Box-Counting Dimension of a Gasket

Covering a Gasket with smaller and smaller boxes we see the pattern illustrated in the table on the right. N(1) = 1 N(1/2) = 3 N(1/4) = N((1/2)2) = 9 = 32 N(1/8) = N((1/2)3) = 27 = 33 and in general N((1/2)n) = 3n. Here is the Log-Log plot to estimate the box-counting dimension of the gasket. In this case, the pattern is simple enough that we can find the exact value of the dimension.

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