The Effros-Shen algebra corresponding to an irrational number θ can be described by an inductive sequence of direct sums of matrix algebras, where the continued fraction expansion of θ encodes the dimensions of the summands, and how the matrix algebras at the nth level fit into the summands at the (n+1)th level. In recent work, Mitscher and Spielberg present an Effros-Shen algebra as the C*-algebra of a category of paths -- a generalization of a directed graph -- determined by the continued fraction expansion of θ. With this approach, the algebra is realized as the inductive limit of a sequence of infinite-dimensional, rather than finite-dimensional, subalgebras. In this thesis, the author defines a spectral triple in terms of the category of paths presentation of an Effros-Shen algebra, drawing on a construction by Christensen and Ivan. This thesis describes categories of paths, the example of Mitscher and Spielberg, and the spectral triple construction.