Definition:Similarity Dimension

Definition

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:

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

Examples

Cantor Set

The similarity dimension $D$ of the Cantor set is given by:

$D = \dfrac {\ln 2} {\ln 3}$

Closed Unit Interval

The similarity dimension of the closed unit interval:

$I = \closedint 0 1$

is $1$.

Koch Snowflake

The similarity dimension $D$ of the Koch snowflake is given by:

$D = \dfrac {\ln 4} {\ln 3}$

Unit Square

The similarity dimension of the unit square is $2$.

Also see

• Definition:Hausdorff-Besicovitch Dimension

• Results about similarity dimensions can be found here.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): fractal
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): fractal