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We construct various self-similar configurations using parallel$\delta$-function plates and show that it is possible to evaluate the Casimirinteraction energy of these configurations using the idea of self-similarityalone. We restrict our analysis to interactions mediated by a scalar field, butthe extension to electromagnetic field is immediate. Our work unveils an easyand powerful method that can be easily employed to calculate the Casimirenergies of a class of self-similar configurations. As a highlight, in anexample, we determine the Casimir interaction energy of a stack of parallelplates constructed by positioning $\delta$-function plates at the pointsconstituting the Cantor set, a prototype of a fractal. This, to our knowledge,is the first time that the Casimir energy of a fractal configuration has beenreported. Remarkably, the Casimir energy of some of the configurations weconsider turn out to be positive, and a few even have zero Casimir energy. Forthe case of positive Casimir energy that is monotonically decreasing as thestacking parameter increases the interpretation is that the pressure of vacuumtends to inflate the infinite stack of plates. We further support our results,derived using the idea of self-similarity alone, by rederiving them using theGreen's function formalism. These expositions gives us insight into theconnections between the regularization methods used in quantum field theoriesand regularized sums of divergent series in number theory.

Casimir energies of self-similar plate configurations