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/9411238INTERNAL 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 2

^{N} - 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 2

^{X} - 1 different variations encoded in the shells.

These 2

^{X} - 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.