Fractal Related Discussion > Fractal Mathematics And New Theories

 The basic rules of the Mandelbrot-Set

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Fraktalist:
this discussion caused me to try to break down the most simple rules of how the mandelbrot set 'behaves'.
the rules that form the image on deeper zoom levels.

I think this deserves its own thread and I'll update the following until we actually have a basic set of rules (and wording) that we can all agree upon.

1st level: number of branches (matches the period of bulbs on the main cardiod)


2nd level:
repetition, lenghtening a branch (by zooming towards points further away from the main period bulb - multiply the number of first main bulb period with number of repetitions  to get the actual bulb number - I'll make an image of this which explains it much better)

2nd level:
curvature of branches/spirals (the deeper into the valley on the oposite site of the main bulb, the more curved the branches are into a spiral)

2nd level:
I just noticed: spirals form around Misiurewicz points and the minibrots are appearing symetrical to these middle points.
And the angle of each branch in relation to its main bulb is mirrored in the angle of the further path of the tip. wow, wasn't aware of that before! fascinating! Best visible in the uncurved, non spiraling branches of the main cardiod.

3rd level: bifurcation(shortcut to shapestacking complex zooms - link to dinkydaus(?) thread at ff1.com, I think dinkydau came up with a method) - mirroring around the symetry axis that you choose as your new mid-point in each bifurcation"area"

4th level: minibrots
once you go towards the minibrot, these 3 levels of shapes are stacked and follow a bifurcation pattern (link to the exact relative distances to the next bifurcation at ff.com?) and the minibrot itself will be surrounded by infinite layers of these doubled shapes.

this will repeat for each minibrot you encounter and in the 3rd generation minibrot the whole pattern of the 4th level minibrot will be surounding every point - best visibule in one image in deep julia sets.


(please help with wording and add if anything is missing or wrong or can be boiled down to even simpler rules)
I'll try to make some images that explain each level much better than words can - once I find a bit more time.

Linkback: https://fractalforums.org/index.php?topic=412.0

Adam Majewski:
https://commons.wikimedia.org/wiki/File:Shrub_model_of_Mandelbrot_set_60_10_labelled.png

www.tic.itefi.csic.es/gerardo/publica/Romera03.pdf


HTH

claude:
Periods are very important, as is the connection to external angles/rays.  Here are some of my most-relevant blog posts about exploring such patterns / structures in the Mandelbrot Set:

https://mathr.co.uk/blog/2013-02-01_navigating_by_spokes_in_the_mandelbrot_set.html
https://mathr.co.uk/blog/2013-06-23_patterns_of_periods_in_the_mandelbrot_set.html
https://mathr.co.uk/blog/2013-10-02_islands_in_the_hairs.html
https://mathr.co.uk/blog/2014-11-18_navigating_in_the_hairs.html

Dinkydau:
This is something I also though about many times but I never tried to solve this problem because it seems so hard.

Here are some thoughts:

Like how you can get away with using less precision using perturbation, you can get away with using an inflection mapping as an alternative to zooming deep, up to some degree. If you apply a couple of artificial inflections, imperfections start to show. The formula for inflections is already known. Far away from minibrots it always works, but this is not what it looks like, zooming close to a minibrot:

...so the difficulty must lie in the shape of the Mandelbrot set itself. It's the only thing the inaccuracies can be caused by.

I think it may have something to do with the fact that minibrots are not 2-fold rotational symmetric. In fact, in a Mandelbrot set with power n, the set itself is (n-1)-fold rotational symmetric.

The 3rd power Mandelbrot set being 2-fold rotational symmetric, is it the result of some (infinite) sequence of inflection mappings? I have discovered that the shape of the 3rd power Mandelbrot set can actually be approximated by a julia morphing (and therefore a sequence of inflection mappings) in the 2nd power Mandelbrot set:
https://dinkydauset.deviantart.com/art/Trees-Revisited-657040568
I'm not sure the shape is entirely the same. It could still be a (smooth) deformation. Actually I'm sure it's not exactly the same because the branches/dendrites connected to the bulbs are missing, but there's definitely something going on here. n-fold and (n-1)-fold rotational symmetry are somehow connected.

Inflections aside, is there a way to know whether the whole Mandelbrot set can be more easily generated than per-pixel evaluation of the formula? Is there another way to express the resulting set, not using the formula? Even if there is such a way, remember that we need to have information about the iteration bands. Most beautiful images rely completely on the coloring of iteration bands. Without them, there's absolutely nothing to be seen besides minibrots.

Claude has some interesting blog posts. Maybe I'll make another thread with some of my thoughts regarding those.

hobold:

--- Quote from: Dinkydau on October 11, 2017, 10:35:00 PM ---The formula for inflections is already known. Far away from minibrots it always works, but this is not what it looks like, zooming close to a minibrot:
--- End quote ---
Would inflection work for Julia set images?

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