Hyperboloid - Figure 1

Object of the Month: Hyperboloid of two sheets of Stoll (No 224)

Object of the Month 10/2023

Only a few insiders would immediately recognise the object of October. The model of a Hyperboloid of two sheets is located in Adlershof, more precisely in the Institute of Mathematics and belongs to the Mathematical Model Collection there. However, the connection to the university goes much deeper. The template for the model arose from the institute’s teaching and research activities. Even though the model is not complete, it demonstrates very well the basic idea of a teaching collection, which is to be seen in its use in academic as well as school teaching. That’s why the objects show signs of usage over the course of time, or sometimes even break. Although they are usually very robustly constructed for touching. But one thing after the other.

Hyperboloid - Figure 1
Specimen of the Hyperboloid of two sheets of Stoll in the mathematical model collection in Adlershof. The upper sheet is missing, indicating the frequent use of the object (Photo: Robert Pässler, TU Dresden).

The Hyperboloid of two sheets, a geometric shape in mathematics, is a second-order surface. In order to think of it as a body, one rotates a hyperbola around its main axis. This creates two separate surface pieces (in the model as a body), whereby in the case of the Berlin model the upper surface piece (the upper body) is missing. The sketch in Figure 2 from Meyer’s Großes Konversations-Lexikon of 1905 shows a Hyperboloid of two sheets with the axes, whereby the vertical axis shown in the picture is the main axis.

Hyperboloid - Figure 2
Meyer's Großes Konversations-Lexikon of 1905 shows a Hyperboloid of two sheets with imaginary axes.

The origin of this model is interesting. It was manufactured by the company Rudolf Stoll K.G. Berlin. It was located at Oderbruchstraße 8-14, i.e. in the Friedrichshain district of Berlin. The company not only took over the production, but also the distribution of the teaching models.

The teaching aids were developed at the Second Mathematical Institute of the Humboldt University in Berlin under the direction of Professor Dr. Kurt Schröder (1909-1978). He held the professorship of Applied Mathematics and was also Director of the Institute. In the first half of the 1960s, he was also Rector of the Humboldt-Universität zu Berlin.

Stoll’s models can be seen in a line of development with the mathematical models produced by Brill, Schilling and Wiener since the 1880s. They appeared at a time when their use in mathematical teaching was already taking place through other media. Nevertheless, they were produced and distributed, and moreover, they were used regularly.

Hyperboloid - Figure 3
The model described in the catalogue "Lehrmodelle für Mathematik" of Rudolf Stoll K.G. Berlin No. 18 (Source: SLUB Dresden).

Only a few traces of the Stoll company can be found today. Apart from the models found in some mathematical collections of other universities (e.g. TU Dresden or University of Marburg), the catalogue “Lehrmodelle für Mathematik” (Teaching Models for Mathematics) by Rudolf Stoll K.G. Berlin No. 18, published in three languages (German, English and French), still exists. The models shown there are divided into teaching aids for elementary mathematics, for geometry and for analysis. Our model is found under the number “Modell 224/114” with the note that “a Hyperboloid of two sheets ” is shown. The weight is 2 kilograms. The dimensions are 20 x 16 x 30 cm.

In this context, it is still worth mentioning that such sales catalogues are not classic library collectors’ items. They are therefore very rare and often only preserved by chance. The price list for the Stoll catalogue cannot be found digitally. We do not know whether a copy has been preserved somewhere.

Dr. Oliver Zauzig


Mathematische Modelle am Institut für Mathematik: https://www.mathematik.hu-berlin.de/de/sammlung-mathematik und https://www.sammlungen.hu-berlin.de/sammlungen/mathematische-modelle/

Mathematik und ihre Didaktik (Completed project for the collection): https://didaktik.mathematik.hu-berlin.de/de/projekte/abgeschlossen/mathematische-modelle/modellhersteller-fa-rudolf-stoll

Zweischaliges Hyperboloid (Stoll) der Mathematische Modellsammlung der HU im Digitalen Archiv mathematischer Modelle: https://mathematical-models.org/de/models/1064

Mathematische Modelle auf der Projektseite Materielle Modelle: http://www.universitaetssammlungen.de/modelle/suche/art/Mathematische+Modelle

Digitalisierter Katalog „Lehrmodelle für Mathematik“ in den Digitalen Sammlungen der SLUB Dresden: https://digital.slub-dresden.de/werkansicht/dlf/90059/1

Hyperboloīd in Meyers Großes Konversations-Lexikon: http://www.zeno.org/Meyers-1905/A/Hyperboloīd