  We develop the foundation of the spectral analysis on Barlow-Evans projective
limit fractals, or vermiculated spaces, which corresponds to symmetric Markov
processes on these spaces. For some new examples, such as the generalized
Laakso spaces and a Spierpinski P\^ate \`a Choux, one can develop a complete
spectral theory, including the eigenfunction expansions that are analogous to
Fourier series. Also, one can construct connected fractal spaces isospectral to
the fractal strings of Lapidus and van Frankenhuijsen. Our work is motivated by
recent progress in mathematical physics on fractals.
