Shapes of bells and shapes of drums

Who first asked “Can one hear the shape of a drum?” When I heard this question, it was attributed to Mark Kac, who wrote (in 1966) a famous paper in the American Mathematical Monthly with this title (this must have been around the time I first heard of Property ]T[ and of expanders, and — name-dropping warning — it might have been from any of Y. Colin de Verdière, G. Besson, P. Bérard or C. Bavard). I have to admit that I only had the curiosity to look at this paper two or three years ago, and I was amused to see the following quote (on top of page 3):

I first heard the problem posed this way some ten years ago from Professor Bochner. Much more recently, when I mentioned it to Professor Bers, he said, almost at once, “You mean, if you had perfect pitch could you find the shape of a drum.”

So the mathematical question goes back further than Kac, and the elegant turn of phrase is really due to someone else!

But this post is motivated by another quote, which I read today in Appendix F of Schlomo1 Sternberg’s book “Group theory and physics”, where he gives a quick survey of 19th Century spectroscopy. The word itself was introduced in 1882 by Arthur Schuster, who goes on to say (top of page 395 in Sternberg’s book, italics mine):

To find out the different tunes sent out by a vibrating system is a problem which may or may not be solvable in certain special cases, but it would baffle the most skillful mathematician to solve the inverse problem and to find out the shape of a bell by means of the sounds which it is capable of giving out. And this is the problem which ultimately spectroscopy hopes to solve in the case of light. In the meantime we must welcome with delight even the smallest step in the right direction.

I didn’t know Schuster before, but he seems to have been a very interesting character, who pioneered the use of harmonic analysis and probability to detect (non)-periodicity in certain data sets (e.g., dates of earthquakes, in 1897), and also speculated about “antimatter” (using that word) in 1898…

(I haven’t found yet the exact source quoted by Sternberg, who doesn’t have a bibliography for this Appendix…)

Footnote: 1 No relation to Schlomo Cohen.

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I am a professor of mathematics at ETH Zürich since 2008.

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