Mandelbrot Set

Definition

The Mandelbrot set $\mathcal{M}$ is the set of complex numbers $c$ for which the orbit of $0$ under iteration of $f_c(z) = z^2 + c$ remains bounded:

Equivalently, $c \in \mathcal{M}$ if and only if $|z_n| \leq 2$ for all $n$, where $z_0 = 0$ and $z_{n+1} = z_n^2 + c$.

Key Properties

• Connectedness: The Mandelbrot set is connected (Douady and Hubbard, 1982). This deep result means it is a single "piece" despite its intricate boundary.
• Self-similarity: Miniature copies of the entire set appear at every scale along the boundary — the same self-referential structure formalised by Iterated Function SystemsIterated Function Systems$$A = \bigcup_{i=1}^{N} f_i(A)$$Constructing fractals via contractive affine transformationsRead more →.
• Boundary complexity: The boundary of $\mathcal{M}$ has Hausdorff dimensionHausdorff Dimension$$d = \frac{\log N}{\log (1/r)}$$Measuring the fractional dimension of self-similar setsRead more → 2 (Shishikura, 1998).

Escape-Time Algorithm

For each pixel (corresponding to a value of $c$), iterate $z_{n+1} = z_n^2 + c$ starting from $z_0 = 0$. If $|z_n| > 2$ for some $n$, the point escapes and is colored by $n$. Otherwise, color it black (assumed to be in $\mathcal{M}$).

Connections

The boundary of the Mandelbrot set has Hausdorff DimensionHausdorff Dimension$$d = \frac{\log N}{\log (1/r)}$$Measuring the fractional dimension of self-similar setsRead more → equal to 2. The filled Julia sets for each $c$ can be viewed as attractors of Iterated Function SystemsIterated Function Systems$$A = \bigcup_{i=1}^{N} f_i(A)$$Constructing fractals via contractive affine transformationsRead more →.