Re: Mysteries in the Cubic Mandelbrot set. Please help!
- From: "Ingvar Kullberg" <fractals@xxxxxxxxxxxxxxxx>
- Date: 26 Aug 2006 03:47:09 -0700
Hi Bagula
Sorry I've caused you a nightmare. I, myself,
have also had some dreams about it! I have no
doubt about what the great mathematicians say,
when they declare that every generic Mandelbrot
set (also called "Multibrots") are connected.
What I call "The Great Cubic Mandelbrot Mystery",
for me, is a visual mystery. The world of fractals
is full of mysteries. But the appointed kind
of way the secondary decorations join the cubic
minibrots on these spots is of a kind that don't
exist in the quadratic Mandelbrot set. Sorry for
the bad jpeg-quality in some images in article 24.
I think the mystery is more visible in the very
last image (Figure 8) in the following article 25.
Or the "Little Cubic Mandelbrot Mystery" are more
easy to grasp. "Another zenbuddhistic koan",
article 26, is a somewhat similar mystery. My
articles:
http://klippan.seths.se/fractals/articles
----------------------
Regards,
Ingvar Kullberg
www.come.to/kullberg
Roger Bagula skrev:
Ingvar Kullberg wrote:
Since more than ten years ago, I have wondered aboutI had a dream about this last night... more of a nightmare.
mysteries in the Cubic Mandelbrot set. I call these:
"the Great Cubic Mandelbrot Mystery", and "the Little
Cubic Mandelbrot Mystery". Now I have illustrated these
mysteries in my chaotic series. Go to:
http://klippan.seths.se/fractals/articles/index.html
and click "24) Mysteries in the Cubic Mandelbrot set".
The first one, "the Great Cubic Mandelbrot Mystery",
I have expressed in the following way:
"Now let's have a look at the upper Elephant Valley
(figures 7 - 9, Mystery Zoom 2b1-3) in order to see
were the secondary decorations join themini copy on
this side. In "Mystery Zoom 2b3" (figure 9) we see
that the secondary decorations do not join the Elephants
trunks. Neither they join in the top of any hierarchy
of "extra heads" as in the body side of the Seahorse
Valleys (both in the quadratic and cubic Mandelbrot
sets). In fact there are no extra heads in the Elephant
Valley. Instead, and hear is what I call "the Great Cubic
Mandelbrot Mystery", they join in completely undefined
spots...."
The articles: "25) Other Views of the "Great Cubic
Mandelbrot Mystery", and "26) Another zenbuddhistic
koan" are following up articles.
Anyone in this group who has any entlighed comment
of these mysteries, especially the first one, with
probably no solutions.
I have the feeling that both these mysteries may
have something to do with the fact that the "Cubic
connectedness Locus", the full analogy to the
Mandelbrot set for quadratics, is NOT what the
great mathematicians call "locally connected".
--------------------------
Regards,
Ingvar Kullberg
www.come.to/kullberg
Have I been giving it the credit due?
So I went back and downloaded your articles.
You are going to have to do "more"
if you want people to believe there is any mystery here.
They appear to be as connected as similar quadratic Mandelbrot pictures
for similar regions.
You need some sort of proof like using the Pc polynomials
f[c,1]=c^3+c
on the nth iteration
Limit[f[c,n]-f[c+c0,n],c0->0]<>0
As long as the limit is zero , they are topologically connected?
Roger Bagula
.
- References:
- Mysteries in the Cubic Mandelbrot set. Please help!
- From: Ingvar Kullberg
- Re: Mysteries in the Cubic Mandelbrot set. Please help!
- From: Roger Bagula
- Mysteries in the Cubic Mandelbrot set. Please help!
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