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Noncommutative geometry provides a framework, via the construction ofspectral triples, for the study of the geometry of certain classes of fractals.Many fractals are constructed as natural limits of certain sets with a simplerstructure: for instance, the Sierpi\'nski is the limit of finite graphsconsisting of various affine images of an equilateral triangle. It is thusnatural to ask whether the spectral triples, constructed on a class of fractalscalled piecewise $C^1$-fractal curves, are indeed limits, in an appropriatesense, of spectral triples on the approximating sets. We answer this questionaffirmatively in this paper, where we use the spectral propinquity on the classof metric spectral triples, in order to formalize the sought-after convergenceof spectral triples. Our results and methods are relevant to the study ofanalysis on fractals and have potential physical applications.

Metric Approximations of Spectral Triples on the Sierpi\'nski Gasket and other fractal curves