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 (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 (which are 7-polyaboloes) are shown in the left figure above (Cipra 1992). The right figure above shows another pair obtained from the original ones by making a simple geometric substitution.
Another example of isospectral manifolds is the pair of polyabolo configurations known as bilby (left figure) and hawk (right figure). The figures above show scaled displacements for a number of eigenmodes of these manifolds (M. Trott, pers.comm., Oct. 8, 2003).
Furthermore, pairs of separate drums (having the same total area) can be constructed which have the same eigenfrequency spectrum when played together (illustrated above). Therefore, you cannot hear the shape of a two-piece band (Zwillinger 1995, p. 426).
