The MLC (for /Mandelbrot locally connected/) conjecture in the article states that all points in the Mandelbrot Set are continuously connected to one another. I have another conjecture of my own: *There are no closed loops in *(the "filaments" of)*the Mandelbrot Set*, i.e, there are no "/white/ islands", but I am unable to formulate this exactly in formal mathematical terms.  A white island would be an area of space not in the Mandelbrot Set, but completely surrounded by a portion of the Mandelbrot Set. My conjecture says that such white islands do not exist. How do you even define a "visible filament", when it becomes something else entirely (and much more complicated) upon zooming into it?  (Mostly, it is simply an /infinitely/ long segment of the Mandelbrot Set between any two points of the set, however, picking the two end points of a visible segment is also difficult, as zooming into such a point also becomes a frilly design, unless, e.g., it is on the /finite/ straight line west of the Mandelbrot Set.) Lee Skinner On 1/26/2024 11:25 AM, David W. Jones wrote:

The Quest to Decode the Mandelbrot Set, Math’s Famed Fractal

https://www.quantamagazine.org/the-quest-to-decode-the-mandelbrot-set-maths-...

--- David W. Jones gnome@hawaii.rr.com exploring the landscape of god http://dancingtreefrog.com

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I haven’t seen any of them - would love to! On Tue, Jan 30, 2024 at 10:27 AM Timothy Wegner <tim@tswegner.net> wrote:

Folks, Lee Skinner has made several attempts to respond to the thread "So read a little bit about the history of complex dynamics". His posts have made it to the list archives, but I can't see them in my list postings.

If you can't see his posts, I'll post on his behalf. Please let me know one way or the other.

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Tim - I can’t remember how to see the list archives. Could you post a link? I have never figured out how to get my own posts to the list to show on my gmail account, so never really know if the message got there - unless someone replies (rare). Thanks! Bill jemison On Tue, Jan 30, 2024 at 10:27 AM Timothy Wegner <tim@tswegner.net> wrote:

Folks, Lee Skinner has made several attempts to respond to the thread "So read a little bit about the history of complex dynamics". His posts have made it to the list archives, but I can't see them in my list postings.

If you can't see his posts, I'll post on his behalf. Please let me know one way or the other.

Tim _______________________________________________ Fractint mailing list -- fractint@mailman.xmission.com To unsubscribe send an email to fractint-leave@mailman.xmission.com

Lee Skinner has been having trouble posting. Here is his response: The MLC (for *Mandelbrot locally connected*) conjecture in the article states that all points in the Mandelbrot Set are continuously connected to one another. I have another conjecture of my own: *There are no closed loops in *(the "filaments" of)* the Mandelbrot Set*, i.e, there are no "*white* islands", but I am unable to formulate this exactly in formal mathematical terms. A white island would be an area of space not in the Mandelbrot Set, but completely surrounded by a portion of the Mandelbrot Set. My conjecture says that such white islands do not exist. How do you even define a "visible filament", when it becomes something else entirely (and much more complicated) upon zooming into it? (Mostly, it is simply an *infinitely* long segment of the Mandelbrot Set between any two points of the set, however, picking the two end points of a visible segment is also difficult, as zooming into such a point also becomes a frilly design, unless, e.g., it is on the *finite* straight line west of the Mandelbrot Set.) Lee Skinner On Sat, Jan 27, 2024 at 12:37 AM Bill Jemison <bill.jemison@gmail.com> wrote:

You've done it again! What an interesting read you've pointed us to.

The following snippet from the article in particular caught my attention in a rather personal way.

Back in the days when we were getting together on the GraphDev Forum using TapCIS and Compurserve, I was fairly involved with the evolution of the sound feature in Fractint. One discussion in particular (as I remember it, it was with Dan Farmer) had to do with what the fractals that I was using for my audio fractal files looked like.

For one thing, they don't look like they sound <s>. But the other thing is that most are generated using a sample of fewer than 1,000 pixels (some are fewer than 200) from the entire fractal image of usually greater than 1024x768 pixels. The audio files are generated using orbit delay values of usually greater than 500 (depending on processor speed) and maxiter sometimes over 500...i.e. to generate the full image while in audio mode would be very time-consuming. There is the additional other thing that since I am not at all interested in the image, just the sounds, the visual fractal is very often quite uninteresting at best and downright ugly most of the time.

I was asked why not just generate the fractal image "quickly" and then use the image as a sound map for the audio? The answer is that my audio fractals generate a HZ value for each iteration of the pixel calculation - often hundreds of tones per pixel - the "journey" - whereas generating audio from the image produces a single tone per pixel - the "destination" - and in no way sounds either like the audio file nor, IMO, "fractal ". In fact, I'm not sure one could distinguish the structural difference between the audio generated from a fractal image and a photograph of a street scene, landscape or family get together.

Thanks for indulging me - here’s the quote

***************** "(…We distinguish “pure math” from “applied math.”) The way math papers are written doesn’t help: Only the final proofs and theorems are usually published, not the meandering process that led to them."

******************

That hit home. For me, the destination is just the end of the journey, and usually quite anticlimactic. Sort of like life.

Bill Jemison

On Fri, Jan 26, 2024 at 11:26 AM David W. Jones <gnome@hawaii.rr.com> wrote:

...

The Quest to Decode the Mandelbrot Set, Math’s Famed Fractal

https://www.quantamagazine.org/the-quest-to-decode-the-mandelbrot-set-maths-...

...

--- David W. Jones gnome@hawaii.rr.com exploring the landscape of god http://dancingtreefrog.com

Sent from my Android device with F/LOSS K-9 Mail. _______________________________________________ Fractint mailing list -- fractint@mailman.xmission.com To unsubscribe send an email to fractint-leave@mailman.xmission.com

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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 On 1/30/2024 1:00 PM, Bill Jemison wrote:

Getting in over my head here, but it seems to me that if the MLC is true - 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

On Tue, Jan 30, 2024 at 11:20 AM Timothy Wegner<tim@tswegner.net> wrote:

...

Lee Skinner has been having trouble posting. Here is his response:

The MLC (for *Mandelbrot locally connected*) conjecture in the article states that all points in the Mandelbrot Set are continuously connected to one another.

I have another conjecture of my own: *There are no closed loops in *(the "filaments" of)* the Mandelbrot Set*, i.e, there are no "*white* islands", but I am unable to formulate this exactly in formal mathematical terms. A white island would be an area of space not in the Mandelbrot Set, but completely surrounded by a portion of the Mandelbrot Set. My conjecture says that such white islands do not exist.

How do you even define a "visible filament", when it becomes something else entirely (and much more complicated) upon zooming into it? (Mostly, it is simply an *infinitely* long segment of the Mandelbrot Set between any two points of the set, however, picking the two end points of a visible segment is also difficult, as zooming into such a point also becomes a frilly design, unless, e.g., it is on the *finite* straight line west of the Mandelbrot Set.)

Lee Skinner

On Sat, Jan 27, 2024 at 12:37 AM Bill Jemison<bill.jemison@gmail.com> wrote:

...

You've done it again! What an interesting read you've pointed us to.

The following snippet from the article in particular caught my attention in a rather personal way.

Back in the days when we were getting together on the GraphDev Forum using TapCIS and Compurserve, I was fairly involved with the evolution of the sound feature in Fractint. One discussion in particular (as I remember it, it was with Dan Farmer) had to do with what the fractals that I was using for my audio fractal files looked like.

For one thing, they don't look like they sound <s>. But the other thing is that most are generated using a sample of fewer than 1,000 pixels (some are fewer than 200) from the entire fractal image of usually greater than 1024x768 pixels. The audio files are generated using orbit delay values of usually greater than 500 (depending on processor speed) and maxiter sometimes over 500...i.e. to generate the full image while in audio mode would be very time-consuming. There is the additional other thing that since I am not at all interested in the image, just the sounds, the visual fractal is very often quite uninteresting at best and downright ugly most of the time.

I was asked why not just generate the fractal image "quickly" and then use the image as a sound map for the audio? The answer is that my audio fractals generate a HZ value for each iteration of the pixel calculation

...

often hundreds of tones per pixel - the "journey" - whereas generating audio from the image produces a single tone per pixel - the "destination" - and in no way sounds either like the audio file nor, IMO, "fractal ". In fact, I'm not sure one could distinguish the structural difference between the audio generated from a fractal image and a photograph of a street scene, landscape or family get together.

Thanks for indulging me - here’s the quote

***************** "(…We distinguish “pure math” from “applied math.”) The way math papers are written doesn’t help: Only the final proofs and theorems are usually published, not the meandering process that led to them."

******************

That hit home. For me, the destination is just the end of the journey, and usually quite anticlimactic. Sort of like life.

Bill Jemison

On Fri, Jan 26, 2024 at 11:26 AM David W. Jones<gnome@hawaii.rr.com> wrote:

...

The Quest to Decode the Mandelbrot Set, Math’s Famed Fractal

https://www.quantamagazine.org/the-quest-to-decode-the-mandelbrot-set-maths-...

...

--- David W. Jones gnome@hawaii.rr.com exploring the landscape of god http://dancingtreefrog.com

Sent from my Android device with F/LOSS K-9 Mail. _______________________________________________ Fractint mailing list --fractint@mailman.xmission.com To unsubscribe send an email tofractint-leave@mailman.xmission.com

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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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