"You Can't Hear the Shape of a Drum", by Carolyn Gordon and David Webb. American Scientist, January/February 1996, pages 46-55.
A drum produces a sound by vibrations in the drum head, which has certain characteristic frequencies that determine how it vibrates and therefore how it sounds. In 1966 the mathematician Mark Kac asked the question, Can you hear the shape of a drum? That is, if you know all of the sounds that a drum can make, can you infer the shape of the drum head? The answer came in 1991, in work by the authors of this article, who proved that the answer is no. They came up with examples of drum heads that have different shapes but have exactly the same characteristic vibration frequencies. This article describes their work.