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About Digital Art / Hobbyist Dinkydau Linteum23/Male/Netherlands Group :iconmandelbrot-portraits: Mandelbrot-Portraits
 
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5th order Evolution of evolution (model)
Ultra fractal, julia morphing model

First: This is not a real deep zoom. It is a model of something that is actually hiding in the mandelbrot set somewhere, but it's unreachable. You may need to download the 13700×10276 image to view it outside of a browser. The size is necessary to see the details.

Featuring a 5th order evolution (evolution of 5 different shapes) where the 5 shapes are, in order:
1. Evolution of trees (itself evolution)
2. Mandelbrot Extremism (morphing of an evolution set)
3. Stardust4ever x-forms (3 layers of X-shapes)
4. Partition (itself evolution)
5. Trees Revisited (itself evolution)

The 5 shapes are distorted as a result of forcing them all together in one morphing. Also I think I kinda messed up the Partition part, but some distorting is to blame there as well.

I made this by rendering a julia set in ultra fractal and transforming the plane several times in the same way as julia morphing in the mandelbrot set. (Info: www.karlsims.com/julia.html) The zoom level is 0 and the julia set is extremely distorted.
Since 2014 I had been thinking of the possibility of a program that could do this and I considered trying to make it myself, but my programming skills are low so I decided to forget about it. I learned that ultra fractal can do this from YouTube user Fractal universe.  Apparently ultra fractal is capable of rendering accurately with such distortion, which is really awesome.

Models obtained this way resemble real julia morphings that can be found in the mandelbrot set, but they are not perfect. The single-colored blobs are not glitches. They're there because pure julia sets (normally) just don't have the complexity of the mandelbrot set. There is no minibrot in the center, so as the symmetry increases, it becomes easier to see that the center is empty. A more carefully chosen and accurate constant seed (the first value of z in the julia formula z -> z^2 + c where c is a point in the complex plane) can solve this somewhat. Also the surroundings normally present in the mandelbrot set are missing entirely. I think we should consider this to be a tool to do research and try things out but not a replacement for real mandelbrot zooms.

Assuming we are going to zoom to it in the least deep and least dense location possible, the zoom level would be somewhere between these two:
Lower bound: 2^5233597273755 (E1575469764625)
Upper bound: 2^12564475169365 (E3782283905754)

A location of this shape in the mandelbrot set is not automatically obtained. The only way to find it is to zoom to it in the real Mandelbrot set, which is impossible because of the depth that would be required. It's also important to mention that such models can't necessarily be found in a low-density (low iteration) part of the Mandelbrot set in the exact same shape (or at least without at least some visible inaccuracy) because the inflection points need to be inside a minibrot normally. The fact that low-density areas have visible iteration bands doesn't guarantee that the inflection points used for the models are inside (or even close to) a minibrot: they could be in an iteration band. Choosing an extremely high density location would solve that problem, at least to such extent that the inaccuracies become unnoticeable. So if the depth alone wasn't enough, we would also need extreme iteration counts for these morphings: double impossible.
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Dragon (surface version)
Mandel machine, Mandelbrot set

I think this looks kinda like a dragon curve.

Magnification:
2^340.486
3.1368886558310062319068272783244 E102

Coordinates:
Re = -0.14887322444320039855687187363505383529818409865610513527808327124509101086805786044085822791395804113726009400
Im = 0.64961120563169810945842289448176674645450328921321721892708148989671893156697427376997727273713986796299514700
Loading...
For more than a year I didn't have any inspiration. Actually I think my last two submissions were even pretty boring. Now I have many new ideas again. I have at least 5 more renders planned right now, deeper and with more iterations than ever before.

I have so much to say related to my latest image submission "Trees revisited" that I decided to write a journal about it. I hope to clarify what I mean by the term evolution. Information about what it is is spread out over comment sections and deviation descriptions. Also there's been a breakthrough in computer-assisted zooming, which is what's helping me to zoom this deep.

This is "Trees Revisited":
Trees Revisited by DinkydauSet

Evolution zoom method


Maybe the title of "Trees Revisited" is misleading because it's not really about the trees. It's the same old trees again. Instead this is a variation of what I have come to call the evolution zoom method. In general, given some shape that lies somewhere in the Mandelbrot set, evolution can be described as:
1. double the shape
2. morph one of the two copies
3. repeat by treating the morphing of step 2 as the new shape

rare_glitch.png (1024×533)
In the left image, 2 points are labeled 1 and 2 respectively. Zooming in on the point labeled 1, which is outside of the "shape" yields the middle image, a doubling. Zooming in on the point labeled 2 yields the right image, a morphing.

Doubling a shape can be done by zooming to 3/4 (as a good rule of thumb - it's a little more complex than this) of the depth of a minibrot of choice outside of the shape. The exact result depends on the choice of the minibrot. A doubling leads to two copies of the same shape next to each other. That's step one. Morphing one of them involves choosing a minibrot INSIDE that shape, so we choose one, but that means it's not inside the second copy of the shape, so the second copy gets doubled, causing both the morphed shape and two copies of the original shape to be present in the result, which is a set of shapes. By iterating the steps, the original shape and every morphing tied to an iteration of the steps are present in the result and all visible at once. That allows one to see how the original shape evolved, iteration by iteration of the steps, into the final morphing. That's why I call the result an evolution set.

Here's what's new: So at each iteration of the steps we have a morphing and two copies of the previous stage. The way I used to do step 1 in pretty much every previous render where I mentioned the word "evolution" was to morph one of those two copies, but I realized many other ways could be used to double. The only requirement is that the chosen minibrot is outside of the shape to be doubled. I tried a few things and this is the most interesting one I was able to find, at least thus far.

Automated zooming


There is also a lot to be said about the computer assisted zooming I have used to get to this shape. Claude on fractalforums.com found an algorithm to determine the location and depth of the nearest minibrot inside a bounded region, involving the Newton-Raphson method. Because doubling and morphing shapes is equivalent to choosing a minibrot and zooming to 3/4 of the depth, knowing where the minibrot is and how deep it is allows one to find the coordinate and the depth of the morphing immediately. The coordinate is the same. The depth (the exponent in the magnification factor required for the minibrot to fill the screen) needs to be multiplied by 3/4. All you need to do is do a few zooms manually to make sure the algorithm searches for the correct minibrot and the computer can do the rest. Kalles Fraktaler has this algorithm implemented and I've been using it a lot. Some links to information about how it works can be found here:
www.fractalforums.com/kalles-f…

This is revolutionary. I think we can call it the best invention since the perturbation and series approximation thing. Zooming manually takes A LOT of time. I have spent days to several weeks just zooming for one image. Once the desired path has been chosen, it's a very simple and boring process of zooming in on the center until the required depth is reached. Note that this is not what the algorithm does. It doesn't need to render any pixels or use any visual reference whatsoever. It's a solid mathematics-based method and it works if you give it an "accurate enough" guess of where the minibrot is. Note also that it doesn't help in choosing a location to zoom to. You really just tell it "zoom into this center" and it finds the minibrot inside it for you, saving a lot of work.

It's pretty fast generally, usually faster than manual zooming, especially in locations with few iterations. Based on my experience with the Newton-Raphson zooming in Kalles Fraktaler thus far, I think it's actually a lot slower than manual zooming for locations with a high iteration count. Usually that's still more than made up for. You can work, sleep, study and (most importantly, of course) explore other parts of the mandelbrot set while the computer works for you, 24/7. If you have a processor with many cores you can let it zoom to several locations at once. Effectively that makes it faster in almost every situation.

The evolution zoom method involves a number of iterations of a few steps and I have found that generally it holds that the more steps taken, the better the result. The way the result looks like converges to a limit as the number of steps goes to infinity. The Newton-Raphson zooming allows me to perform more such iterations without as much effort as before. I always want to push the limits of what's possible, so I will perform those extra iterations, meaning I will be zooming a lot deeper. It will lead to shapes that are even more refined with even more symmetries and patterns.
Trees Revisited
Mandel Machine, Mandelbrot set

I have often used the evolution zoom method to make julia morphings with 2 or more different types of shapes inside, though what I really wanted at first was 1 type of shape. I hid the fact that I didn't know how by choosing the same type 2 times. Now I found a way to get rid of the unnecessary copies, but the overall shape is very different this way.

It reminds me of the shape of a 3rd power Mandelbrot Set. The center is like the main double cardoid and it also has (approximately) infinitely many bulbs. I have been able to mimick the circle-like bulbs of the second power Mandelbrot set before, though without the cardoid. This which makes me wonder: is it possible to make the shape of a 4th power Mandelbrot set? It's definitely not possible to make the shape of the whole set because it doesn't have 2-fold rotational symmetry. The fifth power Mandelbrot set might be possible...

Magnification:
2^9706.3
7.7169718014717701081201473595858 E2921

Re = -1.7688011252221877204709741758179943289719776561698906746323500348383699513591489612873967509274966440413557617241491600329058372805735550148585182138711720726695297443148253463184817954013513962337939472169657134580286294121622150299478675204801828772493631773673987904309055816283198587234189197841056049571802980640265156000065177874740485973621705174732650630769247931314556164494074766611938824067322405243506104378777342718615187425356491320861592233700947335816911589141624787016843941868708374406140859169960100095681558780027696824171951543816248932169324543091691569091287791899396819510536609202651352744135987374400083879072341472022542290361286853009509839846452730453481513265387679576679704654354482489532053723293459079676004717895713664899711957495757332765588472539464918031164836902929586122654796297094770427794366571322707809112332370950804360132673436651821911772042879533877478458718004410635110925632999566574084574235463912943470981721752643771773959558831554589854485604975364621185552698316665558691598027940316801636979000040560621127029663794019613541851673307251878131613238361091458431747078574040791783358730895007793682917277921800488141026306756804472855015620008630819680964376619758271645715301315352030916979509384752961939243018082424561263094775528318739161642895706042971428881780441631696627603372495412977727051511388497154183802329850302603882147199574795085484970649887998929691672019785648299562237807546571551108703613853476476635584556652568811579895157292213613741850776600290618732784205366679406209183839569625343815117957194201655856830438148745539385078552910235978236390095560441089361220724853517548990588385468840182561589121573689707841713231982289676578964566304537759382926839898672600236484643838432273575504175283877683434831283034936477963766914535828475223078368899757971868123519742649163544483797060207905762748263974741206676035342373418596222922297434957597148263365121018319605659231307996294004235202904458380630516068251886040368370758115208145816426943812593846118972673514534516052904742220875050306351568860227521804267873501871666887627451041383334734935761084602750325040614769804761840568041357299680337154522923062905316277109469749988131259315903712995926636115412457507400577726150468154511468443846674422928256891145186072654597069023685731909339210627069566886224118764897114006992931666743587971029936529081795869187074154311186008321350528072353048323119928080244615001852683452018915138610896758225470364123751023258566092514006871941278112232662554373559171314229397877800428215421102331977257179431815411793258149216937585480088953227258261067271164276065880412885595898519409991063276663586816144203494914936288482747501535541621005846894158892662624469998310599333626021627844856527863309756858610732042800629249468455546995415833057344470988207268766674228544321987366798992201893460402434350436358123395575256774446824903498517085456941735650

Im = 0.0023878480881805962820493232458554875037388092952829542156222752945764166987851812825932831879136756035194454484562929129723749322427988417441140327575831621191333413445566201483453075727278255196911626265388377437378928553941592342752390149210875595092895828341550441216137232078130843706694514812411896466242747598047237841899227407137207996153631661165310676900278121081109773454730706218028487944757761815189100251241603686120944865189418544771857366780301296072952615170045095076990744788601062500725092197419557540878567542349119974872148895674529910361655153195175960926770555469301301644923525565259716471074402246266656292684512998421471369841245969419718109233723597846540485542777563021083995697611612300274768455277194690961563407811185353390550470749277084198271845472675080540680170598292583097345103989777045408378143967319947909282864165283151152839968641455189430997723981196299993593987572466737384717840587289933588707393432757726734145910903896328723978531513514818704044611071990775303126673802839955813108666929364596523501121842680741526691737338933843486436192243112643661559745631040930829858257086064458039521611542008653748816940818206207801995237906691860097067120322932112566157731633594592235122962402399303991990210641677394799740426353614590140043776160707267731601373537975599905293399363474940388882572852053965492001959280706180255738237245127327321458755584892753479301937772087551712369041261813263065763604200797565327779240767708248581170092652292506127018525202830263388240593001343477542962460321943107998174534879536290722653460016994101289360707367764170864339880483298771875602916087103654747021702626710543570965779670061788813662643393054075906197477431314232350408058042444336898537707639499259746724526716689862903092860371521093305826388382514627257961717670586447669706353227908699821704206249993663026132845466290889074256678794497679729004222488549040762005041804135438300445236268180071097035346868429301772840208183351656109468691582900448799641336130006700050798475724210207542415707634564703190071036272867374271564705033881962125025327642255479054567095087898614790426657803039774715608219744663580852307172337863461461834873527416092515835896614707350930635299568427145577276278022801039971177502546312139872058890952120043526040902563046387962512879897692678849575962053093732046987800363511115465138658476718912646306300966817227976958372341341033832688719927824607797592973257546782077781443370408783271172848875987410446352252668907268099455600238014668283296120502908065655257368576768542147837197473079816454901442718050330214070753960956199047391284973073528869336377525870293835186654326572883021390296220440232615891627131048813022135789763124753663887361437564467802011993179524788277980585521125569771547775923387161097600941727093963430649526526356251808119775432738991172835499142733538819006271799133002338357073537763377568409858091085076509585387354490650
Loading...
For more than a year I didn't have any inspiration. Actually I think my last two submissions were even pretty boring. Now I have many new ideas again. I have at least 5 more renders planned right now, deeper and with more iterations than ever before.

I have so much to say related to my latest image submission "Trees revisited" that I decided to write a journal about it. I hope to clarify what I mean by the term evolution. Information about what it is is spread out over comment sections and deviation descriptions. Also there's been a breakthrough in computer-assisted zooming, which is what's helping me to zoom this deep.

This is "Trees Revisited":
Trees Revisited by DinkydauSet

Evolution zoom method


Maybe the title of "Trees Revisited" is misleading because it's not really about the trees. It's the same old trees again. Instead this is a variation of what I have come to call the evolution zoom method. In general, given some shape that lies somewhere in the Mandelbrot set, evolution can be described as:
1. double the shape
2. morph one of the two copies
3. repeat by treating the morphing of step 2 as the new shape

rare_glitch.png (1024×533)
In the left image, 2 points are labeled 1 and 2 respectively. Zooming in on the point labeled 1, which is outside of the "shape" yields the middle image, a doubling. Zooming in on the point labeled 2 yields the right image, a morphing.

Doubling a shape can be done by zooming to 3/4 (as a good rule of thumb - it's a little more complex than this) of the depth of a minibrot of choice outside of the shape. The exact result depends on the choice of the minibrot. A doubling leads to two copies of the same shape next to each other. That's step one. Morphing one of them involves choosing a minibrot INSIDE that shape, so we choose one, but that means it's not inside the second copy of the shape, so the second copy gets doubled, causing both the morphed shape and two copies of the original shape to be present in the result, which is a set of shapes. By iterating the steps, the original shape and every morphing tied to an iteration of the steps are present in the result and all visible at once. That allows one to see how the original shape evolved, iteration by iteration of the steps, into the final morphing. That's why I call the result an evolution set.

Here's what's new: So at each iteration of the steps we have a morphing and two copies of the previous stage. The way I used to do step 1 in pretty much every previous render where I mentioned the word "evolution" was to morph one of those two copies, but I realized many other ways could be used to double. The only requirement is that the chosen minibrot is outside of the shape to be doubled. I tried a few things and this is the most interesting one I was able to find, at least thus far.

Automated zooming


There is also a lot to be said about the computer assisted zooming I have used to get to this shape. Claude on fractalforums.com found an algorithm to determine the location and depth of the nearest minibrot inside a bounded region, involving the Newton-Raphson method. Because doubling and morphing shapes is equivalent to choosing a minibrot and zooming to 3/4 of the depth, knowing where the minibrot is and how deep it is allows one to find the coordinate and the depth of the morphing immediately. The coordinate is the same. The depth (the exponent in the magnification factor required for the minibrot to fill the screen) needs to be multiplied by 3/4. All you need to do is do a few zooms manually to make sure the algorithm searches for the correct minibrot and the computer can do the rest. Kalles Fraktaler has this algorithm implemented and I've been using it a lot. Some links to information about how it works can be found here:
www.fractalforums.com/kalles-f…

This is revolutionary. I think we can call it the best invention since the perturbation and series approximation thing. Zooming manually takes A LOT of time. I have spent days to several weeks just zooming for one image. Once the desired path has been chosen, it's a very simple and boring process of zooming in on the center until the required depth is reached. Note that this is not what the algorithm does. It doesn't need to render any pixels or use any visual reference whatsoever. It's a solid mathematics-based method and it works if you give it an "accurate enough" guess of where the minibrot is. Note also that it doesn't help in choosing a location to zoom to. You really just tell it "zoom into this center" and it finds the minibrot inside it for you, saving a lot of work.

It's pretty fast generally, usually faster than manual zooming, especially in locations with few iterations. Based on my experience with the Newton-Raphson zooming in Kalles Fraktaler thus far, I think it's actually a lot slower than manual zooming for locations with a high iteration count. Usually that's still more than made up for. You can work, sleep, study and (most importantly, of course) explore other parts of the mandelbrot set while the computer works for you, 24/7. If you have a processor with many cores you can let it zoom to several locations at once. Effectively that makes it faster in almost every situation.

The evolution zoom method involves a number of iterations of a few steps and I have found that generally it holds that the more steps taken, the better the result. The way the result looks like converges to a limit as the number of steps goes to infinity. The Newton-Raphson zooming allows me to perform more such iterations without as much effort as before. I always want to push the limits of what's possible, so I will perform those extra iterations, meaning I will be zooming a lot deeper. It will lead to shapes that are even more refined with even more symmetries and patterns.

deviantID

DinkydauSet
Dinkydau Linteum
Artist | Hobbyist | Digital Art
Netherlands
My name is Dinkydau. I started using Apophysis somewhere in 2007. I discovered it on a forum. Someone on that forum had an Apophysis fractal in his signature. I asked him how he made that, and he said he did it with Apophysis. So I downloaded Apophysis and started working with it. In november 2008 I started to do animations and I joined deviantart.

At the moment I don't make flames anymore. In early 2012 I started to focus on exploring the mandelbrot set in the program Fractal eXtreme. I knew about the mandelbrot set before, but it's extremely computationally intensive to explore compared to flames, so I focused on fractal flames at first. Technology and algorithms have improved and I saved up money, so I bought a nice computer. Now I'm focused on finding and rendering mandelbrot locations.

Current Residence: Klaud
Favourite genre of music: classical, deep house, electro, dubstep
Favourite style of art: fractal flames
Operating System: Windows 7
Favourite cartoon character: Donald Duck
Personal Quote: The world seems complex, but that's just because we're part of it.
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:iconjhantares:
jhantares Featured By Owner Jul 28, 2016
MenInASuitcase Blower fella (Party) fella's Gobbler (Party) 

Hope you like the cake. :) (Smile) 
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:icondinkydauset:
DinkydauSet Featured By Owner Aug 7, 2016  Hobbyist Digital Artist
thanks!
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:iconjhantares:
jhantares Featured By Owner Aug 8, 2016
:) (Smile) 
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FractalMonster Featured By Owner Jul 27, 2016
:iconhappybirthdaysignplz: :iconbouquetplz: :iconcakeplz: :icondinkydauset: :iconcakeplz: :iconbouquetplz: :iconhappybirthdaysignplz:
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DinkydauSet Featured By Owner Aug 7, 2016  Hobbyist Digital Artist
thanks!
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FractalMonster Featured By Owner Aug 8, 2016
No problem :)
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Meztli72 Featured By Owner Jul 27, 2016  Hobbyist Traditional Artist
Happy Birthday!!! :hug: :dance: Rainbow Striped Spinning Star :yayay: - NaNoEmo 24/30 + Plz a cow can dance better than u 
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:icondinkydauset:
DinkydauSet Featured By Owner Aug 7, 2016  Hobbyist Digital Artist
thanks!
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Meztli72 Featured By Owner Aug 10, 2016  Hobbyist Traditional Artist
You're welcome!!! :aww:
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Nao1967 Featured By Owner Jul 28, 2015  Hobbyist Digital Artist
Happy birthday!
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