  Pearson and Bellissard recently built a spectral triple - the data of
Riemanian noncommutative geometry - for ultrametric Cantor sets. They derived a
family of Laplace-Beltrami like operators on those sets. Motivated by the
applications to specific examples, we revisit their work for the transversals
of tiling spaces, which are particular self-similar Cantor sets. We use
Bratteli diagrams to encode the self-similarity, and Cuntz-Krieger algebras to
implement it. We show that the abscissa of convergence of the zeta-function of
the spectral triple gives indications on the exponent of complexity of the
tiling. We determine completely the spectrum of the Laplace-Beltrami operators,
give an explicit method of calculation for their eigenvalues, compute their
Weyl asymptotics, and a Seeley equivalent for their heat kernels.
