| We select squares of side lengths 1/3, 1/9,
1/27, ... , 1/3n, ... noting the scaling
of the Cantor set. |
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| N(1/3) = 2 + 3 - 1 |
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N(1/32) = 22 + 32 - 1 |
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N(1/33) = 23 + 33 - 1 |
|
| and in general |
| N(1/3n) = 2n + 3n - 1 |
| |
| Knowing N(1/3n) we can compute the box-counting dimension by
factoring out the larger term from the sum: |
| d | = limn→∞Log(N(1/3n)) / Log(1/(1/3n)) |
| = limn→∞Log(2n + 3n - 1) / Log(3n) |
| = limn→∞Log(3n(1 + (2/3)n - (1/3)n)) / Log(3n) |
| = limn→∞(Log(3n) + Log((1 + (2/3)n - (1/3)n))) / Log(3n) |
| = 1 + limn→∞Log((1 + (2/3)n - (1/3)n))) / n⋅Log(3) |
| = 1 |
|