71 The Collection of Mathematical Models student of Monge. With the help of taut threads, the models represent ruled surfaces, that is: curved surfaces created by the movement of straight lines. From 1839, Olivier initially had these surfaces – of utmost interest to civil engineering – made for the Conservatoire des arts et métiers in Paris. Monge, too, had previously depicted ruled surfaces by means of taut threads. However, the special feature of Olivier’s constructions is that the models are movable, which means they represent more accurately the spectrum of possible cases – for example, intersection phenomena. Characteristic of Olivier’s creations are brass frames with mechanically complex joints and, in some models, lead weights for automatically retightening the threads (Cf. the collection at Union College). In today’s collection, there is only one model left – a hyperbolic paraboloid – that due to its construction and materiality can be traced back to the delivery of those first thirteen models. The heyday of mathematical models The heyday of mathematical models in the second half of the 19th century is closely linked to research in the field of algebraic surfaces (Fischer 1986). One of the first mathematicians to be mentioned here is Ernst Eduard Kummer. At the mathematical seminar of the University of Berlin between 1862 and 1872, he constructed nine models of fourth order surfaces, including Steiner’s Roman surface. The models were still praised twenty years later for being “among the most beautiful and elegant that have been produced up to now” (Schilling 1911, p. 20). The most influential protagonists in the rise of mathematical models, however, were Felix Klein and Alexander von Brill. Both were students of Kummer and were appointed to the Technical College in Munich in 1875. Their teaching, which was entirely focused on “Anschauung”, led them to set up a modeling cabinet. In the following years, with the support of a lathe operator and a plaster molder, and using arithmetic and drawing extensively, their students constructed around 100 mathematical models. However, this was not just indulging the “luxury” (Brill 1889) of having teaching models in engineering training; the models were intended to promote specialized studies in geometry. The mathematical models, inasmuch as they represented surfaces of revolution or generalized helicoids, were initially made of wood. For the other models, planar cuts were made into zinc sheets and then soldered together to form a framework. The final form was created in several steps by filling the zinc framework with a special molding mass and by subsequent smoothing. Finally, plaster casts were made from the master models and lines relevant to the design were applied to them. Only such plaster casts were intended for distribution. In this respect, it is remarkable that some wooden models still exist in Dresden. These are very The study of the shape relations, including of structures that are otherwise well known to the geodesist prompts new and often momentous questions. Alexander von Brill 1889 Steiner’s Roman surface with main tangent curves Plaster model of a fourth order surface after Kummer, Martin Schilling publishing house Series IX 3, 1883
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