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About Digital Art / Hobbyist Dinkydau Set23/Male/Netherlands Group :iconmandelbrot-portraits: Mandelbrot-Portraits
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Spiral path morphing
Mandel Machine, Mandelbrot set

A morphing of spirals in a simple embedded julia set

1.5566501813351948897674305883109 E1383

Re = -1.768569992047404316099118177398948366214248997084351867455766443912554642424610530686892509849019082933992291698645409415887934075651323503539908962495846381871159858033389598001691900821959152001942595047393181540092977780066009924759802069509066757053233118834927300959510506917687869580844838430813440953119647283365492743622260524246442106390725657965428362086766487089026955989964457784669772669293246907189466958394404512931455729386662493098720317228609695475839166313950236854921488851572554335765029647920994142599511036642666744997657415307191918380469712864137722798347506942345108056421642271553838228679925105832201584550714562260639291582054999446725806629348252243305091090276818225801453243649577147418523447132557389546391564441713843700800547597404157647419618232594747055106967423053580643451818023986303240791535042832252180206456928192063236884251692156269700691895101860172211689808560510933769874335228224094184019126503579760748070978973765430314490421871345511661376530946834811640092312760245838045041123822201970299377169692687781631222674165352248562044364549088271876738606905512124102296927286728441844076568840062354308429510040852184528027070876273922853572583767568193602049272960564505069025820657200978288821127881383852535083715812746049726117173665892544811769136847220211753524244306487863807779140903570480831991810404812082262314222809057583878718438
Im = -0.001301819447354728908985842817546499677021733482898050411849938605579568835986674312319249463389246598653713257468668136491021674748168657905654796800792138444357247881021340866788944387699324398204018795869005318831696316041496889640211007558639862289604218802976426385226104307281860003590047329824560016632718768627371436687704401939740454103394585755454057380705645019030716466973463744966804530738930761530161791085297995928333203926063233642772758464004873503959309383671920666362476832042828260227330867768490233810572500494485495855706903570301775826626909538187703652376912508590667880904077870207882098021159052470510271892064446764330847715526862233582815275723386500401610171488061597740649942980345484937721690039976632069444404594010277251166925228185538219250523400744654690265065883342191104544428355269395950296778528396979576982273486870688585836461859906130266280152349010499499498042210626390517589001959296038690862645154746712175181827547672236400984212335741643839848616034758739305445083337472151516377874357628725011303975285343607202404105343826787751465059074565197664969131200622624025924123675591368868395066605765521470365738477291059566902477967762196975826605583004602229153604513666412759173097501641516007187309491048593384754340919340403727618639327874837675246678176448234744523533546571780939959815910092002276722033591676271483545261418910856731737141
Golden Ratio Tiling
Mandel Machine, Mandelbrot set

2.9095965591197057426879587427262 E220

Re = -1.756882996047415507177544263880044205560240816681939504337701122205524205232602173448341712703733871053600384797384909496400768301564935457122130644767462662664858651590631630737324402579324531143018225431629992886257332217126295
Im = 0.012178698364084925222456470314532744312959953422670766200219117572544897165197364251729676017200851973935584329711981645307984466766116936909884995575928407038962826724678263924301445881502867259057819318295270283982300902903242
Dragon (Deep version)
Mandel Machine, mandelbrot set

Deeper version of Dragon (Surface version) with simpler shape-stacking and no (simple) copies of the morphing inside itself
I think it kinda looks like a pile of metal, and the dragon curve.

2.3620314359461870687346087637027 E1558

Re = -1.6744017604363755146321878207363732277829749347284068785701734608965981144921453410119443323999158687799867851945466219874709246512305346273282168930531980958800729840377893102352150279625037687841574844214624072622581974461271075961491819429059190722911531107079902199796674174412385330279079237121556434874203703034929956505654043708198104642329655587600968732438457102669056751811390373435757157826087155358970426020848674225598876149106703393889287079843507617512724634173833503932771985633652009142493769376654956533146996601987240466279576443911572471501069705562126548577363872242885979351652944448127995367166377267430243815306795681330886991281857251387441681685759670625070140215176661403132193903557192749053259621917541410160946835916267181634804989789469952778238155170118231310507571224792731005339931483396251937592439734813048231533872403697482745936248407437975718562315612589926686116987739681470223309831571651819299615504158109989859401531493645164150364104002263253474649437133200073310293563209232545255419627741849777766442254150925249477763094641392330736459220208425956395104480145603210838209372635713751849005402756555752704854887202264125285847879300369292961772237250519206331172061782814878902382803907478220666339289025632852018765197460824185775389067406031875253381293002167073398075843724451252805163501763808279987913990766372036242370209707041378256668885538732549817108304286526378588723888846183694924423726101804919063327199655653257275361292249347477554339525597308694369943875104868392107398646956906440235180467774583948678

Im = 0.0000172652336599597724511837895148473559264463315061605044630161768042387311697974454605450810363003038474485833129999239337931457384816928323951116788577232125875667661694579828262179533240328992426817246047644126022212849124260222321150265716478526846129559004355698393096914128168823993756238688804429794034452887116500901590482948043683557756236659793776024641462823667020406814124411821038257777297948937935006613063360523198519989278884672193042715598464606938571227926581519896820280451349525854889166600171501139834744713972263834306208579748894039640934297329458037965940607807328126332273123725083565424433898412265750894621319538482415015968045947274719136592015668817799886893160938069075206688653180764864266144661381079773794784789157064412812364038504906533136467202316226314305988421150521758787739081198122096040773293770454815343005460192185131873900028282639652045864253985318424736950680691667232072770097626281085991091189165917045727548772251297948597021307152468174045880649725180955128681239576132482270990627720741326843214057092864637844228473397416570140004601588169683205523230046790410094631458827205741573556558870014982707410777682790767250276166466737364885401860447468151851704739417852685472517196575657313628004307155397350360228733625068170083841386911930156377114608458054876434278985198905890707469585389428318053145899037057927456955656462720480566803108112136431315567796079061831104576660021694420262107215675391246989082227069215174387018437669619606725221756411727022138330335554192171853246977901126181850858227588161373
Density near the cardoid
Mandel Machine, Mandelbrot set

Trivial location to find

4.4360868791990979941887918394644 E4

Re = -0.5727260685
Im = -0.4513802142
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: 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.
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 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:…

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.


Dinkydau Set
Artist | Hobbyist | Digital Art
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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jhantares Featured By Owner Jul 28, 2016
MenInASuitcase Blower fella (Party) fella's Gobbler (Party) 

Hope you like the cake. :) (Smile) 
DinkydauSet Featured By Owner Aug 7, 2016  Hobbyist Digital Artist
jhantares Featured By Owner Aug 8, 2016
:) (Smile) 
FractalMonster Featured By Owner Jul 27, 2016
:iconhappybirthdaysignplz: :iconbouquetplz: :iconcakeplz: :icondinkydauset: :iconcakeplz: :iconbouquetplz: :iconhappybirthdaysignplz:
DinkydauSet Featured By Owner Aug 7, 2016  Hobbyist Digital Artist
FractalMonster Featured By Owner Aug 8, 2016
No problem :)
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 
DinkydauSet Featured By Owner Aug 7, 2016  Hobbyist Digital Artist
Meztli72 Featured By Owner Aug 10, 2016  Hobbyist Traditional Artist
You're welcome!!! :aww:
Nao1967 Featured By Owner Jul 28, 2015  Hobbyist Digital Artist
Happy birthday!
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