| We need to rewrite the similarity dimension formula |
| ds = Log(N)/Log(1/r) |
| so the scaling factors of each piece (each is r in the cases to which this formula can be applied)
can be separated from one another. Then we could change the
individual values of r into different values ri. |
| Writing d = ds, |
| d = Log(N)/Log(1/r) |
| can be rewritten as |
| d⋅Log(1/r) = Log(N) |
| Pulling the d inside the Log |
| Log((1/r)d) = Log(N) |
| and exponentiating both sides |
| (1/r)d = N |
| That is, |
| 1 = N⋅rd |
| and so |
| 1 = rd + ... + rd |
| where we have one r for each of the N copies of the fractal in the decomposition. |
| Replacing each copy of r with ri, we see the similarity
dimension d must satisfy |
|
| This is the Moran equation. |
| So long as each of the ri
satisfies 0 < ri < 1, in the next section we see that the Moran equation has a
unique solution, and that
solution is the similarity dimension d = ds. |