  Using scaling arguments and the property of self-similarity we derive the
Casimir energies of Sierpinski triangles and Sierpinski rectangles. The
Hausdorff-Besicovitch dimension (fractal dimension) of the Casimir energy is
introduced and the Berry-Weyl conjecture is discussed for these geometries. We
propose that for a class of fractals, comprising of compartmentalized cavities,
it is possible to establish a finite value to the Casimir energy even while the
Casimir energy of the individual cavities consists of divergent terms.
