We need to rewrite the similarity dimension formula
ds = Log(N)/Log(1/r)
so the scaling factors (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
dLog(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
| 1 = r1d + ... + rNd. |
This is the Moran equation.
So long as each of the ri satisfies 0 < ri < 1, the Moran equation has a unique solution, and that solution is the similarity dimension d = ds.
Return to Moran Equation.