| The straightforward approach may appear to run into some trouble. Specifically,
how can we simplify |
| Log(3n + 2n) |
| when the Log does not behave nicely with respect to sums? |
| The trick is to turn the sum into a product: |
| 3n + 2n = 3n(1 + (2/3)n) |
| With this, the calculation is fairly straightforward. |
| db |
= limn→∞ Log(N((1/2)n)) / Log(1/((1/2)n)) |
| | = limn→∞Log(3n + 2n) / Log(2n) |
| | = limn→∞Log(3n(1 + (2/3)n)) / Log(2n) |
| | = limn→∞(Log(3n) + Log((1 + (2/3)n)) / Log(2n) |
| | = limn→∞Log(3n) / Log(2n) |
| | because limn→∞Log((1 + (2/3)n) = Log(1) = 0 |
| | = limn→∞(n⋅Log(3)) / (n⋅Log(2)) |
| | = Log(3)/Log(2)
| |
| Note this is the larger of the box-counting dimension of the line segment (db = 1)
and the Gasket (db = Log(3)/Log(2)). |