Box-Counting Dimension of a (Filled-In) Square

Covering a filled-in square with smaller and smaller boxes we see the pattern shown in the table on the right.

N(1) = 1 N(1/2) = 4 = 1/(1/4) = (1/(1/2))2 N(1/4) = 16 = 1/(1/16) = (1/(1/4))2 N(1/8) = 64 = 1/(1/64) = (1/(1/8))2 and so in general N(r) = (1/r)2.

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