Abstract
Let
F
n
be the
n
th stage in the construction of the Sierpiński carpet. Let
R
n
be the electrical resistance of
F
n
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 ¼
ρ
n
≼
R
n
≼ 4
ρ
n
. 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
).
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