University of Groningen Theory and history of geometric models Polo-Blanco, Irene IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite from it. Please check the document version below. Document Version Publisher's PDF, also known as Version of record Publication date: 2007 Link to publication in University of Groningen/UMCG research database Citation for published version (APA): Polo-Blanco, I. (2007). Theory and history of geometric models. [s.n.]. Copyright Other than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons). The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license. More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne- amendment. Take-down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons the number of authors shown on this cover page is limited to 10 maximum. Download date: 04-10-2021 Chapter 1 Models of surfaces, a Dutch perspective 1.1 Introduction The collection of more than 100 mathematical models in the University of Groningen gives rise to many questions. When were these models made? Who built them and how? From a local perspective, one may also ask whether the models were made in Groningen. If not, where did they come from? Such a big collection must have been rather expensive. Why did the University spend money on this? Who was responsible for buying the models? And finally, what was their use? Many of the models turn out to be sold by the German companies L. Brill and M. Schilling in the second half of the 19th century. Much information on the mathematics and the building of models in Germany at that time can be found in the literature ([F], [Dy] and [Sch1]). We shall next give a short introduction to the topic, and shall dedicate the remaining part of the chapter to the story of the mathematical models in The Netherlands, and more particularly, in Groningen. 1.2 Mathematical models in Germany In the second part of the 19th century, a wave of innovations occurred in western Europe and the United States. Germany together with the United States came ahead of the innovating countries, where science and engineering 1 2 CHAPTER 1. MODELS OF SURFACES, A DUTCH PERSPECTIVE was combined for the first time ([McN] and [Chr]). We find an example of this fruitful combination in the polytechnic schools in Germany. In particular Felix Klein and Alexander von Brill, professors of mathematics at the Poly- technic School of M¨unich, and Walter von Dyck among others, constructed mathematical models to make the visualization of geometrical objects eas- ier. The building of models became rather popular among mathematicians in Germany during the second half of the 19th century and the beginning of the 20th. 1.2.1 Brill and Schilling The great period of model building started in the 1870’s, when Ludwig Brill, brother of Alexander von Brill, began to r and sell copies of some mathe- matical models. A firm under the name L. Brill was founded in 1880 for the production of models. The company was established in Darmstadt and in 1899 it was taken over by Martin Schilling who renamed it. Under this new name, the firm was first located in Halle a. d. Saale, but later moved to Leipzig. Figure 1.1: Cover of Schilling’s catalogue [Sch1]. 1911. By 1904, Martin Schilling had already produced 23 series of models. In that year, the firm published a book [Sch2] with the mathematical explana- 1.2. MATHEMATICAL MODELS IN GERMANY 3 tions of the models existing at the time. Some of these texts have been used to develop chapters 2, 3 and 4. The 1911 catalogue of Schilling [Sch1] describes 40 series consisting of almost 400 models and devices. But by 1932 Martin Schilling informed the mathematical institute at G¨ottingen that: “in the last years, no new models appeared” ([F]). These models often represent objects from differential geometry, algebraic surfaces or instruments for physics. In Schilling’s catalogue [Sch1], the models are given a name and a short mathematical explanation. Sometimes this is accompanied with a drawing, as shown in Figure 1.2. Figure 1.2: Drawings of models: Series XXV, nr. 4 and Series VIII, nr. 1 from Schilling’s catalogue [Sch1]. Another drawing from the catalogue is displayed in Figure 1.3, alongside the corresponding model (in poor condition) in the Groningen collection. Figure 1.3: Model nr. 1, Series XII: drawing from [Sch1] and model in the Groningen collection. Another collection and catalogue is the one of 1892 [Dy] of Walter von Dyck. He was a student of Klein at M¨unich. Under the direction of von 4 CHAPTER 1. MODELS OF SURFACES, A DUTCH PERSPECTIVE Dyck models were created (see advertisement in Figure 1.4 on the left). Still another catalogue is the one of H. Wiener [W2]. 1.2.2 Felix Klein, use and popularization of models. During the great period of model building, many models were constructed un- der the direction of Klein. Klein was appointed at the Technische Hochschule in M¨unich in 1875. He and his colleague Alexander von Brill taught there advanced courses to several excellent students including Walter von Dyck and Karl Rohn. Klein lectured at the University of G¨ottingen from 1886 until he retired in 1913 (see [Yo]). Klein developed his love for geometric models under the influence of Al- fred Clebsch. Klein believed strongly in the utility of models and other illustrative objects for didactic purposes ([Yo]). He not only played an im- portant role in the construction of mathematical models, but also was a key figure in their popularization. At the end of his life he was proud to be able to say that no German university was without a proper collection of mathe- matical models ([Yo]). The idea of model building reached many universities in Europe and the United States. Klein brought the topic to the new world when he presented many of the models (including a model of the Clebsch diagonal surface) at the World Exhibition in Chicago in 1893. During the congress of mathematics (part of this World Exhibition), Klein gave a series of talks [Kl]. In several of them he referred to the models. We quote Klein’s lecture at the Chicago Exhibition. The importance he attributed to the construction of such models, and the purpose he attributed to them is expressed by him as follows: The principal effort has been to reduce the difficulty of mathema- tical study by improving the seminary arrangements and equip- ments... [...]... Collections of mathematical models and courses in drawing are calculated to disarm, in part at least, the hostil- ity directed against the excessive abstractness of the university instruction. [Kl] 1.2.3 Design and building of the models The designers of the mathematical models were often students of Alexander von Brill and Felix Klein (see [Po-Za]). The latter believed that students were 1.3. MODELS IN THE NETHERLANDS 5 not merely pupils, but collaborators ([Kl]), and instigated them to design and build some of the models. An example of this is the series of 20 models representing singularities of cubic surfaces that Carl Rodenberg, a student of Klein, presented along with his 1878 thesis (these models are now Series VII in Schilling’s catalogue). Another example of this collaboration with students is Karl Rohn. He designed several models, among which are three plaster models of Kummer surfaces with four, eight and sixteen real double points (part of Series II in [Sch1]). Rohn also designed the ruled surfaces of degree four discussed in Section 1.5.2 and Chapter 4. For a list of the students who designed models see [Sch1] or [Po-Za]. Once a geometric object was mathematically conceived, how could one realize it as an actual physical model? How is it possible that the results are so accurate and precise? Little has been found about the actual technique used to build the models. The two references that we found are a recipe for modelling in clay ([F]) and a mechanical construction of a model of the Weierstrass sigma function ([Sny]). Nowadays, there exist 3D printers for models which can create a plaster or bronze model of a surface starting from its equation (see [*] and [**]). 1.3 Models in The Netherlands 1.3.1 An overview More than half of the Dutch universities keep some collection of Brill’s or Schilling’s models. We briefly describe some of them, namely the ones located at the Universities of Amsterdam, Leiden and Utrecht. 1. University of Amsterdam. The University of Amsterdam keeps the largest collection of mathematical models in The Netherlands. It con- sists of more than 180 models, most of them from the collections of Brill and Schilling. The University Museum of Amsterdam has made a detailed inventory of their collection. Most of the plaster mod- els and many of the string ones have been restored. More detailed information of this collection can be found at the online catalogue http://opc.uva.nl:8080/.