hearing the shape of a drum
When a drum vibrates it produces a set of pure tones — its spectrum of frequencies. Mark Kac asked, in a famous 1966 article titled exactly this, whether you could run the experiment backwards: if you knew all the frequencies a drum makes, could you deduce its shape? In mathematical terms, does the spectrum of the Laplacian determine the manifold (or planar domain) up to isometry? It is the poster question of spectral geometry, and the surprising answer is no.
Concretely, two Riemannian manifolds are isospectral if their Laplace-Beltrami operators have exactly the same list of eigenvalues with the same multiplicities (the same set of pure tones). The question is whether isospectral forces isometric. A great deal is determined by the spectrum: from the eigenvalues alone, via the heat-kernel asymptotic expansion sum e^{-lambda_k t} ~ (4 pi t)^{-n/2} (volume + t * (a curvature integral) + ...), you can read off the dimension, the total volume, and the integral of scalar curvature; Weyl's law recovers dimension and volume from the counting function. But the full shape is not determined. In 1964 Milnor produced two flat 16-dimensional tori that are isospectral but not isometric, and in 1992 Gordon, Webb, and Wolpert built two genuinely different flat polygonal drums in the plane that sound exactly alike — a clean, elementary 'no' to Kac's literal question.
The story is a perfect lesson in how much, and how little, analysis remembers about geometry. The spectrum is a rich invariant — enough to pin down dimension, volume, and curvature averages, and enough to make 'inverse spectral geometry' a deep field — but it forgets some shape information, and reconstructing geometry from sound is impossible in general. The honest caveats: the counterexamples are special (flat tori, polygonal billiards) and exploit symmetry, and under extra hypotheses (analyticity, generic metrics, certain symmetry constraints) one can sometimes hear the shape after all. Also, 'the same frequencies' must mean the same eigenvalues with multiplicity; weaker matchings determine even less. So the precise statement is: the Laplace spectrum determines many geometric invariants but not the isometry class in general.
The 1992 Gordon-Webb-Wolpert pair of planar drums are two seven-sided polygons, built from the same eight triangular tiles glued in two different patterns; they have identical Laplace spectra under Dirichlet boundary conditions yet are not congruent — you cannot hear them apart.
The Gordon-Webb-Wolpert drums: isospectral but not isometric, answering Kac's question in the negative.
The spectrum does determine dimension, volume, and the integral of scalar curvature via the heat-kernel expansion, so 'you cannot hear the shape' is not 'you can hear nothing' — it is the sharper claim that the full isometry class is not recoverable.