Definition:Similarity Dimension
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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
- 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