Box-Counting Dimension of the Koch Curve

Covering the Koch curve with smaller and smaller boxes

we see

.

N(1/3) = 3

N(1/9) = N((1/3)2) = 12 = 3*4

N(1/27) = N((1/3)3) = 48 = 3*42

and in general

N((1/3)n) = 3*4n-1

Here is the Log-Log plot to estimate the box-counting dimension of the Koch curve.

In this case, the pattern is simple enough that we can find the exact value of the dimension.

Return to Box-Counting Dimension.