It is well known that although the spectrum of a manifold does not, in general, determine the geometry of the manifold, various geometric invariants such as volume and dimension are in fact determined. One of the goals of finite spectral geometry is to study, in an exact way, how information about these invariants is revealed by only a finite part of the spectrum. One is reminded of the fact that geometric detail about an optically detected object cannot be determined to a greater extent than the wavelength of the impinging radiation allows. Shorter wavelengths potentially reveal greater detail. This situation is studied in scattering theory and differs in many ways from the case of trying to use a finite part of the Laplace spectrum to gain geometric information about a manifold, but the general notion that access to higher eigenvalues (in analogy to shorter wavelengths) should give more geometric information still holds. The question is: what geometry can we expect to be missed due to not having information about the high end of the spectrum? If one where trying to estimate the area of a (microscopic) planar region via some kind of pixel counting, then the estimate can only be expected to be reasonable if one knew, a priori, that the region did not include something like a haze of thin tentacles extending beyond some central region (or some other complexity near the boundary). But this is exactly where the analogy with hearing the volume via the Laplace spectrum breaks down since a long thin tenticle, being similar to a long string, may indeed have a major effect on the smaller eigenvalues of the region and so cannot on principle be expected to escape detection. The question is: what aspects of a manifold can be expected to interfere with the finite spectral detection of a given geometric invariant such as volume?
