Fractal Dimension of Cantor Set
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Theorem
The Cantor set is a fractal with similarity dimension of $\dfrac {\ln 2} {\ln 3}$.
Proof
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.
$\blacksquare$
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Cantor set
- 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): Cantor set
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): fractal