"Overcoming Resistance with Fractals: A New Way to Teach Elementary Circuits" is the name of an article by W.K. Ching, M. Erickson, P. Garik, P. Hickman, J. Jordan, S. Schwarzer, and L. Shore in The Physics Teacher, Vol 32., No. 9 (December 1994), p546-551. They discuss an experiment in which students construct a Sierpinksi gasket from resistors and measure its resistance as a function of size. As they write, "Because of the simplicity of the network, the students can derive its theoretical resistance using the properties of series and parallel resistors, and then compare their experimental and theoretical results." But this is more than just a laboratory exercise. They go on to note:
"But how to model a disordered solid? One alternative is to model the connection properties with self-similar fractal geometry. Such a study allows the building of physical intuition for the behavior of objects with complicated geometry. In particular, the Sierpinksi gasket has proven a workhorse for testing physical theories on a fractal geometry. Electrical conductivity, diffusive transport, and thermodynamic properties have all been studied extensively on the gasket because of the simplicity of its connectivity."