A plurality of (non)visualizations: Branch points and branch curves at the turn of the 19th century

This article deals with the different ways branch points and branch curves were visualized at the turn of the 19th century. On the one hand, for branch points of complex curves one finds an abundance of visualization techniques employed. German mathematicians such as Felix Klein or Walther von Dyck were the main promoters of these numerous forms of visualization, which appeared either as two-dimensional illustrations or three-dimensional material models. This plurality of visualization techniques, however, also resulted in inadequate images that aimed to show the varied ways branch points could possibly be represented. For branch (and ramification) curves of complex surfaces, on the other hand, there were hardly any representations. When the Italian school of algebraic geometry studied branch curves systematically only partial illustrations can be seen, and branch curves were generally made “invisible”. The plurality of visualizations shifted into various forms of non-visualization. This can be seen in the different ways visualization techniques disappeared.