From AMS Glossary
An irregular geometric object that is self-similar to its substructure at any level of refinement.
The fractal dimension is a measure of the irregularity of the boundary of the object. Clouds have been shown to exhibit fractal characteristics with cloud perimeters having fractal dimension of about 1.35. The result follows from the observation that over a large size range, cloud area is proportional to cloud perimeter raised to a power slightly less than 1.5 (rather than proportional to the square of the perimeter, as is the case in Euclidean geometry); fractal dimension of the perimeter then is calculated as the ratio of 2 (the exponent in the Euclidean power law) to the exponent in the observed power relation.
Lovejoy, S. 1982. Area-perimeter relation for rain and cloud areas. Science. 216. 185–187.