  We construct spectral triples for compact metric spaces (X, d). This provides
us with a new metric d_s on X. We study its relation with the original metric
d. When X is a subshift space, or a discrete tiling space, and d satisfies
certain bounds we advocate that the property of d_s and d to be Lipschitz
equivalent is a characterization of high order. For episturmian subshifts, we
prove that d_s and d are Lipschitz equivalent if and only if the subshift is
repulsive (or power free). For Sturmian subshifts this is equivalent to linear
recurrence. For repetitive tilings we show that if their patches have
equi-distributed frequencies then the two metrics are Lipschitz equivalent.
Moreover, we study the zeta-function of the spectral triple and relate its
abscissa of convergence to the complexity exponent of the subshift or the
tiling. Finally, we derive Laplace operators from the spectral triples and
compare our construction with that of Pearson and Bellissard.
