Julia plots are Beautiful and Interesting.
See : https://postimg.cc/gallery/QHcFVXN

Plotted on the complex plane, each Julia is specific to a complex C.
If for any complex Z, the magnitude of { Z = [ ( Z + C ) squared ]
iterated n times } does not exceed 2 , then Z is a point in the Julia
for C at n iterations.

If a Julia contains the origin and is connected, then C is part of the
Mandelbrot set.
See : https://postimg.cc/gallery/YqLphGg

The Mandelbrot appears to be a well defined figure with apparent
borders or boundaries MBT-1 . Yet when one zooms in on a border or
boundary, there is an ongoing riot of complexity on all scales, MBT-1,
-2, -3, -4 . Very interesting.

On Sunday, July 30, 2023 at 10:00:39 PM UTC+2, casagi...@optimum.net wrote:
> Julia plots are Beautiful and Interesting.
> See : https://postimg.cc/gallery/QHcFVXN
>
> Plotted on the complex plane, each Julia is specific to a complex C.
> If for any complex Z, the magnitude of { Z = [ ( Z + C ) squared ]
> iterated n times } does not exceed 2 , then Z is a point in the Julia
> for C at n iterations.
>
> If a Julia contains the origin and is connected, then C is part of the
> Mandelbrot set.
> See : https://postimg.cc/gallery/YqLphGg
>
> The Mandelbrot appears to be a well defined figure with apparent
> borders or boundaries MBT-1 . Yet when one zooms in on a border or
> boundary, there is an ongoing riot of complexity on all scales, MBT-1,
> -2, -3, -4 . Very interesting.

Is there a function that maps the unit circle to hyperbolic components iteratively?
So applying the function to the unit circle yields the hyperbolic component of
order 1 (the main cardioid), applying it again yields hyperbolic component of order 2,
the circle with radius 1/4 at (-1,0), applying it again yields hyperbolic components
of order 3, etc..

Here you can see the hyperbolic components of orders {1,2,3} in an interactive
circle inversion (drag the midpoint in the graph to reposition it or drag the slider
on the left side to enable optional rotations):
https://www.desmos.com/calculator/vrzu5fv4na

Kind of similar to this method of generating the Julia set:
https://www.desmos.com/calculator/0qq5my21nu