  We present a new method to approximate the Neumann spectrum of a Laplacian on
a fractal K in the plane as a renormalized limit of the Neumann spectra of the
standard Laplacian on a sequence of domains that approximate K from the
outside. The method allows a numerical approximation of eigenvalues and
eigenfunctions for lower portions of the spectrum. We present experimental
evidence that the method works by looking at examples where the spectrum of the
fractal Laplacian is known (the unit interval and the Sierpinski Gasket (SG)).
We also present a speculative description of the spectrum on the standard
Sierpinski carpet (SC), where existence of a self-similar Laplacian is known,
and also on nonsymmetric and random carpets and the octagasket, where existence
of a self-similar Laplacian is not known. At present we have no explanation as
to why the method should work. Nevertheless, we are able to prove some new
results about the structure of the spectrum involving "miniaturization" of
eigenfunctions that we discovered by examining the experimental results
obtained using our method.
