Pathfinding in the Mandelbrot set - Revisited

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Offline wbarry

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« on: July 05, 2019, 05:38:27 PM »
In reference to some discussions from the old forum:
http://www.fractalforums.com/fractal-related-links/inner-workings-of-the-mandelbrot-set-1/
http://www.fractalforums.com/mandelbrot-and-julia-set/pathfinding-in-the-mandelbrot-set/
http://www.fractalforums.com/mandelbrot-and-julia-set/shortcuts-in-juliasets/

I would like to propose another way of describing what appears to be happening, as I understand it.
No math, just discussion.
Some of the details may need adjustment, but I believe the core idea is on the mark.
I am fairly certain this is part of what is being shown in this paper by Dierk Schleicher (referred to in the earlier posts mentioned above).
https://arxiv.org/abs/math/9411238
INTERNAL ADDRESSES OF THE MANDELBROT SET AND GALOIS GROUPS OF POLYNOMIALS

The Mandelbrot set can be thought of as a heirarchical network of mini-brots.
For each mini-brot, there is a "most efficient path" to the mini-brot from the center of the set, passing through other mini-brots along the way.
Brot to brot to brot ...
Each mini-brot is surrounded by a set of concentric "shells".
The "most efficient path" to any mini-brot is visually encoded into the shells surrounding it.

The shells contain an exact "history" of what shapes are traversed getting to the mini-brot when starting from the center of the Mandebrot Set - a "visual record" of the exact shapes traversed along the most efficient path.
New shells are created whenever the path includes a cross-road where a decision needs to be encoded.
The encoding is done in a "combinatorial" fashion, meaning new moves are spliced into the existing sequences, doubling the number of distinct shell patterns each move.
The first moves are on the outside, the latest moves appear on the inside (the deep end).
When trying to understand morphing, it is essential to imagine building "from the inside out".

From Schleicher, p.6 - "The internal address of c \( \in \) M can be viewed as a road map description for the way from the origin to c in the Mandelbrot set:
the way begins at the hyperbolic component of period 1, and at any intermediate place,
the internal address describes the most important landmark on the remaining way to c"

Each Move creates a set of shell patterns.
The "encoding" of the Move is a recording of each object that was passed through and these recordings are the patterns in each shell.
If N is the number of Moves to reach a mini-brot using the most efficient path, then the mini-brot will be surrounded by 2N - 1 (I think) distinct shell patterns.

From Schleicher, p.14
"The following result complements the interpretation of internal addresses as road
descriptions by saying that whenever the path from the origin to a parameter c \( \in \) M
branches off from the main road, an entry in the internal address is generated: the way
to most parameters c \( \in \) M traverses infinitely many hyperbolic components, but most
of them are traversed “straight on” and left into the 1/2-limb.
"

If there are X moves in the most efficient path, then there will be 2X - 1 different variations encoded in the shells.
These 2X - 1 shell variations will all be shown for 2 arms, then repeated with 4 arms, then 8 arms, then 16, and so on for the powers of 2.

Each move adds a pattern to EACH shell that existed around the node from the previous move.
Each mini-brot that is descended from the same path has some of the same encoded shells.
Because we are zooming, we never leave where we came from, we just look deeper into the structure of the area we were already in, so prior path encodings must be retained - you still need to travel through those parts to reach the mini-brot.

Because these shells are added in this mathematically regular manner, zooming to 3/4 depth finds the magical shell with 2 arms (perhaps halfway through the new stack?) - the next step in the Julia Morph.
Same for the 4 and 8 armed automated zooms being at 7/8 and 15/16.
Those shells are always located at the same depth in the stack.

How This Relates to Julia Morphs
I think that what we call a "Julia Morph (Evolution)" is the last two-arm shell in the current stack, which contains all of the moves up to that point.
When trying to do Julia Morphs in the past, I was always trying to "grow an arm" and the results would surprise and confound me.
But now that I understand that I am actually morphing the shape from the center out, I have much more control over the process.
Take the path from the current central mini-brot of the current Julia Morph to any new mini-brot you choose.
Insert all of objects contained in the path between the two brots into the center of the current shape, inverted, with the chosen mini-brot as the new center.
This results in the new shape you will find when you reach the proper depth (3/4).

Big-Eye Shortcuts
I think Big-Eye Shortcuts may also be explained by this.
Shortcuts are already at the "newest" shells - they don't need to re-create all the shells that came before.
I think it may relate to the "rotational procession" Claude Helland has shown - where similar shells at different depths are identical, but their rotation is slightly different.
Rotation is most likely a product of the orientation of the final mini-brot - the direction the spike is pointing.
Perhaps for Big-Eyes, the rotation is identical to the central brot, so they are "sharing" the stack up to that point. They are already surrounded by the necessary shells.
But mini-brots with a different final rotation might need to re-encode all of the shells in the new orientation, so the stack needs to start all over.
The math has some variables that cancel out or something.
That might fit the observed behavior.
I thought I was reading something that implied this in Dierk Schleicher's paper, but I can't find it now, so perhaps it was wishful thinking.
My brain hurts.

Construction Tool
My final thought is that this means one can use the set as a Construction Tool.
The path you choose to follow BUILDS the shape you will find.
We just need to learn how to use the tool effectively and how to comprehend the way each move will be encoded - what will the result look like.
The results can be predicted.
Walter Barry
Mandelbrot Evolution

Offline quadralienne

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« Reply #1 on: July 05, 2019, 06:27:43 PM »
Yeah, I do that all the time! Zoom into the cusp of the cardioid, find a minibrot, then zoom into the seahorse valley, find a minibrot, zoom into the ... uh, other side of the seahorse valley, find a minibrot.

That minibrot will have layered cusp-seahorse-otherside patterns around it!

Offline wbarry

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« Reply #2 on: July 05, 2019, 08:29:20 PM »
A few tables of images to demonstrate how Julia Morphs are including the new path in the next evolution.

Offline wbarry

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« Reply #3 on: July 07, 2019, 09:53:36 PM »
Re: "That minibrot will have layered cusp-seahorse-otherside patterns around it!"
Exactly.
And if you examine things carefully, you'll find it records the exact cusp, seahorse, and otherside.
It will have the exact number of turns and branches and bulbs as whatever path you traveled.
It's kind of amazing.
A deeper layer of self-similarity.

Offline Fraktalist

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« Reply #4 on: July 11, 2019, 10:45:01 AM »
oh yeah - here we go. :)
my absolute favourite aspect of the mandelbrot set. absolutely undervalued.
and as usual when this topic comes up: https://www.youtube.com/watch?v=Ojhgwq6t28Y


"The path you choose to follow BUILDS the shape you will find."
Exactly! Every decision you take when zooming is recorded and it will actually create the look of every deeper zoom.
Which is pretty cool in itself.

But this also has stunning implications:
It means you can actually STORE information in the Mset.
See image below. You could choose a 2 branched bulb and call the left arm 1 and the right arm 0. and now you can store binary information by deciding 0 or 1. and this information will be embedded around the next minibrot, forever the basis of your future zoom path.
You could say 01001000 01100101 01101100 01101100 01101111 00100000 01010111 01101111 01110010 01101100 01100100: "Hello World"
Or you could zoom a lot deeper and store all the text written in this thread until now.
Which in turn means: there is a location in the mandelbrot set - right now - a pair of coordinates already exists, which has this whole thread stored into its minibrot shells.
And if you think it through: Also any future posts we haven't yet written. Or every book. Or all books. Or a complete binary simulation of our universe. After all, the mandelbrot set is infinite and has no limit of the amount of information you can store.

It's all already somewhere in the Mandelbrot-Set.

This blows my mind!!

But why choose binary? Let's take a 4 branched bulb and call the branches G A T C.
In the image below I decided to go G T then A and C. And it shows perfectly
And then lets write your whole genome this way.
Not the best or most efficient way to store information, the image will be boring, 3 billion turns into branch 1,2,3 or 4  - but it does the job.
Every minibrot at the end of this will be surrounded with the building plan that can build YOU.

And that location already exists in the mandelbrot-set.
Now.
Has always been in it.
And always will be.


I love that.


« Last Edit: July 11, 2019, 11:20:24 AM by Fraktalist »

Offline marcm200

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« Reply #5 on: July 11, 2019, 11:11:46 AM »
Which in turn means: there is a location in the mandelbrot set - right now - a pair of coordinates already exists, which has this whole thread stored into its minibrot shells.
And if you think it through: Also any future posts we haven't yet written.
Wow! Could that be used as a compression algorithm? How many decimals would that location have when written out exactly? And conversely, what's the text like at easy locations like 10^-300 + i*10^-400 (if there's a minibrot somewhere in the vicinity).

And: if one of the C++ programs I've coded is faulty - the correct version is in the Mandelbrot set - now I just have to find it!  :)

Offline hobold

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« Reply #6 on: July 11, 2019, 02:02:58 PM »
No, it cannot be used as a compression algorithm. The so called "counting argument" applies to compression regardless of algorithm details. In practice this probably means you need to store more bits of precision for the mandelbrot coordinates than you encoded as payload data.

The idea of all past and future texts being present in some infinite library is mind blowing, though. Many versions of it are older than the Mandelbrot set, but they are also more hypothetical than the Mandelbrot set. The first version I personally encountered was a relatively recent one, where some science fiction author estimated the length of a bookshelf containing all possible books that have up to 600 pages of text in the Latin alphabet. It was an unimaginably large number of light years.

Offline wbarry

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« Reply #7 on: July 12, 2019, 03:51:37 PM »
Re: "my absolute favourite aspect of the mandelbrot set. absolutely undervalued."
Yes! It seems like this is one of the keys to the fabric of the Mandelbrot set. And that any "guide" to the set should have this as one of the first things mentioned, yet I felt like I had to discover it for myself (I had not seen your video). "Shape stacking" hints at it, but does not make it clear how deep this goes. I think the mathematicians mention it, but in hyperbolic isomorphic pedantic semantics.
I suppose I should read more and zoom less, if I really want to understand. But zooming is much more fun.

Offline claude

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« Reply #8 on: July 12, 2019, 04:04:17 PM »
"history repeats, only twice as fast" is my take on approaching mini-sets and period doubling shape stacking.

Offline Fraktalist

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« Reply #9 on: July 12, 2019, 11:20:36 PM »
I suppose I should read more and zoom less, if I really want to understand. But zooming is much more fun.
disagree. learning by doing, the pleasure of finding things out - yourself! when you learn something from own experience, the knowledge and understanding is so much deeper and more satisfying!
you found out yourself, just like the first one who discovered it and academically cryptic wrote a paper about a few hundred people will ever read AND understand.
so your achievement is the same. you won't get credits, but you rock! you did it! who cares who was first.
 :horsie:
 :joy:
also, explaining it in your own words, with practical examples can potentially reach far more interested people, no "academic talk barrier".

Keep zooming & exploring!!!

Obligatory disclaimer: I love reading ;)

Offline wbarry

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« Reply #10 on: July 13, 2019, 02:09:02 AM »
Wow, how kind of you! Thanks.

And following that positive trend, I definitely should have shown more appreciation for the people who contributed to all of the articles and guides that I found. There is so much valuable info out there, mostly contributed by fellow enthusiasts who do it just because it is freakin' cool stuff. This forum has been hugely helpful. And I feel so fortunate for the tools we have to explore the set. In my day (get off my lawn!), we had to wait days to get a single image out of the mainframe. Now we have nearly instant images from deep in the set on our personal machines. We live in amazing times.

For a bunch of idiots, people are actually pretty cool.

Now, back to the chimps in the typing room ... they're demanding more cigarettes ...

Offline tavis

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« Reply #11 on: July 13, 2019, 09:51:57 PM »
This also works for the bulbs, not just minibrots. So for example, you can zoom into a bulb of a bulb of a bulb etc and it will store the unique location in the attached tendrils in a sequence, without ever visiting a minibrot. I actually think this approach could be much more efficient for the binary storage idea, requiring far less zooming. But honestly either way it's pretty computationally prohibitive (even though it's a super cool thought exercise)
Check out the Mandelmap poster

Offline wbarry

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« Reply #12 on: July 13, 2019, 11:28:08 PM »
First off, I have given no mathematical proof, so I could totally be wrong.
 
Meanwhile, I think we are actually agreeing.

I am saying that WHEREVER you zoom will be recorded into the fabric of what you will find.
Zoom through three bulbs you will find "shells" of recordings of those three bulbs - as you say, it will be stored.
I am really talking about the storage mechanism.

The shells of recordings are woven around minibrots, which act like the galactic black holes at the center of all galaxies. These central minibrots are surrounded by recordings of the shapes you would pass through to arrive at that specific minibrot, whether you passed through bulbs, spirals, tendrils, or anything. Each minibrot is surrounded by recordings of its own unique path (or address, as in Dierk Schleicher's paper).

The fabric of the set itself is a network of minibrots surrounded by recordings. The recordings, woven into the original, are the fabric, a set of self-similar copies of the path to each exact location. The encoded paths are the fabric of the Mandelbrot set.

Or not. :)

Offline Fraktalist

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« Reply #13 on: July 14, 2019, 02:47:27 AM »


The shells of recordings are woven around minibrots, which act like the galactic black holes at the center of all galaxies.

Welcome to a new episode of "mind blown":
https://www.zmescience.com/science/physics/black-holes-store-information-0023534/
 :awe:



This also works for the bulbs, not just minibrots. So for example, you can zoom into a bulb of a bulb of a bulb etc and it will store the unique location in the attached tendrils in a sequence, without ever visiting a minibrot. I actually think this approach could be much more efficient for the binary storage idea, requiring far less zooming. But honestly either way it's pretty computationally prohibitive (even though it's a super cool thought exercise)

good point.  :yes:
reason is because the "recording" of the  previous path starts at the first bifurcation/doubling, and every branch growing from a bulb is "behind" this first bifurcation.
« Last Edit: July 14, 2019, 03:08:56 PM by Fraktalist »

Offline marcm200

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« Reply #14 on: July 14, 2019, 11:57:18 AM »
I've read that higher degree monomials in the parameter-space go to the unit disk in the limit, so the "fractal nature" somehow decreases (un-mathematically spoken).

Does this recording of paths also apply to other monomials of higher degree or to polynomials in general (or is the z²-Mandelbrot the "master fractal"? And only to the parameter-space or also to the dynamical entity?

Questions over questions...


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