We need to rewrite the similarity dimension formula 
d_{s} = 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 r_{i}. 
Writing d = d_{s}, 
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⋅r^{d} 
and so 
1 = r^{d} + ... + r^{d} 
where we have one r for each of the N copies of the fractal in the decomposition. 
Replacing each copy of r with r_{i}, we see the similarity
dimension d must satisfy 
1 = r_{1}^{d} + ... + r_{N}^{d}. 

This is the Moran equation. 
So long as each of the r_{i}
satisfies 0 < r_{i} < 1, in the next section we see that the Moran equation has a
unique solution, and that
solution is the similarity dimension d = d_{s}. 