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).

Chapman, S. J. "Drums That Sound the Same." Amer. Math. Monthly102, 124-138, 1995.Cipra, B. "You
Can't Hear the Shape of a Drum." Science255, 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. Monthly73, 1-23, 1966.Trott,
M. "The Mathematica Guidebooks Additional Material: Isospectral Polygons."
http://www.mathematicaguidebooks.org/additions.shtml#S_1_11.Zwillinger,
D.(Ed.). "Eigenvalues." §5.8 in CRC
Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, pp. 425-426,
1995.