| To emphasize this common pattern, and to give a short-cut for computing the dimension of a self-similar shape, we define the similarity dimension. |
| For a self-similar shape made of N copies of itself, each scaled by a similarity with contraction factor r, the similarity dimension is |
| ds = Log(N)/Log(1/r). |
| To help develop some understanding of what ds means, here are some relatives of the Koch curve. |
| This formula can be generalized to account for the posibility that different pieces are scaled by similarities with different contraction factors. We shall see the more general formula is the Moran equation. |
| The similarity dimension equals the box-counting dimension, but the box-counting dimension is defined for a wider variety of shapes. |
| Then why mention the similarity dimension at all? |
| It is much easier to compute than the box-counting dimension. |
| Comparing the next section's similarity dimension exercises with the box-counting dimension exercises of an earlier section should convince you of that. |
Return to Similarity Dimension.