I may have misspoken - or more likely don't understand. Rather than saying the entire Mset is a single line, I probably should have said that the boundary line (or shoreline) should be a single line if MLC holds. There is no mention of multiple boundary lines or shorelines that I remember. Let's say that the fractal is the lake. My understanding of MLC is that there can be no islands in the M-lake, since that would bring into play more than one shoreline. Here is a quote from the article: *************** In the Orsay notes, Douady and Hubbard proved several major theorems that were motivated by the computer images they’d seen. *They showed that the Mandelbrot set was connected — that you can draw a line from any point in the set to any other without lifting your pencil*. Mandelbrot had initially suspected the opposite: His first images of the set looked like one big island with lots of little ones floating in a sea around it. But later, after seeing higher-resolution pictures — including ones that used color to illustrate how quickly equations outside the set flew off to infinity — Mandelbrot changed his guess.* It became clear that those little islands were all connected by very thin tendrils.* The introduction of color “is a very mundane thing, but it’s important,” said Søren Eilers of the University of Copenhagen. **************** And here is another: ***************** Density of hyperbolicity deals with the Mandelbrot set’s interior. But* MLC would also enable mathematicians to assign an address to every point on the set’s boundary.* “It gives a name to every dot. And then, once you have been able to name every dot of the boundary of the Mandelbrot set, you can hope to really understand it completely,” Hubbard said. ****************** It seems to me that naming every point on the set's boundary strongly implies that there is indeed only one boundary. If there is an island in the fractal lake (or outside it?) that would introduce another unbroken boundary. Obviously, I am struggling to understand some of the concepts and theories, but am fascinated nonetheless. On Tue, Jan 30, 2024 at 6:44 PM Timothy Wegner <tim@tswegner.net> wrote:
Posting for Lee Skinner again (having trimmed a lot of the quoted text from the thread):
Assuming that the Mandelbrot Set is black on a white background, I'm talking about surround a white space, not a black midget. I don't think that the Set cannot be a single line, as it also goes through midgets (areas) where multiple lines depart. Also, does the line to the west of the Mandelbrot Set include a limit, or does it approach a limit? I think it is the former, a truly calculated point, but I'm not sure. Lee
Getting in over my head here, but it seems to me that if the MLC is true
On 1/30/2024 1:00 PM, Bill Jemison wrote: -
and my understanding of the article is they are close to saying so - then it would be impossible to surround an interior lake, since the entire set would be a single line.
Bill
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