). The conformal mapping of the upper half plane onto the interior of the geodesic triangle generalizes the Schwarz–Christoffel transformation. Through the theory of the Schwarzian derivative, it can be expressed as the quotient of two solutions of a hypergeometric differential equation with real coefficients and singular points at 0, 1 and ∞. By the Schwarz reflection principle, the discrete group generated by hyperbolic reflections in the sides of the triangle induces an action on the two dimensional space of solutions. On the orientation-preserving normal subgroup, this two dimensional representation corresponds to the monodromy of the ordinary differential equation and induces a group of Möbius transformations on quotients of solutions. Since the triangle function is the inverse function of such a quotient, it is therefore an automorphic function for this discrete group of Möbius transformations. This is a special case of a general method of Henri Poincaré that associates automorphic forms with ordinary differential equations with regular singular points. In the special case of ideal triangles, where all the angles are zero, the tessellation corresponds to the Farey tessellation and the triangle function yields the modular lambda function.

Later that was modified when another editor inserted content about the hypergeometric ODE tight at the beginning of the lead. As a result crucial conditions on the Schwarz triangles were omitted (that the angles should have the form 0 or π over a positive integer) and the geometric connection with tessellations was lost. It is probably a good idea to merge the first version of the lead with the additional content on the hypergeometric function. The densely written paragraph of my original lead can certainly be written in a way that makes it far more approachable to readers.

Clearly the final section of the article was incomplete. Unfortunately on wikipedia that is often what happens. On the other hand, the theory of the hypergeometric ODE is easy to summarise briefly with exact page references for the books of Caratheodory, Hille and Nehari (already listed in the article). The case of ideal triangles can also be briefly summarised from the 2nd edition of Ahlfors book on Complex Analysis, where the modular lambda function appears. In this way missing material and page numbers can be added, matching up content on the uniformization problem (conformal mapping of Schwarz triangles) and the hypergeometric ODE. Mathsci (talk) 00:43, 8 October 2021 (UTC)

I strongly agree with Apocheir's suggestions for reworking. Gumshoe2 (talk) 18:56, 25 February 2022 (UTC)

• Unlikely to happen when new material is being added by me at the moment. That takes time, care and patience. Mathsci (talk) 19:50, 25 February 2022 (UTC)

Please be aware you are misusing the "in use" tag, which as per [1] is meant "for a short period of time, no greater than a few hours at a time". You've had it up for around 11 hours now.[2] Gumshoe2 (talk) 20:26, 25 February 2022 (UTC)