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!