2. C. Similarity Dimension

Box-Counting Dimension Patterns

For self-similar fractals such as the Koch curve and the Sierpinski gasket, the box-counting dimension is easiest to compute if the box sizes are taken to be powers of the scaling factor, 1/3 for the Koch curve and 1/2 for the Sierpinski gasket.

For example, we observed the gasket can be covered with

3ⁿ boxes of side length 1/2ⁿ,

so the box-counting dimension computation is

d_b = lim_{n -> infinity}Log(3ⁿ)/Log((2ⁿ))

= lim_{n -> infinity}(nLog(3))/(nLog(2))

= Log(3)/Log(2).

Note n cancels out of the computation. This is not an accident, but a consequence of the self-similar scaling.

This observation can be exploited to simplify the computation of the dimension, in the case of self-similar fractals.

Return to Similarity Dimension.

d_b	= lim_{n -> infinity}Log(3ⁿ)/Log((2ⁿ))
	= lim_{n -> infinity}(nLog(3))/(nLog(2))
	= Log(3)/Log(2).