Interpolation functions $f:[0,1] \to \mathbb{R}$ of the following nature are constructed. Given data \[ \left\{ {\left( {t_n ,x_n } \right) \in [0,1] \times \mathbb{R}:n = 0,1,2, \cdots ,N} \right\}\] with $0 = t_0 < t_1 < \cdots <' t_N = 1$, f obeys \[ f(t_n ) = x_n ,\qquad n = 0,1,2, \cdots ,N.\] Furthermore, the graph of f is the projection of a set G in $\mathbb{R}^M $ (M an integer greater than or equal to 2) that is homeomorphic to $[0,1]$ and is the attractor for an iterated function system consisting of afflne maps in $\mathbb{R}^M $. The latter characterization ensures that f can be computed rapidly while possessing many ”hidden” variables, on which its values continuously depend, which allow great flexibility and diversity in the interpolant, making it potentially useful in approximation theory. Estimtes and exact values for the fractal dimensions of G and the graph of f are obtained.