  To unify general relativity and quantum theory is hard in part because they
are formulated in two very different mathematical languages, differential
geometry and functional analysis. A natural candidate for bridging this
language gap, at least in the case of the euclidean signature, is the
discipline of spectral geometry. It aims at describing curved manifolds in
terms of the spectra of their canonical differential operators. As an immediate
benefit, this would offer a clean gauge-independent identification of the
metric's degrees of freedom in terms of invariants that should be ready to
quantize. However, spectral geometry is itself hard and has been plagued by
ambiguities. Here, we regularize and break up spectral geometry into small
finite-dimensional and therefore manageable steps. We constructively
demonstrate that this strategy works at least in two dimensions. We can now
calculate the shapes of 2-dimensional objects from their vibrational spectra.
