Box-Counting Dimension
Practice Problem 1 Answer
We can cover this fractal with 5 boxes of side 1/3. Self-similarity guarantees we need 25 boxes of side 1/9, and in general, 5
n
boxes of side 1/3
n
.
   
From this, we fill in the table
r
n
N(r
n
)
1/r
n
Log(1/r
n
)
Log(N(r
n
))
1
1
1
0
0
1/3
5
3
.477
.699
1/9
25
9
.954
1.398
1/27
125
27
1.431
2.097
1/81
625
81
1.908
2.796
1/243
3125
243
2.386
3.495
Here is the
log-log plot
.
Here we compute the
exact value
of the dimension.
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Box-Counting Dimension Practice Problems
.