Box-Counting Dimension
Practice Problem 4 Answer
We can cover this fractal with 6 boxes of side 1/3. Self-similarity guarantees we need 36 boxes of side 1/9, and in general, 6
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
6
3
.477
.778
1/9
36
9
.954
1.556
1/27
216
27
1.431
2.334
1/81
1296
81
1.908
3.113
1/243
7776
243
2.386
3.891
Here is the
log-log plot
.
Here we compute the
exact value
of the dimension.
Return to
Box-Counting Dimension Practice Problems
.