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Drums that sound the same, i.e., have the same eigenfrequency spectrum. Two drums with differing Area, Perimeter, or Genus can always be distinguished. However, Kac (1966) asked if it was possible to construct differently shaped drums which have the same eigenfrequency spectrum. This question was answered in the affirmative by Gordon et al. (1992). Two such isospectral manifolds are shown in the right figure above (Cipra 1992).
References
Chapman, S. J. ``Drums That Sound the Same.'' Amer. Math. Monthly 102, 124-138, 1995.
Cipra, B. ``You Can't Hear the Shape of a Drum.'' Science 255, 1642-1643, 1992.
Gordon, C.; Webb, D.; and Wolpert, S. ``Isospectral Plane Domains and Surfaces via Riemannian Orbifolds.''
Invent. Math. 110, 1-22, 1992.
Gordon, C.; Webb, D.; and Wolpert, S. ``You Cannot Hear the Shape of a Drum.'' Bull. Amer. Math. Soc. 27, 134-138, 1992.
Kac, M. ``Can One Hear the Shape of a Drum?'' Amer. Math. Monthly 73, 1-23, 1966.