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!

• And did you try to make more iterations to see what happens? Mar 12, 2022 at 1:45
• You need more iterations and a $<$ not an $=$ desmos.com/calculator/hvrhuvaue5 Mar 12, 2022 at 17:36
• David P : This is exactly what I was looking for! Just a question : Can you please explain the "commas and periods" used in these functions? Thanks! Mar 13, 2022 at 18:01
• David P: I posted a follow-up question over here based on the excellent work you have done math.stackexchange.com/questions/4402462/… - could you please check it out if you have time? Thank you! Mar 13, 2022 at 18:17
• José C Ferreira : It would appear as though more iterations was the answer! Mar 13, 2022 at 18:17

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.