Well the question is as simple as that and what I really want to see is if there is a mathematical proof that can tell whether every fractal can be constructed as with an iterative sequence of one underlying shape. A reference will ve very helpful.

PS: One way to try is to consider the box counting method which gives a formula for the dimension and it seems, by reversing the argument, that any dimension (a non negative number) can also be written in terms of that formula. Am I correct?