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Can you hear the shape of the graph?


How can you tell which instrument someone is playing just by hearing its sound? This is a simple variation of an inverse problem.

We often want to determine the properties of a physical object, even if we cannot measure them directly. What can we learn about a system, looking through the glasses that nature provides?

This exhibit explores inverse questions that are related to sounds and the vibration patterns that give rise to them. Visitors can investigate what one can deduce about an object, just by listening to the sounds it produces or by counting the number of areas of vibration.

How does it work?

This exhibit focuses on quantum graphs, which can be thought of as a network of guitar strings tied to each other. When a system like this vibrates it produces a series of sounds with different pitches. This exhibit investigates what happens when we have two different graphs which sound exactly the same. In this case we would not be able to distinguish between such graphs just by listening to the sounds they produce. However, it is possible to deduce some of the graph’s properties – such as the total length of the graph – just by listening to it. .

One can also observe the vibrating strings of the graph and note that some special points of the strings do not move at all. These nodal points are a fascinating phenomenon and discovering their patterns is still an unsolved challenge.

See all exhibits from 2011


The following videos demonstrate some of the science of this exhibit. The first video demonstrates how graphs can be considered as musical instruments. The second video is an interview with one of the scientists working on this exhibit. These pdf documents also demonstrate part of this exhibit.