Numerical investigation of two-dimensional projections of random fractal aggregates

Three-dimensional random fractal aggregates of variable fractal dimension D ranging from 1 to 2.5 are built on a cubic lattice using a hierarchical computer algorithm. The fractal dimensions of the surface ${\mathit{D}}_{\mathit{s}}$ and the perimeter ${\mathit{D}}_{\mathit{p}}$ of their two-dimensional projections have been computed. From an extrapolation of finite size results for aggregates containing up to 8192 particles, it is found that ${\mathit{D}}_{\mathit{s}}$ and ${\mathit{D}}_{\mathit{p}}$ are both equal to D for D2. The numerical results are consistent with a nontrivial value of ${\mathit{D}}_{\mathit{p}}$, varying continuously with D, for D>2.