Around the middle of the 19th century, the French philosopher Comte speculated that knowledge of the chemical composition of stars would be forever beyond the reach of science. He was wrong, and soon afterwards the related field of spectroscopy was created. Molecules in interstellar space are identified by their natural vibration frequencies, i.e., by the pitches at which a molecule naturally rings. Given a set of vibration frequencies, an important research topic is to ask what can be inferred about the system's structure.
In 1966, the Polish-American mathematician Mark Kac asked a related question about whether one can always hear the shape of a drum. In other words, if you close your eyes and listen to differently shaped drums being played, Mark Kac wanted to know if you could distinguish the shape by the sound or vibration frequencies you hear. A mathematical drum is not a standard musical instrument; it is any shape in the plane that has an interior and a boundary, such as a circle, a square, or a triangle. The interior vibrates with each strike while the boundary frame remains rigid. Imagine that we had a machine that could tell us the exact frequency of the sound of the drum vibrations. Then we could check and see whether the machine could always distinguish the sounds of differently shaped drums. In 1911, Hermann Weyl proved that one can always hear the area of a mathematical drum. It makes sense that we can hear the area since the bigger the area of the drum, the lower the tone. Later, Minakshisundaram and Pleijel proved that one can always hear the length of the boundary, or the perimeter of the drum. It was thought that the sound of a drum might contain enough geometric information to specify its shape uniquely, and Mark Kac asked whether this was true.
The problem challenged researchers for nearly three decades. Finally, in the spring of 1991, Carolyn Gordon, her husband David Webb, and Scott Wolpert came up with the answer: No! One can sometimes, but not always hear the shape of a drum. They found two mathematical drums that have different shapes, but still had the same vibration frequencies, therefore making the same exact sound.
A mathematical proof does not need to be constructive. In fact, in this case, mathematicians would not have been satisfied with experiments to show that the drums sound the same because calculations and experiments cannot be exact and a very small frequency difference could escape experimental detection. Instead, Carolyn Gordon and her collaborators used mathematics to prove that her drums sounded the same without actually testing them in real-life. Later on, physicists created the drums and tested them in real life. They found that the drums sounded nearly the same with an error attributed to the experimental procedures. Carolyn Gordon's research on hearing the shape of a drum shows us there is not always just one conclusion that can be reached from a complete set of measurements.