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, 5ⁿ 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 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

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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