Category:Definitions/Similarity Dimensions

This category contains definitions related to Similarity Dimensions.
Related results can be found in Category:Similarity Dimensions.

Let $S$ be a self-similar fractal embedded in a space of dimension $d$.

Then $S$ can be assigned a similarity dimension $D$, such that $0 \le D \le d$ as follows:

Let there be $N$ similarity mappings with scale factors $r_1, r_2, \ldots, r_N$ that map $S$ to itself.

Then $D$ satisfies the equation:

$\paren {r_1}^D + \paren {r_2}^D + \cdots + \paren {r_N}^D = 1$