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 second-order normalized factorial moment in three
dimensions is found to exhibit power-law scaling, indicating that the
fluctuations are isotropic. This is confirmed by a detailed study of the
corresponding one-dimensional moments. Such scaling corresponds to a
self-similar fractal, i.e., the associated branching process is
self-similar. However, when two-jet subsamples are analyzed, the
branching is found to be self-affine. These features are at least
qualitatively reproduced by the Monte Carlo models JETSET and HERWIG.