## Math Beyond the HorizonCan you hear the shape of a drum?

Using spectroscopy, astronomers determine the chemical composition of stars from the frequencies of emitted light. We address a similar problem concerning frequencies of sound.

Imagine a possibly exotic-shaped drum. The drumhead can be any shape, not necessarily round. A vibrating membrane such as the drumhead has a fundamental tone and an infinite but discrete collection of possible overtones, each associated with a characteristic mode of vibration. In general drums of different shape will sound different, but is this always the case? In the words of Mark Kac, "Can you hear the shape of a drum?" We will see that the answer is sometimes "yes"; for example no other shape produces the same sound as a round drum. However, we will also see that the answer is sometimes no; we will construct pairs of drumheads of different shapes that sound identical. We will also listen to the sound of these exotic-shaped sound-alike drums, using a simulation produced by Dennis DeTurck.

The question generalizes to other settings besides drumheads; one wants to obtain information about an unseen object by analyzing frequency data. The problem has applications to medical imaging, for example.