Box-Counting Dimension of a Gasket

From the relation N((1/2)ⁿ) = 3ⁿ we can compute the exact value of d_b for the gasket.

d_b = lim_{r_n->0}Log(N(r_n)) / Log(1/r_n)

= lim_n->infinityLog(N(r_n)) / Log(1/r_n)

= lim_n->infinityLog(N((1/2)ⁿ)) / Log(1/((1/2)ⁿ))

= lim_n->infinityLog(3ⁿ) / Log(2ⁿ)

= lim_n->infinity(nLog(3)) / (nLog(2))

= Log(3) / Log(2) = 1.58996 ... .

Return to Box-Counting Dimension of a Gasket.

d_b =	lim_{r_n->0}Log(N(r_n)) / Log(1/r_n)
	= lim_n->infinityLog(N(r_n)) / Log(1/r_n)
	= lim_n->infinityLog(N((1/2)ⁿ)) / Log(1/((1/2)ⁿ))
	= lim_n->infinityLog(3ⁿ) / Log(2ⁿ)
	= lim_n->infinity(nLog(3)) / (nLog(2))
	= Log(3) / Log(2) = 1.58996 ... .