3rd Order Evolution

[Dinkydau]

Mandel machine, mandelbrot set, z = z^2 + c

What to do when something is possible for any natural number? Just call it "order". In May I said that I had found a way to extend the "evolution" zoom method to work with any natural number of shapes. Thus far I have never made anything containing more than 2 different shapes. This render shows the effect of doing evolution with 3 shapes. There are normal trees, the layered thing that appears in the center and long stretched objects, all visible at once. The reason why I hadn't done this before is that the depths at which such "higher order evolution julia sets" lie are ridiculous. At a magnification of 2862 zooms, this is the most simple 3rd order evolution julia set that I could come up with and it is made with just a few iterations per shape.

This render gives just enough information for a new conjecture. Recall the conjecture about 2nd order evolution sets that I mentioned here:
http://www.deviantart.com/deviation/399217546
"When the method to create this structure is iterated infinitely, the overall shape approaches the shape of a julia set."

New conjecture: When the method to create an nth order evolution set is iterated infinitely, the overall shape approaches the shape of a julia set of the type found in the (n-1)th seahorse valley like here:



Magnification:
2^2862.649
5.5362546428023729586009972615094 E861

Coordinates:

Code:

Re = -1.7686389258582457602591798109664610559676104901531905534612371862156545682969746321939724950213366661636057536396779252859389632742413403244591187693751857699628584822959009920039586511918331032717466981617102477216955452515931265069265459682575124225929982484549136821686672341957090084798297901945350456714473246976282931047515059052131380350014617491424546432109112221506785196917078386726176965468588606774134572580668444740086109512807232439800764077947169834856997061535689836542138166769909236750987732240414274002288285785607256803383237379566639531930762047062673439517943372528066912105695847685374443102191667563509517486566769236505413400380565051983826496246388789426936587553197854928983707975272349056386817513728191815761344305079832337564878329836898161980996260364000073928407804100971048825945654928390119537091987437292216008763261326016208316985670
Im = 0.0014695398454686002602235602267435872742696288949454819641251917327001871646802222375794043554808984181050527747507024464475338925367544255504298138781497343514560484733471422567832875318623115371701089842454270266265594519203981996708780997969745056297187067310740754284635798427586526851073269358903637794497695778867387359644780106433549147575964695111500996174081311715975724522468033785608575894350758842715149520432125463466989127128011373497235051106233995040400616686607537287015952944156433221067736412912764567454009987622575582846143831794466663212601750260788535536819398196051616431301579942687208400722137429236421828930580251321137582633974441013377045265864719031580086405254319096195351614253832019998050137179725141523664399006493496816091681497072318461909569635448046827726248602343503824462069744543470519557285574917761743072242567051244893569548

[claude]

This is pretty incredible, very nice!  I need to learn more about this evolution technique, I'm sure you've posted about it before - got any links handy?

I'm thinking it might even be possible to partially automate the technique using properties of external angles - would take a while to research it though (tracing external rays takes O(iterationcount^2) time, maybe the spider algorithm is quicker but I don't have it working at high precision)!  Here's some links to my blog posts related to external angle properties:

http://mathr.co.uk/blog/2011-12-06_gruff_labels_and_rays.png (rays are labelled in decimal which is not so useful)
http://mathr.co.uk/blog/2013-02-01_navigating_by_spokes_in_the_mandelbrot_set.html
http://mathr.co.uk/blog/2013-06-23_patterns_of_periods_in_the_mandelbrot_set.html
http://mathr.co.uk/blog/2013-10-02_islands_in_the_hairs.html
http://mathr.co.uk/blog/2014-11-18_navigating_in_the_hairs.html
http://mathr.co.uk/blog/2014-12-20_automatically_finding_external_angles.html

EDIT: the minibrot island at the center of your image has period 287359, and tracing external rays to find their rays is O(period^2) because the iteration count of where you need to start from is usually a few times the period.  I've only traced rays up for periods up to about 1000, and that's already pretty time consuming!

[cKleinhuis]

indeed incredible, it is showing some order, the order is infinite, i am just laughing because no matter what optimizations are thrown at the good ole mandelbrot, it eats them just alive

[quaz0r]

interesting stuff.   

[Dinkydau]

Thanks for the reations. I have posted about evolution before, but I have not explained how it is done. It's difficult to explain. I have tried it a few times but I really couldn't do it. Instead, I have given coordinates here to investigate:
http://www.fractalforums.com/images-showcase-(rate-my-fractal)/evolution-of-trees/

Before that was done, I posted about what evolution is:
http://www.fractalforums.com/images-showcase-(rate-my-fractal)/prototype-of-evolution/

This image shows the evolution of mandelbrot tree structures. On the far right (almost not visible in the image) the tree has only few branches. The trees next to it have more and more branches, although they all appear twice. The tree in the center has an s-shape. In this image you can see exactly the intermediate steps before reaching the s-shape. This is the evolution of the S with tree-structure in one single image!

That's the idea behind it. It is possible to kind of "record" the process of morphing by copying each stage and continuing the morphing in one of the two copies. At first I recorded how an S with tree-structure evolves. With that I hit a problem, because every stage of the shape was visible 4 times, and I thought that was a bit too much. Later I gained more understanding about how the zoom method really works, and I found that it was possible to record the evolution of 2 different shapes in one single morphing, with a way to extend that to any number:
http://www.fractalforums.com/images-showcase-(rate-my-fractal)/parallel-evolution/

Unlike in your publications I don't really have and proofs or formulas, but I hope the coordinates can give insight in how evolution works. It's pretty simple, in a way.

By using the method of evolution it is possible to create julia sets filled with other shapes (that is true if my conjecture is true). Julia sets are themselves the base object to create an S from, so it is possible to morph an evolution julia set into an S, as I have done here:
http://www.fractalforums.com/images-showcase-(rate-my-fractal)/mandelbrot-extremism/
That method as well can be extended up to infinitely many levels/layers of shapes, but the zoom depth would become too extreme to be done by a human. I'm also afraid that an S filled with 3 different shapes (from a 3rd order evolution set) is pretty much out of reach for now, although not necessarily impossible in the future. I expect about 40k zooms is required, possibly less if we sacrifice some details and consistency.

What is it that you're doing Claude? Do you work on a method to let a computer zoom automatically?

[claude]

Thanks for all the links and explanation, will take me some time to go through it all.

Unlike in your publications I don't really have and proofs or formulas, but I hope the coordinates can give insight in how evolution works. It's pretty simple, in a way.

I'll render some zoom sequences and see where you made the decisions to go "off-center" each time.

What is it that you're doing Claude? Do you work on a method to let a computer zoom automatically?

Sort off, but not fully-automatically.  More like, analyse the first few times you go "off-center", and then automatically extrapolate the same action several more times, so manual zooming/point-selection work is reduced.  But I'll see how you did it in the zoom sequences and see if it is feasible to do in evolution.

The second video here uses an extrapolation technique for a simple "pass near a julia" idea: https://archive.org/details/ClaudiusMaximus_-_Down_The_Rabbit_Hole

[Kalles Fraktaler]

Totally awesome.
Shape stacking reached a new dimenstion!

[Dinkydau]

Thanks for all the links and explanation, will take me some time to go through it all.

I'll render some zoom sequences and see where you made the decisions to go "off-center" each time.

Sort off, but not fully-automatically.  More like, analyse the first few times you go "off-center", and then automatically extrapolate the same action several more times, so manual zooming/point-selection work is reduced.  But I'll see how you did it in the zoom sequences and see if it is feasible to do in evolution.

The second video here uses an extrapolation technique for a simple "pass near a julia" idea: https://archive.org/details/ClaudiusMaximus_-_Down_The_Rabbit_Hole

Reducing manual zooming would be very helpful. From the video it looks like automatic point selection would be much more a challenge. It would have to be very accurate and the computer would need to have the same interpretation of "doing the same action" as the user.

You can save yourself the time to render a zoom sequence of evolution of trees because I have rendered a video of that:
<a href="https://www.youtube.com/v/c0C8DTbfj3g&rel=1&fs=1&hd=1" target="_blank">https://www.youtube.com/v/c0C8DTbfj3g&rel=1&fs=1&hd=1</a>