Mandelbrot contours

Question

I'm not a mathematician so please forgive me if I have a poor understanding of some math concept.

I see a lot of tutorial on how to generate fractals and Mandelbrot images from a pixel-wise perspective but I would like to know if it is possible to find some kind of parametric equation to see contours of image portion with the same color.

Let's explain with images because I understand that I can't find the right words, given this image taken from wikipedia section on Mandelbrot:

Is it possible to find out some kind of equation for my poor-hand-drawn green lines?

What is the most appropriate name for this? Contour? Isoline?

Answer 1

The way those colors are given to the Mandelbrot set is to count the number of iterations of the Mandelbrot map it takes to get to a certain magnitude (which seems to be $2$ in this case). That is, if we define the map $f_c(z) = z^2 + c$, then for each point $c \in \mathbb{C}$ we determine how many iterations $f^{(n)}_c(0) = f_c(f_c(\cdots f_c(0)\cdots))$ it takes before $\lvert f_c^{(n)}(0) \rvert \geq 2$.

The outer circle consists of those points on the boundary where this occurs on the first iteration, i.e. those points $c$ with $\lvert c \rvert = 2$. This is the circle with radius $2$.

The next contour is when the first iteration yields points of magnitude $2$, i.e. when $\lvert f_c(f_c(0)) \rvert = \lvert c^2 + c \rvert = 2$. Writing $c = x + iy$ we see that $f_c^{(2)}(0) = (x^2 - y^2 + x) + i (2xy + y)$, and so in real coordinates the second contour consists of those points satisfying $$ (x^2 - y^2 + x)^2 + (2xy + y)^2 = 2^2.$$

One could continue to write down the successive contours in this (implicit) fashion.

For example, if we write the real and imaginary components of $f_c^{(2)}(0)$ as $x_2 = x^2 - y^2 + x$ and $y_2 = 2xy + y$ (continuing to write $c = x + iy$ as above), then we see that $$f_c^{(3)}(0) = f_c^{(2)}(0)^2 + c = (x_2 + iy_2)^2 + (x + iy) = (x_2^2 - y_2^2 + x) + i(2x_2 y_2 + y).$$ And thus writing $f_c^{(3)}(0) = x_3 + iy_3$, we see that $x_3 = x_2^2 - y_2^2 + x$ and $y_3 = 2x_2y_2 + y$. This continues iteratively.

This is the logic behind the plots of the boundaries in this desmos plot.

The resulting equations are explicit but quickly untenable.

Answer 2

These curves are called boundaries of level sets, equipotential lines or lemniscates

Definition :

$L_n= \{c: \operatorname{abs}(z_n)=ER \} \,$

where

• $ER\,$ is Escape Radius, bailout value, radius of circle which is used to measure if orbit of $z_0\,$ is bounded
• $z, c$ are complex numbers (points of 2D planes )
• $z$ is point of dynamical plane ( z-plane)
• $c$ is point of parameter plane ( c-plane) $c = x + y*i $