Computer Creativity
Iterated Function Sequences
Let $f:\Bbb C\rightarrow \Bbb C$ be some complex valued function.
The $n^{\text{th}}$ iterate of $f$, denoted by $f_(n)$, when applied to $z\in\Bbb C$ is the complex number, denoted by $z_n$, that results by applying $f$ repeatedly (to the initial value $z_0$). Thus:
\[f_{(n)}(z)=\begin{cases}z & \text{if }n=0\\f\left(f_{n-1}(z)\right)&\text{if }n>0\end{cases}\]Escape Radius
The complex plane ca be divided into two disjoint parts relative to some parameter $r$ called the escape radius of $f$:
\[\begin{aligned} \text{in}(f)&=\{z\in\Bbb C:\forall n\vert f_{(n)}(z)\vert\leq r\}\\ \text{out}(f)&=\{z\in\Bbb C:\exists k\vert f_{(f)}(z)\vert>r\} \end{aligned}\]The escape radius can often be set as $r=2$.
Orbits
An orbit is a sequence that loops back on itself:
\[f(z_k)=z_0;f(z_j)=z_{j+1}\forall 0\leq j<k\]Popular Sets
Interesting patters are generally of the form:
\[f(z)=z^2+c;c\in\Bbb C\]The Mandelbrot Set classifies those $c$ for which:
\[\vert f_{(n)}(0)\vert\leq 2;\forall n\geq0\]If $c$ is fixed the subset (using radius 2) $\text{in}(z^2+c)$ is called a Julia Set while the set $\text{out}(z^2+c)$ is called the Fatou Set.
Computing and Displaying Sets
Iterating a fixed number of times gives a good enough approximation.
The colour mapping of these sets often represent the number of times required to iterate before a point exceeds the escape radius.
Fractal Dimension
In Euclidean dimension, the dimension is based on whole numbers. Fractal sets have a dimension that is in the rationals ($\Bbb Q$).