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).
Fig 4.41
Fig 4.42

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