Hearing the Shape of a Drum: From Digital Signal Simulation to the Shape of the Universe

The well-known “Hearing the Shape of a Drum”

In 1966, Mark Kac, a mathematician, posed a question which left a name behind in the science circles: “Can One Hear the Shape of a Drum?” After all, drums of different shapes produce waves of different frequencies, and therefore make different sounds. But is that enough to recognize the shape of a drum? At that time, Kac thought the answer was no. But later on, a series of studies pointed out that Kac was wrong.

Now, two of the major players in mathematics have reconsidered the question- in the past two years, Walter van Suijlekom and Alain Connes have gone further and gained new insights.

Van Suijlekom and Connes, who received the prestigious Fields Medal in 1982, studied how vibrations of a finite number can provide information about the shape of an object. For this purpose, they developed a new mathematical theory describing how to accurately estimate the complete shape of a drum by “listening to part of it”.

Now, Walter van Suijlekom wrote an explanatory article in Letters in Mathematical Physics on July 14. “Let us compare it to MP3. Musical works are generated by instruments as analog signals with an infinite number of sound frequencies. However, this signal can be digitized by limiting the analog signal to a finite number of frequencies; i.e., an MP3 file.”

What does he mean by that? The frequency range of the whole musical work is infinite, but the overall frequency range of the digital signal of an MP3 is limited, and we can still make an MP3 file of any piece of music- although the sound quality is degraded, it doesn't prevent us from recognizing the music. Similarly, despite the drum sound cannot convey complete information, it is enough to make sure the general shape of the drum.

Estimating the Shape of the Universe

Just as the concept that digital MP3 can still provide a simulated source, our limited observations of the universe also provide enough information for a better understanding of simulated source. Using the frequencies of light emitted by stars and other celestial bodies, scientists can figure out the appearance of these objects.

Van Suijlekom: “We are developing the mathematics that makes it possible. In this way, we described how to calculate the shape of an object when only a finite number of vibrations are available. It is just like a physics experiment.”

This means that, in principle, it should also be possible to use a finite number of frequencies to estimate the shape of the universe.