Mandelbrot Set Equation

Question

Is there a graphable equation that graphs the Mandelbrot Set? It seems like an interesting design, but I want to find a way to show all of the details via a graphing calculator.

Answer 1

The Mandelbrot set is much too complicated to just graph using a graphing calculator. Since its details get smaller and smaller until they are infinitely small, there is no graphable equation for the Mandelbrot set.

Answer 2

See: Wikipedia article. A typical color graph I believe is formed from taking iterations of the polynomial $P_z(x) = x^2 + z$ at $0$, so that output lies in $\Bbb{C} \cup \{ \infty\}$. If it converges, the output of our function $f$ will be $f(z) \in \Bbb{C}$, and if it diverges $f(z)$ will be set to $\infty$. Then you need a map from the output space to a finite set of colors $C = \{c_1, \dots, c_n\}$. This can be done in a number of ways, one such being taking the magnitude $x = |f(z)| \in \Bbb{R}$, then taking the inverse tan of $x$ then mapping the range $[0, \pi/2]$ to $C$.

The size of $C$ will limit the number of iterations of $P_z(x)$ that are required to find what color in $C$ the output would be mapped to if you can take the iterations infinitely.

What you want to prove is that for every bounded subset $S \subset \Bbb{C}$, there is a minimal number $N_S \in \Bbb{N}$ such that the graph of $f(z) = P_z^N(0) = P_z\circ \cdots \circ P_z(0)$ ($N$ times), on the domain $S$ equals the graph of the infinitely iterated function. Then the function that you're graphing, for all intents and purposes has a polynomial expression (i.e. the output colors are equal on a finite grid).

The bounded domain comes from the fact that when you usually view a graph, you view a boxed region of the $\Bbb{C}$ plane with the points colored with the function output values.