  Compactifying a higher-dimensional theory defined in R^{1,3+n} on an
n-dimensional manifold {\cal M} results in a spectrum of four-dimensional
(bosonic) fields with masses m^2_i = \lambda_i, where - \lambda_i are the
eigenvalues of the Laplacian on the compact manifold. The question we address
in this paper is the inverse: given the masses of the Kaluza-Klein fields in
four dimensions, what can we say about the size and shape (i.e. the topology
and the metric) of the compact manifold? We present some examples of
isospectral manifolds (i.e., different manifolds which give rise to the same
Kaluza-Klein mass spectrum). Some of these examples are Ricci-flat, complex and
K\"{a}hler and so they are isospectral backgrounds for string theory. Utilizing
results from finite spectral geometry, we also discuss the accuracy of
reconstructing the properties of the compact manifold (e.g., its dimension,
volume, and curvature etc) from measuring the masses of only a finite number of
Kaluza-Klein modes.
