Roughly speaking, isospectral manifolds are 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 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 (which are 7-polyaboloes) are shown in the left figure above. The right figure above shows another pair obtained from the original ones by making a simple geometric substitution.
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