Box-Counting Dimension of the Cantor Set

The Cantor Middle Thirds set is generated by the IFS
T1(x) = x/3   T2(x) = x/3 + 2/3
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. That is, we use boxes of side length 1/3, 1/32, 1/33, ... . We find the values shown in the table 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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