Three layers of curly elephant morphings

[Dinkydau]

Mandel Machine, Mandelbrot set

The overall shape was made by using the same technique as in Trees Revisited. It gives a kind of self-similarity that I like a lot. Notice that the "tree" in the center also appears all around itself. The accuracy of that effect is only limited by the depth and render time I could reasonably accept. The render took more than a month (including glitch correction) because of heavy shape-stacking.

There are 3 layers of stacked shapes here. It is the first time I have applied 3 layers in a "deep" zoom so that makes it something special, at least for me personally. The first layer is what the trees are filled with: curly morphings that started with an elephant julia set originally. The second layer allows the trees to have clear boundaries and the third is the boundary of the main shape itself. Maybe you can see that the first two layers have the same shape. The first layer has a lot less contrast than the second which is something that happens in general. It makes it hard to find applications of 3 or more layers. Making the last layer look good can easily make the first layer almost invisible. A lot of shape-stacking AND depth also drives the render time up like crazy which is why I have never done this before.

It is simply because of how shape-stacking works. One way to do it is this: If you have an idea for a zoom, you can first go to a minibrot, think of it as being the Mandelbrot set itself and perform the planned zoom from there. That's it. The shapes resulting from the zoom into the minibrot are then filled with patterns that were seen before the minibrot. By choosing a minibrot the stacked pattern can be influenced. The recurring patterns are the iteration bands of the minibrot. The number of iterations required normally is therefore multiplied by the (max) difference in iterations inside the pattern - the "iteration count" of the minibrot - called the period. You can think of it as composing two zoom paths, one to the minibrot and then another one (to, say, a deeper minibrot). The period of the composition is the product of the periods. That's why it's so hard to do shape-stacking at great depths and it's the reason deep zoom videos never continue after reaching a deep minibrot. It's just too computationally intensive.

Shape-stacking can be done because minibrots have exactly the same features as the whole Mandelbrot set. It's one of the things I love about the Mandelbrot set. Once you find a minibrot, you can, at least theoretically, start all over again.

Magnification:
2^3216.392
1.7000817826179882903331171189311 E968

Coordinates:

Code:

Re = -1.749365089130713355136630080718347789559672431892481991549752887987257790534288948852686176092823166880490367584365537241402861301051987572756134885766638315030733184102356683745567733704732242336955716689374220053239462271783286969695153138595960968290613096949326592408278551176423296957665693781278222135957297409064817652373422135574657112969716927947485486819016815962443472752011902737416089338071774841485991253273631345957163307488029852241047470109385772255147111632339006553554769628100546684622495780340381355203792634585520532761334503955692314908107291929614735326860054327591263812484743920329421708025534820190742048371429639420053196044998160293387459457173088444684708887203035147442894047082313460968372046848337015731658271372561131887628465964557653463317650999148237981158362626539221835897659252524262512209501785961590731030815348889046456268071858340927774884222761306721255869200087387892697348047269984941762337148007330658012362815577473768915429790
Im = 0.000000749338763025441077635067897843290324328364823725106652774163469955659130000553036152692348499586259211773488938327142632275497110083700087625810153356917154351213845759552072775573138870307623452220908006924373386358851662101475560714902845049069557710671495531000214771157708102117014124261161460873019872562499024280227792829384747412179007566059304454037672086576578431671511987764318814190590560786798393590730745886310081445382254877137292162499001064519808832805042668998490623157561409593714128196422705485718006062939517024762567595955043641585945566334004032551810905620950364643187700183288907723485916230573975063135198454571171623145315843288678780241131693068481538731006056464310869184858150811508962009441974266124120904462670554045286020287151700359927450985717710570585083311930785250338126621250447636233137838208798719630214780937239875875238050692819292432690046739868306452339999403349451922613923505714140121592722817731635276657757981771752651230

[Chillheimer]

shapestacking - not a very usefull skill and even in the fractal community recognized be few.
but those who do can see the mastery that goes into your renders.

you truly are a master of that artform!!


(also my favourit feature of the mandelbrot-set and escape time fractals in general.. it magically gives these kinds of fractals the ability to store data, your previous path. I see some deeply philosophical aspects in this.. worth another long post)

[Dinkydau]

Thank you for your kind comment.

The point of my whole gallery was originally to show that the Mandelbrot set is more diverse than most people would expect, but then it got out of hand and I was myself amazed how diverse it was.

(also my favourit feature of the mandelbrot-set and escape time fractals in general.. it magically gives these kinds of fractals the ability to store data, your previous path. I see some deeply philosophical aspects in this.. worth another long post)

That is indeed a very interesting consequence. You can literally store bits of information in a zoom path by, for example, zooming in on a spike of a seahorse julia set and choosing to go either left (a zero) or right (a one). The information is then encoded in every shape you get to see from there on. Here's an image of such shape-encoded information where 0 is "down" and 1 is "up":


It means that, just like in the number pi, any arbitrary information is present in (just) the shape of the Mandelbrot set.

[claude]

just like in the number pi, any arbitrary information is present

This "fact about pi isn't proven to be true (though it seems likely):  https://en.wikipedia.org/wiki/Normal_number "It is widely believed that the numbers √2, π, and e are normal, but a proof remains elusive."

I experimented with similar branching near 0+i, going left or right outwards from the set. https://mathr.co.uk/blog/2013-02-01_navigating_by_spokes_in_the_mandelbrot_set.html

[Dinkydau]

Oh yeah, I forgot that it's not proven to be true. You're not the first one to tell me that. I keep forgetting it.

Your article seems interesting. I will read it.

[Chillheimer]

And unlike Pi, where the occurence of information is (probably) random and by chance, the information you store into a zoompath is there because you selected exactly this path and made all the decisions necessary.
so in the mandelbrotset indeed ALL information IS present. and has ever been!
it just has never been computed or put into meaningful context.

I love your visualization with 0 and 1s! That's exactly what I meant but I haven't seen it displayed so well!

hmmm.. how much does the iteration count rise for each of these decisions? It's more than just one, isn't it? (put on my todo list to try out)
else this would actually be an fascinating new way to store large amounts of data in just the coordinates. kind of the ultimate compression..

imagine the universe is actually a huge quantum computer capable of infinite simultanous calculations. calculating the mandelbrot set to near infinite depth. and we are exactly THAT branch where all decisions lead to what is exactly happening.
storage of information in DNA isn't that different, just 4 bits instead of 2.
instead of choosing a 2 branch pattern you could also choose a period 4 bulb to get ACGT.

(I know how far out I'm always with these thoughts - but it's mindblowing how well all these things fit together..)

edit: claude, that's fascinating, you've ben there already. thanks for sharing!

[Dinkydau]

Thought experiment: One "zoom" (magnification ×2) requires one extra bit in each of the coordinates, so 2 bits in total. For coordinates to be good enough to actually compress data instead of inflate it, the power in the Mandelbrot formula needs to be chosen such that the amount of rotational symmetry (= the number of options to choose a new direction) is higher than 2^(extra bits needed in the coordinates). I don't think that works. It clearly doesn't work with low powers and I don't expect it would work with high powers because high symmetry makes the spikes to zoom in on smaller and therefore requires more zoom depth. Zoom depth is very valuable because each extra bit decreases the compression potential by a factor 2...

Assuming it does work actually leads to an impossible situation because then this algorithm could compress all strings of information. As I've learned, no compression algorithm is able to compress all information. It would certainly be an unexpected way to encrypt information though.

Another way to do it is to look at the antenna of the Mandelbrot set and consider the interval bounded by the two largest minibrots. Do the following:
Find the largest minibrot inside the interval, then choose to make it either the new right (a one) or the new left (a zero) bound. Repeat.
Several zooms are required for each bit of information but at least the imaginary coordinates are always 0 and therefore don't need to be stored. Maybe usefull if you want to fill your harddrive with extremely bloated and very hard to decode data, but still think the julia set method was a bit too much.

[Dinkydau]

New coloring:

This is the original coloring that I made months ago. I regret changing it after rendering.