40. Referring to Figures 4.41 and 4.42, the Cantor set structure is
apparent. The length of the two intervals remaining in Figure 4.41 is determined by
the intersection of y = s*x and y = 1, and by the intersection of y = s*(1 - x)
and y = 1. Both intervals have length s. That is, the Cantor set is covered by two
intervals of length s. A similar calculation shows at the next step we have four intervals
of length s2. So we have N(s) = 2, N(s2) = 4 =
22, and in general, N(sn) = 2n. The box-counting
dimension is the limit as n becomes large of the ratio log(2n)/log(sn)
= log(2)/log(s).
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