Fractal Dimension of Cantor Set

Theorem

The Cantor set is a fractal with similarity dimension of $\dfrac {\ln 2} {\ln 3}$.

Let $C$ denote the Cantor set.

$C$ has the following self-similarities:

$\ds x$

$\mapsto$

$\ds \dfrac x 3$

with scale factor $r_1 = \dfrac 1 3$

$\ds x$

$\mapsto$

$\ds \dfrac 2 3 + \dfrac x 3$

with scale factor $r_2 = \dfrac 1 3$

Thus we have:

$\ds r_1 = r_2$

$=$

$\ds \dfrac 1 3$

$\ds \leadsto \ \ $

$\ds \paren {\dfrac 1 3}^D + \paren {\dfrac 1 3}^D$

$=$

$\ds 1$

Definition of Similarity Dimension: $\paren {r_1}^D + \paren {r_2}^D = 1$

$\ds \leadsto \ \ $

$\ds \paren {\dfrac 1 3}^D$

$=$

$\ds \dfrac 1 2$

$\ds \leadsto \ \ $

$\ds D \ln \dfrac 1 3$

$=$

$\ds \ln \dfrac 1 2$

taking logs of both sides

$\ds \leadsto \ \ $

$\ds D$

$=$

$\ds \dfrac {\ln 2} {\ln 3}$

Hence the result.

$\blacksquare$