# Material models of Riemann surfaces

It is known that during the last quarter of the 19th century there was a flourishing of the production of material models (from plaster, strings, card-board etc) of curves and surfaces in Germany (but obviously not only there, but also before that in France); see example the collection in Göttingen (here). My question concerns the models of Riemann surfaces. Obviously Riemann defined these surfaces and the notion of the branch point already in 1851 in his thesis: "Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse", and I assume the notion of a multi-valued function became therefore clearer mathematically.

My question is the following: were these models considered, along them being an additional way to visualize the branch points and how the different sheets 'change their position', necessary and essential for the mathematical understanding of how the Riemann surfaces 'look like' / behave? Or was this (this = what happens in the neighborhood of the branch point) already clear mathematically and the models were just a secondary resp. additional way to help the teaching of these surfaces?

I also hardly found any literature specifically about these models and their connection to concetualization of Riemann surfaces, so any help would be great. Thanks!

Kharkiv University (Ukraine) subscribed to all models made M. Schilling, who probably was a student of Klein, and who run a company making and selling these models. Currently they photograph them and place on internet: http://touch-geometry.karazin.ua/list

This page also has Schilling's catalog of his production. Recently I learned that they have some Riemann surfaces made of cardboard. The corresponding items are described in M. Shilling's catalog. But they are not sure how to photograph them properly. This summer I will visit Kharkiv and hope that I will be able to help them. According to the description, one set of cardboard models is based on the PhD thesis of W. Ihlenburg, Klein's student who was apparently killed in WWI. I have Ihlenburg's thesis posted on my web page, including the pictures: http://www.math.purdue.edu/~eremenko/dvi/ihlenburg3p.pdf Presumably they have in Kharkiv these Riemann surfaces made of cardboard. (Schilling's catalog mentions them).

Remark. In 1970s, in Lviv University (Ukraine), prof. A. A. Goldberg taught a course of Riemann surfaces where students were assigned to make these models as a part of their HW. I've seen many of them, but I don't think the models from this course are preserved.

EDIT. Specifically, the Riemann surface models have numbers 303-310 p. 159-180 in Schilling's catalog.

EDIT2. Here are some photos made in Kharkiv on June 22, 2017: http://www.math.purdue.edu/~eremenko/Ihlenb/

• Thank you for the answer! Do you happen to know how these models were regarded in Germany, at the time of their production? Commented May 26, 2017 at 8:02
• I do not quite understand what does it mean "how they were regarded" but some universities bought them, and in few places they are still preserved. They say, in Gottingen, for example. I suppose Klein was a big enthusiast of them, but this is only my guess based on Klein's surviving writings. Commented May 26, 2017 at 11:04
• What i meant was the following: did these (material) models had a real contribution for the mathematical understanding, how Riemann surfaces "behaved" / looked like around a branch point, or was behavior already well understood when Riemann (and his followers) presented them? Commented May 26, 2017 at 12:19
• This is difficult to answer, who understood what. I can only tell that the course of Goldberg mentioned in my ans did help my understanding. Commented May 26, 2017 at 14:14
• At the university of Leipzig a collection of mathematical models of Felix Klein exists and is currently renovated. archiv.uni-leipzig.de/blog/… universitaetssammlungen.de/sammlung/543 Commented May 30, 2017 at 18:07