In the mathematical exhibition “ix-quadrat”, which was founded and directed by Richter-Gebert and has been on display since 2002 in the Department of Mathematics at the Technical University of Munich in Garching, visitors can literally “grasp” mathematical correlations by building their own models in seminars. Everyday objects such as coat hangers, skewers, pipe cleaners, tongue depressors or pencils are used for this purpose: “It is an interesting mental exercise to see how these materials can give rise to exciting mathematical structures.”

Mathematics as the science of structures

For many people, mathematics is only about numbers, calculations and formulas, but at its core is thinking about structural effects, Richter-Gebert continued. “One can create, recognize, and analyze these structures.” Building models involves searching for places where structures come together in interesting ways. Coat hangers and similar objects unite two structural principles: symmetry and cyclical interlocking.

To explain symmetry, Richter-Gebert cited a definition by the Physics Nobel Laureate Richard Feynman: “A thing is symmetrical if there is something we can do to it so that after we have done it, it looks the same as it did before.” Symmetrical structures often lead us to perceive something, such as a snowflake, as highly aesthetic. The arched configuration sketched by Leonardo da Vinci, now known as the “Leonardo Bridge”, is a well-known example of cyclic interlocking: The components support each other merely on the basis of their ingenious joining principle, without the need for additional elements.

“Worlds behind glass” – mathematical visualization software

Software can likewise contribute to the sensualization of mathematics. “Ideally, these things behind the glass should feel like real or even abstract objects,” the mathematician said. The mission, or vision, of visualization software is to establish a setting for making abstract relationships phenomenologically experienceable. In this connection Richter-Gebert presented the “TUM interaktiv” app, which he developed for the 150th anniversary of the Technical University of Munich in 2018. This provides animations, simulations, and experimental fields on key scientific topics from all 14 of TUM’s faculties. Users of the app can also become researchers themselves within small demarcated spaces – the micro-laboratories.

Using the example of mathematical catastrophe theory, Richter-Gebert showed how abstract processes can be visualized be means of software. “A mathematical catastrophe takes place as follows: A parameter in a dynamic system is continuously modified – and the system then suddenly tips over into an entirely different state.” He illustrated this with the catastrophe machine constructed by the British mathematician Erik Christopher Zeeman, comprising a disc with an axle embedded in a baseplate, that can be rotated by means of two rubber straps stretched over the base and meeting on the disc. Richter-Gebert transferred Zeeman’s “minimal playing field for viewing catastrophes” into a virtual version, which can be used to vividly simulate the physical principle underlying this effect.

“Between digital and real”

Richter-Gebert described his drawing app “iOrnament” as the meeting place of real life and software. He was inspired to develop this program by the art nouveau ornaments on display at the symmetry exhibition in Darmstadt. As patterns that exhibit symmetries of displacement in at least two directions and can thus fill the entire plane, such mathematical structures can be creatively studied on the basis of ornaments. This connection between art and mathematics is established in the app, which allows users to design geometric patterns based on their own ideas.

Richter-Gebert concluded his lecture with an insight into the exhibition “La La Lab. The Mathematics of Music”, which he co-curated and was staged in Heidelberg from 2019 to 2021. Topics such as rhythm, scales, dissonance, harmony, and frequency spectra were visually presented in the interactive exhibits, so that the visitors could experience the connections and similarities between the fields of mathematics and music by experimenting for themselves.