Box-Counting Dimension of the Cantor Set

The Cantor Middle Thirds set is constructed by

To compute the box-counting dimension of the Cantor set, we cover it with smaller and smaller boxes, taking the box scaling based on the natural size structure of the fractal. We find the values shown in the tsble on the right.

N(1/3) = 2 N(1/9) = N((1/3)2) = 4 = 22 N(1/27) = N((1/3)3) = 8 = 23 and in general N((1/3)n) = 2n.

The pattern is simple enough that we can find the exact value of the dimension.

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