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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.

Jason Chen
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    Please go into detail on how you define "graphing a set". – Daniel Donnelly May 08 '14 at 05:51
  • I seriously doubt a graphing calculator can do much more than a very rudimentary visualization of the Mandelbrot set in a reasonable amount of time, given the typical processing power of such a device. There may exist some cleverly optimized implementations for specific calculator models, but it's unlikely that a naive approach will be computationally efficient. – heropup May 08 '14 at 06:24
  • @EnjoysMath, I mean graphing the equation of the Mandelbrot Set, not really graphing a set. – Jason Chen May 08 '14 at 20:26

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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.

Daniel Donnelly
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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.