Old Math Problem About Music Geometry Solved After 70 Years

A professor, along with his team, have solved a difficult math problem. This problem is about understanding the shape of a drum from the sounds it makes. Professor Iosif Polterovich from the Université de Montréal is very interested in these types of questions. He used a part of math called spectral geometry to learn more about how waves move.

Polterovich, along with researchers Nikolay Filonov, Michael Levitin, and David Sher, figured out a special case of a well-known problem in spectral geometry. This problem was originally proposed by a famous mathematician, George Pólya, in 1954. The problem relates to calculating the frequencies of a drum. Until recently, this problem had only been solved for shapes like triangles and rectangles, but not for the shape of a disk.

Polterovich and his friends published an article in 2023 where they proved that Pólya’s problem could indeed be solved for the shape of a disk. This was a challenging task. Though their work is mostly of theoretical interest, it might also be helpful in practical areas such as computer math and computation. They are now looking into this.

Polterovich said that while math is a basic science, it is also like sport and art. Like in sport, solving difficult math problems is challenging. Like art, finding a simple solution is beautiful. And sometimes, beautiful math solutions can also be very useful.