In 1965, Mark Kac
[6]
asked, `Can one hear the shape of a drum?', so
popularizing the question of whether there can exist two non-congruent
isospectral domains in the plane. In the ensuing 25 years many
examples of isospectral manifolds were found, whose dimensions,
topology, and curvature properties gradually approached those of
the plane. Recently, Gordon, Webb, and Wolpert
[5]
finally reduced
the examples into the plane. In this note, we give a number of
examples, and a particularly simple method of proof.
One of our examples
(see Figure 1)
is a pair of domains that are not only isospectral but *homophonic*:
Each domain has a distinguished point such that
corresponding normalized Dirichlet eigenfunctions
take equal values at the distinguished points.
We interpret this to mean that
if the corresponding `drums'
are struck at these special points, then they
`sound the same' in the very strong sense that every frequency will
be excited to the same intensity for each.
This shows that one really can't hear the shape of a drum.

**Figure 1:** Homophonic domains. These drums sound the
same when struck at the interior points
where six triangles meet.