Both three- and one-dimensional studies of local multiplicity fluctuations in hadronic Z decay are performed using data of the L3 experiment at LEP. The normalized factorial moments in three dimensions exhibit power-law scaling, indicating that the fluctuations are isotropic, which correspends to a self-similar fractal. A detailed study of the corresponding one-dimensional moments confirms this conclusion. However, two-jet subsamples have anisotropic fluctuations, correspending to a self-affine fractal. These features are, at least qualitatively, reproduced by the Monte Carlo models jetset and herwig.