On obtaining the fractal dimension of a 3D cluster from its projection on a plane-application to smoke agglomerates

The finite size of a fractal cluster of dimension D causes a reduction in the dimension of its projection to D* on any finite observation scale. A theoretical argument is presented which relates the apparent dimension D* to D and to the observation scale as a fraction of the outer fractal scale. The theory is tested for computer-generated fractal clusters. When applied to a sample of electron micrographs of diesel soot particles whose apparent dimension is 1.83+or-0.06, the theory gives 1.90+or-0.07 as the true fractal dimension of the soot.