On the resistance of the Sierpiński carpet

Let Fn be the nth stage in the construction of the Sierpiński carpet. Let Rn be the electrical resistance of Fn when the left and right sides are each short-circuited, and a voltage is applied between them. We prove that there exists a constant ρ such that ¼ρn ≼ Rn ≼ 4ρn. The motivation for this result came from the problem of establishing (a) the existence and (b) the value of the ‘spectral dimension’ of the Sierpiński carpet. In this and a subsequent paper, we settle (a) and give bounds for (b).