Box-Counting Dimension

Practice Problem 2 Answer

We can cover this fractal with 4 boxes of side 1/3. Self-similarity guarantees we need 16 boxes of side 1/9, and in general, 4ⁿ boxes of side 1/3ⁿ.

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 4 3 .477 .602

1/9 16 9 .954 1.204

1/27 64 27 1.431 1.806

1/81 256 81 1.908 2.408

1/243 1024 243 2.386 3.010

r_n	N(r_n)	1/r_n	Log(1/r_n)	Log(N(r_n))
1	1	1	0	0
1/3	4	3	.477	.602
1/9	16	9	.954	1.204
1/27	64	27	1.431	1.806
1/81	256	81	1.908	2.408
1/243	1024	243	2.386	3.010

Here is the log-log plot.

Here we compute the exact value of the dimension.

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