Making the Mandelbrot Fractal in Desmos Online Graphing Calculator I would like to make the following "animation" of the Mandelbrot Set using the Online Desmos Graphing Calculator app - as seen over here: https://www.youtube.com/watch?v=naqgsOOEHJs

I couldn't clearly see the equations being used in this video, so I followed the instructions from link provided in the references ( https://www.youtube.com/watch?v=P23UI9cPCQk&t=0s). I manually entered the 8 equations from the video into the Desmos Graphing Calculator:

However, not only does the graph (set of equations) I have made not look like the first graph - when I zoom into the graph I made, I don't see any repeating fractal patterns.

*

*Can someone show me what I am doing wrong?


*What can I do to fix this, so that I also get a fractal pattern I can see through zooming?


*If possible, could someone try to make a Mandelbrot Set on Desmos that shows "zoom fractals" and post the link to the graph on Desmos?
Thank you!
 A: I use to play with the iterated sequence
$$z_0=0+i0, \quad c=c_1+ic_2,\qquad \quad z_{n+1}=z_n^2+c,$$ and drawing the approximation to the Mandelbrot set, in which $|z_n|\leq r$, to some $r>0$. Here you find how to draw some things with Phyton.
If you write $z=x+iy$, you can see that $$z_n^2+c=(x_n^2-y_n^2+c_1)+i(2x_ny_n+c_2).$$
This gives you a glimpse to define the function
$$f(x,y)=(x^2-y^2,2xy)=(f_1(x,y),f_2(x,y))$$ and play with things like
$$f(f(x,y))+(x,y)=(f_1(x,y)^2-f_2(x,y)^2+x,2f_1(x,y)f_2(x,y)+y),$$ that can be written as $$g_{n+1}(x,y)=f(g_{n}(x,y))+(x,y),$$ with  $$g_1(x,y)=f(x,y).$$ Then you can draw the implicitly curve $|g_n(x,y)|=r$ as you can see here in Desmos.
With the implicit given curves in mind you can use your preferred programing language to draw the curves and animate it.
You can see here the second part of the video you mention in your question.
An interesting discussion on how to parametrize the boundary of the Mandelbrot set that I found on SearchOnMath is this thread.
