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# DinkydauSet

Dinkydau Set

Connected tiled four-armed dragon 4 0 Trees Revisited... Revisited 13 1 Misaligned Butterdonut 10 1 Elephant shape-layer alternating evolution 13 6 Density near the cardoid 4 10 0 Special S edition 13 5 Density near the cardoid 3 6 0 Special wood carving 14 2 Recursive sequence 9 2 Elephant forks 18 6 Density near the cardoid 2 5 5 Many stars 7 2 Trees Revisited square version 10 5 Self-similar Dragon 14 4 Cross Complex 6 3 Lattice evolution 14 6

## Groups

 Admin of 1 Group Member of 5 Groups

## Activity

submitted a deviation
Connected tiled four-armed dragon
Mandelbrot set
Computed with Kalles Fraktaler

colored with Mandel Machine

This is a mix of Dragon (Deep version) and Lattice evolution. Like Dragon (Deep version) it's a 4-armed dragon (if that makes any sense) and like Lattice evolution it's a completely connected tiling. That's unlike 3rd Order Evolution Of Tilings, for example, which is a collection of separated tilings.

The new idea here is to make a 4-armed dragon from a 4-fold rotational symmetric tiling and fill it with more 4-fold rotational symmetric tilings. Because of that there are 4-fold rotational symmetric elements all over the place. In theory the same can be done with 8 but that's far deeper than what can be achieved. Unliked Dragon (Deep version) the dragon here is not as complex. More morphings would have turned some of the 4-fold points into 8-fold, breaking the pattern. Usually the patterns that I make are finite approximations of an ideal that requires infinitely many (repetitons of the same sequence of) morphings, but not this time.

Magnification:
2^10719.5
8.26714980770 E3226

Coordinates:
Re = -1.750347190489936502150776819735342819367436255782331484300992001789621287742573135579208092945218408758780763280360580574824056633225773087486242020418233361418921119794385572917168230986412715575596424807586071857601144586498789781209707703283291920030307007171722623320267868743036128973472578752605753954857111617949118641559530242325275448313837739459120395301538631318212469265652054838570659938041549467772375786270344115736606012230651258611882172657357838590780124486394864463010756600562855953490560654938735186035174061035256198324087494919798049947962104211640548558809262548544812456100991835967089876938741763614656124850000167650473982394977881578418585857253216648784744951398168200618339676767983359419577161370595341652147851470839109642515214198486059688935638295197819683480433262822250357411075490872749710226227028275275098763200460666654239433399513728827328394481149664242852108615882238211715277848699045697115093454372392328234099701152854779178961870915256145169081107953209674990889012087824814571069657468017674806373611939511212778132365276389931660232712896901080076306333598491591705808496339270155561411340251764279642850707010839266972595916359278830687615934957745219391405123614236756462542053219360910274166878165169668220618264264653497220772606553361322912795798676857123703140874679948058428168075874902260517135424930140829510820809625344789347777347238446962557684265038036259756667372400887260527582071003298542706695593813318259904489546305523570494347313368160839100822760106299717475780488298836785815557683031050679151697741961949358950404547320826304689869450149654564720040738862657481883828139424865983192126024864922725137728111263912846113368573564511349905299907393139862019051558922231650831401956571931360720298093834512115070353154172267936561065181614328764498841366027133700743193046385309933194166030515084946867967741538990366233727237347223880889529671670223631576247768101955033507904222491387720504619908744096836569236411446629094733330453770375476328218625057547162090778521536404080951820067532758466314279099165368551373925650357889497061021917554658725840454348285999131719915062586591809383803866603869887182233507104481393250405253639841298034585569088226747319462926263898552577247693366187083721878868639568866907604069220562203066590755358001465888169902173400311652854341093103101390832953938982220586878179073457982559766156595018980286956247107826554171040806025198372746512234663569086867040340002572864648865522574714983643507065720845989371951091861730225442495270356586441755377943658002975390783159280810169525205112449352595938714422732177686360824159173553077542079445311191874665206889635640476747481118689397838462107642481913133347636336564465718916699637510555609150209610924910977702822385364020364593691405185094052828801825056103074915221246243415649640714850882933852566268183697044507381574646648975259317559163297528144384277230812066762808152001155982321658325878031669358656210807146721554077970491327667852821220732469477789879277148549757683098850640174563089692741200698593801071327126727399454852694090709474124272521177666290168915393615420833113805207454967315229489560103196596782655492506397794059905264643931282912628443250779118709047344925726741813594086405656223050249860634616987873682532176496078078386546523434528992516858788653220793296339312362759337426839954872474958003500535484589981658312181249351647128336007220988360615242794984699836076323436450884947934731517289332353412006780731253735985388515554805237675813241347638283357658893245852947893538007349382765238463097612606407397158983887839912870885873888491275348845726081306398253399233092260605375642718130739021163871218130403526664922668973736934062700043769505710019290238958730149139537877081072344101930586423451714973969542936326062753177243810476352106492191412367428317354809271790156551249913729359125200171690563997161967754853165973596845531287558070540550408510500769660645820943645146968272857351112146577696205756092146514943898804690098952942889095340963820507829838703432325576881145582884827204904474077227340529968804241208812808321787556600108394082872693081405643556676127020039456004380488948497580723594555144925570031327752350092402394848607455057427936633442713296696464994014416445851518331478430237221053716838910378932419373531876501682817132536431353452785807684190853286404224391673954571945905214229576793478210441967602240813556588409359073878248059799482686430480780848934487244161822902143667079870763189876156130948608345203596537624320746981222484031374776390948152610380483197726531833044407176585923336180135628597603713468151021325200704835774320911186350183011209632968338000005999978999849999865999977000036000125
Im = -0.009671623381213996602469565274562720603527196891055554047410306620151847546786838267956636536204205234590183965455013906108212709155402533951980074354499138531330443080683977633160742401038190739848914014738935474826498497552132521177001687373346462823431847323955558661356880315201625504337618018856629588220911441005586850658277571471469277723562900258372706073478963348593407534904396212598390590966900697921511253061103033389392829688840140931101905883661383867955339555201028180445632899425630885354945580880114142898953461733740092375478971742054023045647589371567332210527683002771044291460045717280563992629845351372483581540347686913007715754473152765598161854224418974483375991879976960337701787553293916733802380786407649482668648646886625514672894943163614401940512908178469241276182132333062196493713677744653600034785237909263630473955504907702149631684161730352195165741415999764461556252599159193141599380785003306840098984912413644236412916369340147385498695532745862476054468041673609474811070436645722737982262056747400371172817067596671312045813592902720039399812752259051801249596005277153072111896486543716397297610088943899982832762114093243425021310572060325279681451451069856146121323899041041417908252958867032497570389553537345189671329196971133878146386604597827104211868862721591334165847987223301253835305748432924246359303059263368020739458652961715952603439439201943051863108118233650214024797290543587977789273448353161028138493434984433438421037685148576830733366706254767562715738279289224112571769853117555192654706431350590507200761644202027359344168683529123148191750414350141793574368539001343998781799280965254524967028168583621435302778413048364152475151630114695389814751875206213012824853920788712297996870150000687861361560403108557021338964665047236320287835801315221076771784109142638736659078478053279819639351527381837010673583518040268793699183984047036724909682702063449331901780799081138915472995092977647216724217814823766619566681996329692794029667809600338263597955783792785808790602243481239270387364755673177959636217931887554742141942335158208991337554924864038096747786625603121916010307101152433435052806882340877950337262502705082956762724554382211227702072179482438199478403331697525832360466064447234473024471924240820989768636117574822662714372221894859962016460167715400538732875907537864362718839303578690537885093730717756806659969848434731736819867078256436224549944786022782426125063040155559371887613717820176586543192471815443113073031629400122507215034930824679141033462670738166966805893555229440669452880519428601945800305123939043099618593641213888602912843208870573375710491045806045970030473554780857705919376240155903392444661715929169264616065864427342516363028622996400948444591546691433571400885797589667315807499925527488039117077046443329768026031340147762168972594019893527738844665104546557515431772088032190650330651205600439514169857333781255765017310142795048617953455878225282140584876208038589278088396030055723605627724926454386265442505844951578370818700569687384956148729594684757967940624187444816284709795187071911745930696552653368563280152335806759463291445616305815635020119823325540401883441957651844750343638144096841335948259848770526833126881265143413329261783837590127633840513713216184519958638223117136482905399309599805483293937834956904092422281087037521920922937700307186372113906344124898059656785977607560084430167671144650244860534018392315473907073079820120751246987962508240927681544904189303493274609161947168150187391581321734635942965529503931058274748445866374685509271743153021560577378732307757255717238747987547257160649871363594167248319782025917635656246443753125037789370532880549504980251886068548949060590937998470292734275607113905641422892380984258263029234426262088919571518244932195478737397176573871445682702614490861975178742075516754733324587072203373758304994560106262390176006165395633410594701494799341390124978344884465749436634528822568868401653007551758051849229481478067017601902972384807279767752598765856160365789092274771000011985234415411196357098561672153309864964593322409334448112585872448474267720381881034114639346347807598228876412066054582514705864794713357428686815114781833742744686657972796040181487482492015475035549903655637747683084989454055492950340240062530249594408749735677969661540172359221943000977097777370962209821652305003449737505948253905412028913062393520782348010210574526259837612829616579247042150912908330755267860262215539556964577924143582448013819824636681075009217712883240808436212615826999205800811707553442891507626789004369399121871864912218585338088511986900721130240958787597315789847560814236879307715000000000000000001000000000000000000000000
submitted a deviation
Trees Revisited... Revisited
Mandelbrot set
Computed with Kalles Fraktaler
colored with Mandel Machine

The image Trees Revisited that I uploaded 2 years ago shows the evolution of trees in such a way that the overall shape looks like a 3rd power Mandelbrot set. This image shows the same process, evolving a whole "Trees Revisited" morphing instead of only a tree. Therefore this image shows evolution of an evolution.

Because Trees Revisited was already so deep that it was close to the limit of Mandel Machine (and in fact I designed it that way), an evolution of it can only be much deeper still. I looked at Kalles Fraktaler and at which depths the most significant slowdowns take place due to increased accuracy and different data types, but it couldn't go deep enough with enough speed. In January 2018 Claude (who maintains a fork of Kalles Fraktaler that is now the best version of the program) got the use of scaled long double working, allowing for renders up to a depth of E9800, before floatexp, the slowest datatype, needs to be used. So I designed this julia morphing with a depth just below E9800 to get the most out of it. Almost a year later now, the render is done. Without the scaled long double improvement it would have taken about 5 times longer.

I also used a new approach to glitches. There were thousands of glitches at sizes 1×1 up to 6×6, and they were all in areas that did not contain details that would be seen after applying anti-aliasing. I rendered at 30000×15000 and used 4×4 anti-aliasing, so a glitch of 4×4 pixels in size would be reduced to 1 pixel only. I figured that it would be good enough to replace the glitches by their surrounding color. For that I cut the glitches out of the image and used Photoshop's content-aware fill to fill the gaps. Areas of a single pixel (or 2 or 3) that are wrong don't really matter if they're very (and I mean very) close to the true value, which is the case here. Fractal renders are always an approximation anyway, with a finite resolution. This doesn't always work out! I could get away with it here only because the glitches were identified by Kalles Fraktaler as such (otherwise I wouldn't know which pixels to cut out) and because they were in environments without significant details at their size, given the resolution of the image.

Magnification:
5.13704035997 E9346
2^31049.1

Re = 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Im = 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submitted a deviation
Misaligned Butterdonut
Mandel Machine, Mandelbrot set

This is related to the idea to transform a tile-pattern into a tree. I described the idea in the description of Lattice evolution. Because the required depth is too extreme, it can't be done.

My new idea is to skip the part of building a Julia set approximation to make a tree out of, by using existing the properties of the tiling to make a tree out of it instead. What I realized it that a Julia set is not necessarily an absolute requirement to create a tree. While a tree can be created from a Julia set, there might be other structures that can be used. If such structure can be the tiling itself, the depth can be low enough. So that's what I tried to do. It worked and the result is this. It's a tree where the space between the branches is filled with tiles connected to the branches.

It doesn't look quite like I hoped though. It's hard to see where the branches are. It just looks like a chaos of tiles.

Magnification:
2^2273.6
2.6411808614749109046433703811336 E684

Coordinates:
Re = -1.76858393197743163040017958058875994716517000824190554354850661508642578475926800873151578575844048102401749186717846896260789640989507236353012591014191972755405519512328864313685151526161683438012991686732886921453771284270520773754990164571478559449298503352981954015945646557743523248657751804741471268447424271356043623849869880907080187926412754986912764944724325990913577941550702724488640238322256554552030405111321484762637414030347245850490669753599224706832607187698775149655687777122696154843034746174698752429151401899208619252257194467547494517765453948976029116380584861545121682483372402947867088491018003852101212660999767014664199009491310679052924989851699850936345990843100
Im = 0.00151753693981184637772620794847760595031542820643416112974324888497212267095256895417024645362312821781411091783761838207278940652885630348991914702322803184477613960573586037115263736000780221339566930906988427265271904330950789979690547601126940193902788878978222641928892486066820994591752346864787262454000237494802346923210397456807132397586196014865533460535375770394765897942646450343993986285262338492095821107390084208738278357449183830490757103217388006032031906638659852108355166153512653960552443499789277558254297926156348707040141738377852985064921978204852689141362999731321980447008068312145587758784022812773368068669177412427249149195145669476850880962057592612773255356400
submitted a deviation
Elephant shape-layer alternating evolution
Mandel machine, Mandelbrot set

There are 2 layers of stacked shapes. I perform evolution with 2 shapes, but between the evolution steps I also morph one of the extremely high ratational symmetric occurrences of one of the types of stacked shapes. The overall shape reminds me of 3rd order evolution.

Magnification:
2^7993.931
2.5882979488266437158498027473844 E2406

Coordinates:
Re = -1.74977473351308433946554680858057926568158071962806160157214596371394660395954890733614783436198144061946589313969954340796254161880794454689337556285417463144013304464568756825580144831469508137399241841846853405652268495084037241678543155873830106912862758117173842102427645753320347956610749579695828611801888613085987692473731422845158485311471337800123346524751523900956411418820290960750198521987744677384553939777389923715814143242562544518646491321234563918638819880798261813853940176877415873353717414992781230719323281775363535116768421496725266926977305705311729666678366475288999610429835785081682595724947183975915073579988740881312032883006113437042941788380054626116898973267286346011468339112193785123596576551566883878599657138669813177273350646777890884695856473346081750117129215011752630687728434813929262021623722141701584349846969551000904275938388038819870378863091701597219496970886193305612411023875493662451219355614883548083677193402253188634839539218657029639911682362660812179387157775932148423907412679387142611136931116387546434776752998840067263254262784587996470708729070456036816788411701678432569166181905336680395758426355636001121857746543143724619906240096440480321311973040754007196772246062638424667000270437473292182229264999098816785937960123045939545759260151990545095443637517248848906547060875174085457102319888845198698037647076657209332089054317367431007266204562956153206299329886908253505870848638864505836417545826973799842310856514297534264690928799780042346761251160888738915585857048103165081035252451637816570921042553744219811956498087087498408933478821717508115410614982796957430608862049199565565357985849840521898706269800496122639762655047047711859453220916408231401226070721830179827523025011303842939062158834641940041541176768772915064517372082099896521511309008868935348296185090958874357345045237943680722214018063927812345669411546883355402982997417281037949854907017208819364604758218020977663450153689339840170005236720509178192706164612367564371246633313418427872918837081824599672721442606611444343880694015210483296926046978010267239407978799032151061761392974841062765592482996040347439312709937246813684009325313683731259139726995199777024095276798618092045109170435644634514051352490297779427306095463039433984112157746588628684105844313288296807414929052423181273496370561002939064205710121729819867447237518218909242404285522457867126335300
Im = 0.00000000000000000000000000003514649090446868627177216845119135660468001726201493113147038904035846451602900274750466566486238052159244841375004186569707989971361966833098676431246312500185630274037110476959288692665293725054499767122158939889227242358364854409698676869130118070159435589794982124647480023336611292929195346229168479755755838898773037513424848600241462754815451434431337808100152964715089852726923708493525940352251060076504147569047509091951693130034402257061276902828664974980158956075098003479736180212107043525119754656775505126762154069204396777577480909057623822760261375982986105564814529063617487791144422022856980919151835072072142917992282378741591447649256542222620348772435071238249264205391554933088758899596980391600486233136454514785073758023659074640175993975706483049631761391464995180589591976427077464467837690023746834101277720020263868055554476743959355445625773843683032347670130508035129552753985433532817508507332590682498545769032843339074414105018995956260184836145171561977867214061179313580565232412283511678337342910790506720169840041675413021365693320282527222506670455942393623322957536654148193310997233091807029766917534574564179664818102888404646588093721111052202757641758716122995006335770816173377926081450819982086498999656608945580927292718490819027747320178451531744795751216818803378120531300805071029084736280637993486854065572363386574515250675108353819464933408126456071401174316830411198405105735025592506994729085028676845994948474627996816658811321793580164753792092704122620700018474119705813489598264066328160972214005175834490747195563970946642527523412925851133241211689591889482007635918670929153170102962492672244919352767715311563097687500446227645529068548814160128152046605013674572849731642313527342153573187546414213842482113488767585094392634058759234229790809621595951579155649762271701733842140008991237221176557910581280480616357727739866006268081601776424912710736862412184951898739460299144995918647540713315420906009466846069202179487116471532494471333077022735423372703010612296688899936394046019372275717362157444544981601842138599749194495354963889725267046904494702844438096762518474818722809333479490064671671745811420207338372437770179416591286842086961397728122769050657435137190163579890448695026618732799369664634568537182489470494001543497184919927109654952804768327538574957421884570337986998380744901611876535078926713600
submitted a deviation
Density near the cardoid 4
Mandel machine, Mandelbrot set

A dense area rendered in 9 tiles to achieve a resolution of 79200×59400. Still it's not perfectly smooth.

Magnification:
2^24.2
1.9392560947398473961355372594916 E7

Coordinates:
Re = 0.18764408324767
Im = 0.55256192550900
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":

## 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

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

Dinkydau Set
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: KlaudFavourite genre of music: classical, deep house, electro, dubstepFavourite style of art: fractal flamesOperating System: Windows 7Favourite cartoon character: Donald DuckPersonal Quote: The world seems complex, but that's just because we're part of it.

Featured By Owner Feb 12, 2019
Thank you for the of
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thanks for the favs.
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Happy Birthday!!!
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Thanks
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You're welcome!! ^^
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