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 8/31/2023 12:56 PM, casagiannoni@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.

Plot DLA on a fractal for fun. :^)

here is one of my examples:

https://www.facebook.com/photo/?fbid=1023598555465809&set=pcb.1023598748799123

:^)

casagiannoni@optimum.net used his keyboard to write :
> 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.

Do Julia Sets have 180 degree rotational symmetry?

On Thursday, August 31, 2023 at 1:06:17 PM UTC-7, FromTheRafters wrote:
> casagi...@optimum.net used his keyboard to write :
> > 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.
>
> Do Julia Sets have 180 degree rotational symmetry?

The 180 degree rotational symmetry of THESE Julia sets is a consequence of the functions used to generate them:

f(z) = (z + c)^2

for various complex c. These are even functions, so that f(z) = f(-z). So if a particular z escapes, so will -z.
And for the complex numbers, multiplication by -1 is the same as rotating by 180 degrees.

For non-even functions, say,
f(z) = (z + c)^2 + z
or
f(z) = (z + c)^3,

we won't get that same symmetry

On 8/31/2023 2:04 PM, FredJeffries wrote:
> On Thursday, August 31, 2023 at 1:06:17 PM UTC-7, FromTheRafters wrote:
>> casagi...@optimum.net used his keyboard to write :
>>> 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.
>>
>> Do Julia Sets have 180 degree rotational symmetry?
>
> The 180 degree rotational symmetry of THESE Julia sets is a consequence of the functions used to generate them:
>
> f(z) = (z + c)^2
[...]

Yup, 2-ary, add pi to a point wrt polar form, we get the the opposite,
its fun and useful.

On 8/31/2023 2:04 PM, FredJeffries wrote:
> On Thursday, August 31, 2023 at 1:06:17 PM UTC-7, FromTheRafters wrote:
>> casagi...@optimum.net used his keyboard to write :
>>> 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.
>>
>> Do Julia Sets have 180 degree rotational symmetry?
>
> The 180 degree rotational symmetry of THESE Julia sets is a consequence of the functions used to generate them:
>
> f(z) = (z + c)^2
>
> for various complex c. These are even functions, so that f(z) = f(-z). So if a particular z escapes, so will -z.
> And for the complex numbers, multiplication by -1 is the same as rotating by 180 degrees.
>
> For non-even functions, say,
> f(z) = (z + c)^2 + z
> or
> f(z) = (z + c)^3,
>
> we won't get that same symmetry

Reverse iteration is the way to go. It creates a lot of information.