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About Digital Art / Hobbyist Dinkydau Set23/Male/Netherlands Group :iconmandelbrot-portraits: Mandelbrot-Portraits
 
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Three layers of curly elephant morphings by DinkydauSet Three layers of curly elephant morphings :icondinkydauset:DinkydauSet 9 3 Elephant spiral tiling with central transformation by DinkydauSet Elephant spiral tiling with central transformation :icondinkydauset:DinkydauSet 10 4 Dragon Tree with Crosses by DinkydauSet Dragon Tree with Crosses :icondinkydauset:DinkydauSet 5 0 Triangle tiling with layered julia morphing by DinkydauSet Triangle tiling with layered julia morphing :icondinkydauset:DinkydauSet 13 3 Elephant waves by DinkydauSet Elephant waves :icondinkydauset:DinkydauSet 8 3 Sequence evolution by DinkydauSet Sequence evolution :icondinkydauset:DinkydauSet 5 0 Dense deformed trees by DinkydauSet Dense deformed trees :icondinkydauset:DinkydauSet 9 3 Hyperbolic tiling with golden ratio tiling pistil by DinkydauSet Hyperbolic tiling with golden ratio tiling pistil :icondinkydauset:DinkydauSet 8 0 Hyperbolic Tiling by DinkydauSet Hyperbolic Tiling :icondinkydauset:DinkydauSet 6 0 Spiral path morphing by DinkydauSet Spiral path morphing :icondinkydauset:DinkydauSet 9 0 Golden Ratio Tiling by DinkydauSet Golden Ratio Tiling :icondinkydauset:DinkydauSet 9 0 Dragon (Deep version) by DinkydauSet Dragon (Deep version) :icondinkydauset:DinkydauSet 12 1 Density near the cardoid by DinkydauSet Density near the cardoid :icondinkydauset:DinkydauSet 4 0 5th order Evolution of evolution (model) by DinkydauSet 5th order Evolution of evolution (model) :icondinkydauset:DinkydauSet 1 0 Dragon (surface version) by DinkydauSet Dragon (surface version) :icondinkydauset:DinkydauSet 4 0 Trees Revisited by DinkydauSet Trees Revisited :icondinkydauset:DinkydauSet 11 9

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Three layers of curly elephant morphings
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:
Re = -1.749365089130713355136630080718347789559672431892481991549752887987257790534288948852686176092823166880490367584365537241402861301051987572756134885766638315030733184102356683745567733704732242336955716689374220053239462271783286969695153138595960968290613096949326592408278551176423296957665693781278222135957297409064817652373422135574657112969716927947485486819016815962443472752011902737416089338071774841485991253273631345957163307488029852241047470109385772255147111632339006553554769628100546684622495780340381355203792634585520532761334503955692314908107291929614735326860054327591263812484743920329421708025534820190742048371429639420053196044998160293387459457173088444684708887203035147442894047082313460968372046848337015731658271372561131887628465964557653463317650999148237981158362626539221835897659252524262512209501785961590731030815348889046456268071858340927774884222761306721255869200087387892697348047269984941762337148007330658012362815577473768915429790
Im = 0.000000749338763025441077635067897843290324328364823725106652774163469955659130000553036152692348499586259211773488938327142632275497110083700087625810153356917154351213845759552072775573138870307623452220908006924373386358851662101475560714902845049069557710671495531000214771157708102117014124261161460873019872562499024280227792829384747412179007566059304454037672086576578431671511987764318814190590560786798393590730745886310081445382254877137292162499001064519808832805042668998490623157561409593714128196422705485718006062939517024762567595955043641585945566334004032551810905620950364643187700183288907723485916230573975063135198454571171623145315843288678780241131693068481538731006056464310869184858150811508962009441974266124120904462670554045286020287151700359927450985717710570585083311930785250338126621250447636233137838208798719630214780937239875875238050692819292432690046739868306452339999403349451922613923505714140121592722817731635276657757981771752651230
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Elephant spiral tiling with central transformation
Mandel Machine, Mandelbrot set

The whole thing is a tiling. In my previous tiling image "Triangle tiling with layered julia morphing" I attempted to make the best suitable julia morphing to be inside a tiling. An important aspect was that it had to connect to the tiling in a visually understandable way. In this image, however, I transformed the tiling itself. I'm using the word transformation for something I would usually call a julia morphing because what I've been morphing here looks more like a tiling than your typical julia set.

The result is something that kinda looks like a cross. Originally I wanted to make a tree-shape. It's something that I have wanted to do for quite a while but there are some obstructions:
1. Transforming a tiling shrinks it by a factor 2 with each transformation. For it to be screen-filling, the initial steps to build the tiling need to be done twice as many times, doubling the required depth. Reaching the desired result of a transformation already requires 1.5 extra depth normally, so each transformation requires 3 times the depth in the tiling situation. That is a very significant difference that gets bigger and bigger as the number of transformations increases. Depth is a problem because it's harder to reach and render at much greater depths.
2. The size of the tiles becomes too small to see them properly after too many transformations.
3. Currently my best idea to do it is this: I know how to make a tree out of a julia set, so I need to make a julia set out of the tiling first. It needs to have enough details to be able to build a tree out of it, requiring equally as many morphings as the number of times new branches split off the center of the tree. It means the number of transformations to build the tree is double that, effectively increasing the depth by a factor of 3^2 = 9 (!!) for each extra increase in number of arms. You can imagine how much larger 9^n is than 1.5^n (n being the number of transformations), as would be the normal depth increase when making a tree. Without any improvements to this, the idea is impractical.

I've said something about making a julia set out of a tiling. The way I was going to accomplish that is by doubling the center of the tiling, the doubling the morphing, doubling that etc. which results in something that, for my intentions, is similar enough to a peanut julia set. It just doesn't have (infinite) self-similarity. What you see here is an early phase of that idea, doubled for 4-fold rotational symmetry.

Magnification:
2^9870
1.4657408896420228648538645519085e+2971

Coordinates:
Re = -1.7498439088634276800460948042342922498295334226571513188740914298952160154495160835921441755475654893999982369876215851617592098468229564680702922351168965760097475776102438594727651509787077718910539436125381937707617605884622639199601316173201755267877495940801859921008741036365835317535325754579310659698172718927559068739259993943489903422220415937003111035235632482550767932444453657416776260831329826364139257446420512980873151922763831367430838804091679690611736967486334921632295099429770359931163075192348893492407688808513153245277731668770979345334874317532468385607661936063534917048525309601960566949936316979336866617749321597297823714013372646899964899346285783803172833516418670799966711933325159465508213919277200803000055872852120819029687577982703573729797193182111971306136776267365805118136976570306298588531306446692744199554252596954263700577089578723507789675760521447513869517740980613852561229822340340821960377577072823654587832275540464160845654172941864633386678150003773464150372439849205478866686753163400567627899379488621753733939322237710510489899894319695620114081220960799108362857739101378201141015236614994736212399978478462228663563168926677891397591332060282422583766336444493991592761516899516720428428445030542837098451502818031894096231891126049834098808975971389759975985150362816201102121018083628059439863570774916034325209057641891673750907649855902580808823385917908184577563638857597924200360805831466148522376361731640904864693988420015313810216766041336750368569166250692222461154246966278977859304313592575028322631156576324942021305183391989500220481100230016404570236474879345415061504192883601549161673363117250118782954658561767077600417297183475910010828666415715253229639189256362950223239974866161213210866713886300308480188590857334497621417446574144421055762236175453201119570339202660482102753858422401131043468838188246221038734014328251300488439192100059396153720293433575490048680511435309255321529303432822265191296671629725830605376213014511810829167186060245587406244714168349595569030001662916290777088899112276716205440876804982372349280290334339551559023028248022816778418359021526473978858832783517847281048800951679546036790338353953143256548965295380780837896255691843446714391071948040002697770654394052015260255393376175280788923261119479110963358582732044643161872472885492978505953850286721671320296384955791562868032502403710449580790524128525636701394699880064688726954218490244554281678600023139268604824255407806421895967872781732308596780588474743680047670637781940624315346439381802681954581244045582241975959875957962316604643837152815035116622127450845860253963974889814159946447197039023379170539609194628016770286801768992266381824648034095425613913539495478717100958571514945248496902412048555941047782774510767649781516106282821761253790079756292170876360361748187034655726742995944555407678379822885287868173604717194566230010229214603428202446987503287921106362107355612486404563715497

Im = 0.0000000159959473146319642717540099365976948500705065715209798252640461993465538628308271291417567856676375259782625663886020914208896933202641339653886737470769520025797007857355688277340916809887701394149355736780982888336303707426011948432980348233270700327920071207166294721964801722915416135341446136599969054710440421562301919810640516949611055655445374278378523875044932672841451066063600543720534132018364998118432787011667094357767592746119445662386613217689596616495795570879453511631475015465202878892577516514921740593502750764153271780790082960896474165423118429738903206147242877552749256873871105093358252352732331060728483105073241167843748437518392524351874358787417493003841145166331703983481875389753117117932032962953506584753054080076767788009823281476265482712320371045802342552403470963012571639687849320512232492175287872997902858556379366234132626695909943079162218988205311016002657446116689747111210533111481206942309139308952184605691152141960826333395982721345476132751975971022146535204595433352512002578302892490105214373912229371581958063660115836891169373147116580207143925553002261364624545240328038200540148829737033585513962225949452826609947213453364282239248466862885661935973676913754680865468875965089162465135074498164822314446163324863073935405497487064708583916816516533975726680127663202426511499033222075508877083212530649296209358665321153111812420249785544611998025674846336691047595198408371194264939589594429132157351553726174718486809904515040993978014361545147263981626591617347758542367019696332480685643916496305189291127422116306514284473360959979610585543698541400957830806278941724934040493095816324802704019794878310546321765246167763900378695196267121408050678909760576027231496049406396550408670379362262182178544798472264457901620793091459280836757386717166694446359647130402178640031711905205225841807086759410586590898742578188102841587698798631546687215830493331902873137159054492715093065045167280084467604666680883585150536344069812477725101393596817253505556349472358259816524030849996642801406956427167170327092676365024688733492601876964692824759894626974050081365969753000708139172781769560750226186916342763822262989132793630073189830957326040774791732039169455213647452293020342005664871046362252739644996046672804407485968984950340837280695544530862471085846992558012998912101664588802674745392701373309979629845938837179272067628387974487488874154567647656115499120884893301050640923839135312465224721530140263802091571317219010359968405330584283214583937484038726235020015489667750541678639777104491176726074347886713693804393387049866338689190684271772913934340107566951726283444179319632633810736907114231121232694263979534185977644430817648323272004289857100248104618629806930539306672302326896495469553220010205700230257475740272421099410035434715924022827075312970916872859580939642662854699129713633533001625605559242928706357843554207880763875716758293274507690637883513572704426297634947130620506
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Dragon Tree with Crosses
Mandel Machine, Mandelbrot set

This is very much like a normal tree except it has a "connected" interior with some crosses inside it. The crosses are achieved by making a stripe out of a low-depth embedded julia set by zooming into one of the main spikes of it, before doing anything else whatsoever. Then, if you deviate from the current zoom path (=the way to the closest minibrot? I think that's how it works), the first things you will see are the julia set and the stripe again because of the rule: If you deviate from the current zoom path, you will see everything you've seen thus far again.  This causes the stripe to be all over the julia set.

Each time I increased the number of arms of the tree by morphing it, I zoomed into a strip inside it, thereby morphing the stripe into a cross. After n morphings, the number of crosses has reached 2^n - 1. This technique is an exploitation of the fact that "what happened thus far" is just 1 thing if only one morphing has been performed. Shapes other than crosses are possible but with the severe restriction that they require only 2 julia morphings to be made.

I wanted to publish a new render sooner but I couldn't due to computer and internet problems.

Magnification:
9937.724
3.5728322825644750137997369306096 E2991

Coordinates:
Re = -1.674364274409247056603320660316824291970062080466541705865560857514382711264381692391451883778167828857329201573088543356588985766062306089895493333839285042788226726677595275366306242750145216885463147565567474862604416261650846284872472465600450893875524615783023617525710353201753108424509867471987138259001030929193190257819073677175100723852940699115248913293168821487104061449545491631051406293054976737459367481850008115092269818719734865312103974302210015932153840455404040157021485400702101168210188771065769722284590129215517165387174500770793312078593263338312594734548134878619509663429586307547732042821766089499919742900707219606915115849846801107087520881411133151676030353408340982573470025825761630441460328129901466249305886338781941212603012374275159494065916072249954878131716732287280011025965861954279370136215182084540844059744932438037348130437853167311846775225704650375858138851127822075084116361941938914845130880136934326125923195131408387017248499188412482964345903899167305387562133010549689893116398282067318135507950676416282694063841483392803688797051776281618715564589820037717309455570188774581997239588388350010896731676949877670747009414625621358296846538667790799726441198100364300343046328393666959818708022982397960166464337974286727140690404934174725525367050768641098770911362560825038751962502526899632804940268295378718753613181256415892883387487578373020210726434716909533954219074351036105687479708986970938006524819499490191710248908640269568407347625381390710334138559123133945780970959415656043832501672797883930756333910229726824291965947826807054218203617943739661160966129744228079904915302657670156019053530875304491508694706669262099444154674173996825443294545330740312620297851767049976911599032063855417151818935495914521815410009393008276881759426938274326547667749262733114170356888447566163396116220555016322626579988097414618977911188245538396896242976964208367430484740914354354528808307152787999057668887103517988627124267089471758240320611809387082004487035345009122755385997984145162488913787106975210603811467950410491633850332653831663071408005434622635171169985863573277688673027292642172734040779969307366509521300841222857437913686314792646647028800345640911788424421367051102200944116602994432427570544317350529438499599780444345183597397644823580098767289539464663268209626604647709544185679901285362670310529049634322640031020909033611307225896851076488091273514348023315970064480536354408463272202124192636623090058926964005367999418811975933355629375354240368054332341818538310934418703362565445715152002093582223578198857156572708021877000145696658169354591325063201633608232156674959072843115622394357111645826411812810925045168630404874260497850744961467429362371018169829441663779912873383465377392315629330724767906707072002408246102902518571639116851530501918164958403283652496104465910387686114717039643792567655290370494523755672818407386924171118632398933629073020312842337862256353043936666033255729927952494209336

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Triangle tiling with layered julia morphing
Mandel Machine, Mandelbrot set

This is somewhat like my render "Golden Ratio tiling" except there are triangles instead of squares, roughly, and there is a morphing at the center that is connected to the tiling. Notice how the tiling looks odd close to the sharp peaks. This image shows what a julia transformation can do to the plane.

Magnification:
2^7561.284
1.4903550178318449484728738200504 E2276

Coordinates:
Re = -1.25539673868026755865583057881474899884324965805338977974834931161909270753708598428507955786132265725790787341032505996383790425294770009682926760678760038756350899655215102205878644724210739492794886576603695584822056326228944405322883876128043521419784068398650048082189903328094223666928063926518382846840131859485476315478235880711721658003833441256818569046853194430529350784682119536773108833463488393163697761402192429756097387714961858678744874458199522006130576804388906748570341814504336685457587167749548288322872130218471448274839817042698717905831296282896184572296128713908967449798472116056514881888801683939223851000287431323336894442496881378705777070710371054440806027399735639396692322947494320719306470574708085702346217592836955660079971867889777798610664668864742574634950499717744718773283207042687948226229603847906214983193112664398233916204454147724975238550562547821344963081493668638526943588769755260131804899721402074907287731519191451163524738558324798559761025301908168987100951527274712280617758661403864423907751426459851506821125537839398255354437258208725148760906354197699944849968108933608465947163436828331417300771517124475934726602173762300539357803205221245585415774897112143250119426714150301309801319608451015128386035191726911921409455877878074524972060731613571786401216370540706975177651621092906520896665796647089230257511709648826321414635400998938455772429223454925573977621930697636796075847672588969185212087053053108181349608410897392328263641074037370638422842713171865370885525752501270851054922936215247758843441612554173733605068983133288967390278512576317125435856073016415738023888947607111601785315411777998870973224588870831157000745915650340811818424894326691568333333287310663932565094814901141779668577091589977159231317802556812362499811007994432885830314171973035269633162611613719150560321254470550921982493890373265712289506368372242619429278367534066301408804736183112417208921239690694152920285716471700108933703166679275908879135641167270283866791874989692797333223912642355592898942261239801754647548769348903336461818747933068920743148283632266306881018573511463953740275679828863365826198203134232482131820640409927700991821156544305676637683125693206284943518734414060423813465798754462992799430697523597233

Im = 0.38224857379114731727780160618658277870068837021245609010295408004674654682792434651526067447152075016911712054055366594385153982530268591629940094738282878381005830478368034558265856542451001686964767561086991338229754750137888164432857760503555842214222748025269569910342171779788637618093216751365225088244974732240931656948930320932483314069560030252947914416671414075723933875670577050947611642505633422489719007040329288770428757402243349059412605593643265949963772513244211135482494958912615966962376348986136117109044491259528402746005899258602303394927333974607538059934632921409448592026483942987894561898331313170398847213274560540208498080978196884537151786484867735486753128144430866094488699815209010954095152294908726663605582313599516699642043260982989596201950521373657665888156922245012034452507667911384714233976586046957767863867279164854686440895021022140335576849458186739533079907091119760435026551937347806292204466683044092574239012840186428481524219398943802887008041935819677278927676704522741056396257948015133645827160792737869947636479090032335098323726897300010931435172681809140933116797070843783091397880472133475422388098864490610439426060149139216159026208588057288395717243484806167182597013551312010188295824752786932984887460712578604990706454971450496155631761424072659784071238345487466805278971489867754392033632748812447312563394897601361535407192856390587221545771654081274649187234430101177287585350211747493805773283204901741582650483692607104969109485572537275107060195960439335093314127023513241367490068166620554906791774001717991666159176505400523230151821683091826328729376557926320188766146883076918559315535454082956773400405689949643379087530045232955917580766583998251709877442400316338096059478353544725752163596160773833593059946203114921949849385337925897088224267096835539727189957953695650604725670584459478345615025117820704668603117497969156810146795706772328220444509536691189407404386319819509366133216601247025480866130704549612899154869401996166840596106974136261677011932709383792444954406525867356896445713414748391884204484183065722694821005218717482082087096172332556051444206891571881664322252754694587273228380586639882319729340727003245417557444363395896390486793186299739084951434426861761486920154278237706587
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Elephant waves
Mandel Machine, Mandelbrot set

Magnification:
2^4182.477
1.124690883091685866186702988919 E1259

Coordinates:
Re = -1.74975917037800050882559106905118750347894214705113542310138791588283442362476794679447379697098337897360380421747537132084177320502872411235004407548291182180220013815784315475119718890858259656345535782423842150705516620434263761115431902470969102299880098926517978912243548772936862151476086759300271704388126344726012474333639866928628752210535408485545059595327550722505752203093841907450401826993892093789118576467344955542020395570425414462370414055524690121110917933282128973112601913616234372074060239459335785573755715506635897305574938124965241397942822443513866282490361782964557196272987423432943989613563484609378000234123636460371172469332910496908840112367523249650308846385894020387183664455996071760284596565209943953396746546991668526196540545907704963402114759408854092385724106953664362453122799519641510073320189430787797737692402590502745389320262056722558822620115778743534192455984051258116177551002351462337179828974581234931469341402834449247747148613854365941333373404789508697903427259196456082684107101864376356383411125025541612063313422327656067398776838299189210018921060489815307622553023409775956982585571816789705098897391324969757769978964451887553404728997176032184406315075365024771720926778298394168977346537844147588162605229

Im = 0.00000065494321464889382395974433548871503861114693049267081515861761243775606941335923766366505355757907021252213122344880630973753203925749497175311893412307237560794258013506711204865294085458745997756305392378064184786596023324128022855641754079645576938607519951982259780438763729788334075452120238755072608348553720523565328171546892700838522789925063190670592587502234964120757097262904912008197340907217242063047632748247153144932253629421239182929452834832405234711442216705522169867738484632487692296978366791361910635637442128464391575578753412821439830273798551500960009077448557732252708009579372346725123703729993352767677363408828600982651140875896265497435630538832509926132678178011558566914911752363563027266475171854228449032212224282831660095165993611404467019808433602873368347067095976691348182625375509713461911953891646629928543220165098623138887791782006927265737975906706782192103382769986044941530583462560200703142636177894286413245330140929999169372093753516169686080903702948047663760448820578181151981451428626324919991051944807585171714992242287793310021623618003330578825893164347407043190662280917017920211141450056073171323522898256168810102684391651125295361344625916133328594687384738376690067347633459855395375232360091982145566
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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 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: 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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Comments


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:iconfractalmonster:
FractalMonster Featured By Owner Jun 11, 2017
.. and also for the :+fav: of Alien Forest :wave:
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:iconfractalmonster:
FractalMonster Featured By Owner Jun 9, 2017
:wave:
Thank you for the :+fav: of Border against Chaos :)
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:iconkuzy62:
kuzy62 Featured By Owner May 13, 2017  Hobbyist Digital Artist
Thanks for the watch!
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:iconflyingmatthew:
FlyingMatthew Featured By Owner Mar 1, 2017  Hobbyist Digital Artist
Thanks for the watch 
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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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:iconfractalmonster:
FractalMonster Featured By Owner Jul 27, 2016
:iconhappybirthdaysignplz: :iconbouquetplz: :iconcakeplz: :icondinkydauset: :iconcakeplz: :iconbouquetplz: :iconhappybirthdaysignplz:
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:icondinkydauset:
DinkydauSet Featured By Owner Aug 7, 2016  Hobbyist Digital Artist
thanks!
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:iconfractalmonster:
FractalMonster Featured By Owner Aug 8, 2016
No problem :)
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