  In the path integral formulation of quantum mechanics, Feynman and Hibbs
noted that the trajectory of a particle is continuous but nowhere
differentiable. We extend this result to the quantum mechanical path of a
relativistic string and find that the ``trajectory'', in this case, is a
fractal surface with Hausdorff dimension three. Depending on the resolution of
the detecting apparatus, the extra dimension is perceived as ``fuzziness'' of
the string world-surface. We give an interpretation of this phenomenon in terms
of a new form of the uncertainty principle for strings, and study the
transition from the smooth to the fractal phase.
