Mandelbrot symmetries


Here's some stuff to try: (symmetries of z^2 in Mandelbrots)
z'=-z^2+c
z'=I*z^2+c
z'=-I*z^2+c
They give rotations by 90 degrees.

If we want an 120 degree rotation:
z'=(-1/2+I*Sqrt[3]/2)*z^2+c

Suppose we want to do resonants to angular rotations:
z'=Exp[k*I*Arg[z]]*z^2+c
Since ( this polar coordinate method is as some of you know a pretty fundamental way to look at these iterations)
z^2=Abs[z]^2*Exp[2*I*Arg[z]]
one can get a simple cubic like resonance by:
z'=Exp[I*Arg[z]]*z^2+c
Canceling the angular part as ( crescent moon output) :
z'=Exp[-2*I*Arc[z]]*z^2+c=Abs[z^2]+c
An oval output that doesn't look at all fractal at:
z'=Exp[-3*I*Arc[z]]*z^2+c
The next integer is:
z'=Exp[-4*I*Arc[z]]*z^2+c= Conjugate[z]^2+c

Try using a golden mean:
gm=(1+Sqrt[5])/2
z'=Exp[gm*I*Arc[z]]*z^2+c
to get a bent cubic like output.

This is a relatively simple way to look at rotations
as they change how a Mandelbrot type iteration goes on.
What can be learned from this line is that the angular part of the Mandelbrot is the tale that wags the dog.
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