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# DinkydauSet

Dinkydau Set

Connected tiled four-armed dragon 13 6 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 19 9 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

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## Activity

posted a Journal
Almost all of my images from the last few years are the result of using Julia morphings that occur in the Mandelbrot set. It takes a lot of time and effort to find their location and render them. Therefore it is extremely beneficial to have a good idea of what the result is going to look like. If you explore the Mandelbrot set a lot you can do a simulation in your head. You can then zoom to the location and see if it looks any good in reality. In this article I give information about Julia morphings and how to simulate them with a computer instead, to get a better idea of what to expect when zooming in on the Mandelbrot set. I also talk about whether such a simulation can replace real renders of deep locations.

## Julia morphings

Mathematically, I define a Julia morphing as a transformation of the complex plane of the form
c -> f(c) + coordinate
where f is a polynomial function. For the Mandelbrot set this is:
c -> c^2 + coordinate

It follows that each coordinate in the complex plane then defines a Julia morphing.

I call such morphing a Julia morphing because, by definition of the Julia set, when the same Julia morphing is repeated infinitely many times, those c that have not been sent to infinity form the Julia set of f(c) + coordinate. A Julia morphing is like a single iteration.

There's also another way to look at it. This video shows what repeating a Julia morphing does to the plane. It shows the Julia morphings being applied one by one:

As you can see, a Julia set is approached as the number of morphings increases.

## Julia morphings in the Mandelbrot set

Julia morphings occur inside the Mandelbrot set. All the variation of shapes that you can find is caused by it.

First let me say what I mean by a region. In fractal software you always have a particular region visible, usually determined by the center coordinate and the image width or magnification factor. It's really just a rectangular subset of the complex plane.

Let a region with zoom depth 2^n (and that contains a part of the Mandelbrot set, otherwise it's silly) be given and call it R. Then you can choose a minibrot inside it. Let the depth of the minibrot be 2^m. Then at the location of the minibrot and at the average of the depths 2^ ( (m+n)/2 ), it looks like the region R with a Julia morphing c -> c^2 + coordinate applied to it where the coordinate is the coordinate of the minibrot.

What that means in practice is: if you see some shape in the Mandelbrot set, anywhere, and you zoom further in on somewhere that leads to a minibrot, then when you're half way there you get a Julia morphing of the shape. Here's an example image of Julia morphings occurring in the Mandelbrot set, taken from my previous journal Newton-Raphson zooming and Evolution zoom method.

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.

## Simulating Julia morphings instead of zooming deep!?

I just mentioned two paragraphs ago what a Julia morphing is, with mathematical precision. You can find them in the Mandelbrot by zooming deep, requiring high-precision calculations and therefore long render times. Does the fact that the morphings seen in the Mandelbrot set correspond to Julia morphings mean that they can be pretty easily simulated without having to zoom deep at all? All you have to do is apply c^2 + a coordinate of a minibrot before iterating c!

Unfortunately the answer is no. The reason is that morphings in the Mandelbrot set are not exactly the same. It's only approximately true, and the approximation is better the deeper you zoom. The problem is that minibrots (including the entire set) deform the plane as well. The effect is so strong at low depths that the technique of pre-calculating a Julia morphing result is completely worthless, in fact. Note that repeatedly using the same minibrot to get another Julia morphing effect means that the depth of the result is averaged with the minibrot depth again and again. Eventually the depth of the morphing will be roughly the same as that of the minibrot. The morphing is then just a ring around the minibrot, highly deformed by the minibrot. Clearly a simulation doesn't work there.

To which extent simulation of Julia morphings can be used to reduce render time is something that still requires research. As long as the minibrot is so small that its influence on the shape of the morphing is negligible it should be possible to use a simulation, but it's not clear to me when that is, and what happens when multiple Julia morphings are simulated.

## Simulating Julia morphings to test ideas

As said, Julia morphing simulation is not a replacement for rendering deep zooms, but it can still give a good impression of what the real result looks like. For that I made my own program.

No other fractal software currently implements Julia morphing simulation. Ultra Fractal is actually flexible enough to do it with layers of transformations but that's not user friendly and doesn't allow fast exploration. I wanted a program that makes testing the effect of Julia morphings as easy as possible: with a single click.

ExploreFractals.exe   518 KB
mega.nz/#!Z88Q1QiI!pwRXrL6c3UN…
CRC32: 76357D65
MD5: 2915B31F8CC4C15B0F3EA293AFC63A22

With this program you can test the effect of inflections / Julia morphings on the Mandelbrot set with powers 2, 3, 4 and 5. This works by transforming the plane and then actually rendering the fractal. Each click adds an inflection at the location of the cursor. You can also work with Julia sets. You can go somewhere in the M-set, then use "Toggle Julia" which uses the center of the screen as the seed for the Julia set.

fractalforums.org/other/55/exp…
Also when I release a new version I will post it there.
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
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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
Almost all of my images from the last few years are the result of using Julia morphings that occur in the Mandelbrot set. It takes a lot of time and effort to find their location and render them. Therefore it is extremely beneficial to have a good idea of what the result is going to look like. If you explore the Mandelbrot set a lot you can do a simulation in your head. You can then zoom to the location and see if it looks any good in reality. In this article I give information about Julia morphings and how to simulate them with a computer instead, to get a better idea of what to expect when zooming in on the Mandelbrot set. I also talk about whether such a simulation can replace real renders of deep locations.

## Julia morphings

Mathematically, I define a Julia morphing as a transformation of the complex plane of the form
c -> f(c) + coordinate
where f is a polynomial function. For the Mandelbrot set this is:
c -> c^2 + coordinate

It follows that each coordinate in the complex plane then defines a Julia morphing.

I call such morphing a Julia morphing because, by definition of the Julia set, when the same Julia morphing is repeated infinitely many times, those c that have not been sent to infinity form the Julia set of f(c) + coordinate. A Julia morphing is like a single iteration.

There's also another way to look at it. This video shows what repeating a Julia morphing does to the plane. It shows the Julia morphings being applied one by one:

As you can see, a Julia set is approached as the number of morphings increases.

## Julia morphings in the Mandelbrot set

Julia morphings occur inside the Mandelbrot set. All the variation of shapes that you can find is caused by it.

First let me say what I mean by a region. In fractal software you always have a particular region visible, usually determined by the center coordinate and the image width or magnification factor. It's really just a rectangular subset of the complex plane.

Let a region with zoom depth 2^n (and that contains a part of the Mandelbrot set, otherwise it's silly) be given and call it R. Then you can choose a minibrot inside it. Let the depth of the minibrot be 2^m. Then at the location of the minibrot and at the average of the depths 2^ ( (m+n)/2 ), it looks like the region R with a Julia morphing c -> c^2 + coordinate applied to it where the coordinate is the coordinate of the minibrot.

What that means in practice is: if you see some shape in the Mandelbrot set, anywhere, and you zoom further in on somewhere that leads to a minibrot, then when you're half way there you get a Julia morphing of the shape. Here's an example image of Julia morphings occurring in the Mandelbrot set, taken from my previous journal Newton-Raphson zooming and Evolution zoom method.

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.

## Simulating Julia morphings instead of zooming deep!?

I just mentioned two paragraphs ago what a Julia morphing is, with mathematical precision. You can find them in the Mandelbrot by zooming deep, requiring high-precision calculations and therefore long render times. Does the fact that the morphings seen in the Mandelbrot set correspond to Julia morphings mean that they can be pretty easily simulated without having to zoom deep at all? All you have to do is apply c^2 + a coordinate of a minibrot before iterating c!

Unfortunately the answer is no. The reason is that morphings in the Mandelbrot set are not exactly the same. It's only approximately true, and the approximation is better the deeper you zoom. The problem is that minibrots (including the entire set) deform the plane as well. The effect is so strong at low depths that the technique of pre-calculating a Julia morphing result is completely worthless, in fact. Note that repeatedly using the same minibrot to get another Julia morphing effect means that the depth of the result is averaged with the minibrot depth again and again. Eventually the depth of the morphing will be roughly the same as that of the minibrot. The morphing is then just a ring around the minibrot, highly deformed by the minibrot. Clearly a simulation doesn't work there.

To which extent simulation of Julia morphings can be used to reduce render time is something that still requires research. As long as the minibrot is so small that its influence on the shape of the morphing is negligible it should be possible to use a simulation, but it's not clear to me when that is, and what happens when multiple Julia morphings are simulated.

## Simulating Julia morphings to test ideas

As said, Julia morphing simulation is not a replacement for rendering deep zooms, but it can still give a good impression of what the real result looks like. For that I made my own program.

No other fractal software currently implements Julia morphing simulation. Ultra Fractal is actually flexible enough to do it with layers of transformations but that's not user friendly and doesn't allow fast exploration. I wanted a program that makes testing the effect of Julia morphings as easy as possible: with a single click.

ExploreFractals.exe   518 KB
mega.nz/#!Z88Q1QiI!pwRXrL6c3UN…
CRC32: 76357D65
MD5: 2915B31F8CC4C15B0F3EA293AFC63A22

With this program you can test the effect of inflections / Julia morphings on the Mandelbrot set with powers 2, 3, 4 and 5. This works by transforming the plane and then actually rendering the fractal. Each click adds an inflection at the location of the cursor. You can also work with Julia sets. You can go somewhere in the M-set, then use "Toggle Julia" which uses the center of the screen as the seed for the Julia set.

fractalforums.org/other/55/exp…
Also when I release a new version I will post it there.

## 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.

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thanks for the favs.
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You're welcome!! ^^
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