2. D. The Moran Equation

The similarity dimension equation can be applied only when the all the pieces are scaled by the same amount.

Yet many self-similar fractals are made of pieces scaled by different amounts. Here we learn to compute the similarity dimension of these more general self-similar fractals.

First, here is an example of a self-similar fractal whose dimension we can't compute from the similarity dimension formula. Derivation of the Moran equation. We re-express the similarity dimension formula in a way that allows us to compute dimensions of fractals made of different size pieces. Though it may not be obvious from its form, the Moran equation has a unique solution. The proof of this uses a small amount of calculus. Here are some examples of solving the Moran equation. We give a numerical approach that always works, and an abstract approach that works for a special class of fractals. Exercises in computing the similarity dimension using the Moran equation.