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, 6ⁿ 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 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

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.