Spectral Geometry: The Art of Seeing Sounds and Hearing Shapes
Imagine a violin bow moving across the edge of a metal plate covered by sand. The bow movement creates vibrations that produce sounds, forcing the sand grains to dance.
At first, the sand picture looks shapeless and the sound is annoying. However, as the frequency of vibrations increases and reaches a certain threshold, the sound suddenly becomes pure and the sand forms a beautiful symmetric pattern. Within seconds it gets destroyed. As the next frequency threshold is reached, an even more fascinating pattern emerges, accompanied by another pure tone.
When the German musician and physicist Ernst Chladni performed this experiment in front of Napoleon Bonaparte in 1809, the link between the formation of sand shapes and pure sounds was not understood. The French emperor was so intrigued that he offered Chladni 6,000 francs to continue his investigations, and established a prize of 3,000 francs for finding an explanation of the patterns—claimed in 1816 by the French mathematician Sophie Germain.
Today, this remarkable phenomenon—that pure sounds can also be seen—is well-known in spectral geometry. The thresholds of "pure sound" correspond to the resonance frequencies of the plate, and the grains of sand, trying to minimize their energy, concentrate along the zero sets of the amplitudes.
Despite the ground-breaking works of past scientists, Dr. Iosif Polterovich believes our understanding of vibration and sound propagation remains incomplete. The Canada Research Chair in Geometry and Spectral Theory will research various areas of geometric spectral theory. One of them, called isospectrality, is motivated by the famous question "Can one hear the shape of a drum?", asked by Mark Kac more than 40 years ago.
If one can see sounds, why not hear shapes?