Box-Counting Dimension of a Gasket
From the relation N((1/2)
n
) = 3
n
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→∞
Log(N(r
n
)) / Log(1/r
n
)
=
lim
n→∞
Log(N((1/2)
n
)) / Log(1/((1/2)
n
))
=
lim
n→∞
Log(3
n
) / Log(2
n
)
=
lim
n→∞
(n⋅Log(3)) / (n⋅Log(2))
=
Log(3) / Log(2)
≈ 1.58996
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Box-Counting Dimension of a Gasket
.