Projections of three-dimensional random percolation clusters onto a two-dimensional plane give roughly the same fractal dimension 1.3 (for the perimeter versus area relation) as Lovejoy's observation of real clouds and cellular automata simulations of Nagel and Raschke.
Real clouds in our sky are usually observed from above or from below, and thus their ffactal properties are those of their two-dimensional projections or pictures. Lovejoy [1] measured in real clouds, and Nagel and Raschke [2] found in computer simulations, that the perimeter P of a cloud projection varies with some power D / 2 of the area A of that projection:
In a computer simulation on a lattice, the perimeter P is the number of empty sites which have at least one occupied cloud site as a neighbor, and the area a is the number of occupied sites forming the connected cluster representing the cloud. Both P and a are defined for the two-dimensional projection, not for the three-dimensional original cluster.
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[2] simulated such clouds by a cellular automata approximation which gives also dynamical information. For the static fractal dimension D, Nagel and Raschke found 1.38_+0.04 in good agreement with the experimental [1] D = 1.33. They suggested already that the static properties might have come out also from a simpler percolation model [31. The present note confirms this suggestion.
Thus we simulated on an Amiga computer, programmed in C, with the Leath algorithm [4] the formation of a single random percolation cluster on a simple cubic lattice at the percolation threshold 0.3116. This cluster was projected onto a (100) lattice plane, then the perimeter P and number a of