*Continued* SuperFractalThing: Arbitrary precision mandelbrot set rendering in Java.

[stardust4ever]

One can skip here over 99% of iterations. Very skippable.  ;DThe whole region is dominated by the central minibrot of period 82432 so it's trivial to go further in that direction and using this as reference the image can be rendered cleanly without further references according to my program (for the resolution I chose which was < 4K ). Minibrot has size ~5E-1197. There is no problem in such a case if you want to go towards the central minibrot. There is only a problem when the dominant minibrot is further up. Here something based on image features might be useful if you need to go down in the center of the current circular structures. Maybe the period of the upcoming minibrot can be computed from the current position before it's dominant. There must be some rules...

How does one determine "dominance"? I imagine it's something totally different from iteration bailout or the general shapes or complexity of Mandelbrot features which we find so interesting.

[hapf]

How does one determine "dominance"? I imagine it's something totally different from iteration bailout or the general shapes or complexity of Mandelbrot features which we find so interesting.

Each pixel has a "period" like the big cardioid and its bulbs and all the copies/minibrots with their bulbs and cardioids have. It is the iteration number at which the iterated value has its lowest absolute value. If all the pixels in the image have the same "period" as the cardioid of a minibrot in the region this minibrot is dominant in that image/region. The pixels either have the period of a dominant minibrot further down or further up or they have different periods because several (mini)brots have their own regions of dominance in the image. There can be thousands and thousands such as in some spirals spiraling towards a Misiurewicz point.

[Kalles Fraktaler]

Skippable due to Series Approximation I assume?

I loaded the location in KF in 640x360:
Max-iteration: 165163
Min-iteration: 163292
With 10 terms 154561 iteration can be skipped (94.7%), image rendered in 37 seconds
With 20 terms 159713 iteration can be skipped (97.8%), image rendered in 19 seconds
With 40 terms 162289 iteration can be skipped (99.4%), image rendered in 14 seconds
Using 100 terms unfortunately gives incorrect result and only gives a small number of additional skipped iterations.

Next doubling of "chromosomes" is this location, following your method and going just left of the leftmost chromosome:

Code:

re:-1.76973927107570278411520061419065946542347799372595152472101192040064094711336613121195702934273509531110920827719735988557827595999806330371323751859876701146210458806064544177069996999828391655185561042469808511402300223535883114455472484511423365855441119164893414414188906330197301018546484636691096364711626721208178249174072183033484198989852428081400589644144574714652215363016893192108533799955241566248299427809467589643090726924845245039740319491540054239489718441137692571568233952550429974345579671201229362878862729943368276039880766440708454565350593901104555321239593383546867095019659471571166180242494183400421258098231638606655519421918832801505565036377462175161679479913432266064848634617218812509508674693057633239911906027116361934291810218686634514232406561410364816866722072165101789409275784262461387890519359482022010390695157846380903429796008812574713407054196678588644723528860910520590905196670451139094295823495172577538778175845923533617587491427006033016602930900278404641562369796189524200550352924161304993887520131641880565034516078770708206943090725951473272863101719316646550470305393493596164656491513320724401548205844175371960034916997591335500423936357608167240915110205662919250126158696276192747025878258149810015608721712554623391357094690983844241174709909015252279070297968544176041518336849095481262345439207029158860533966756384084999639478493365075861692547428353717288319345008285956733788737950087166990901738963854816259215373861305911053377248373347327533386764341081400371266836934831405817979283935531422790742412459811321804473839844435089769762155990155250178014931357778215347716254080455258628723421585799192229526300512890358421750654418464175339338700097626083812582737428156995792247813443649353878917473590509904660167197369608670966587464420205758779389875106
im:0.00479575593386521785056156632204869711543048868866739207812561117523981061072803666759892986917797971367373649618307585237412544802866649204604791951282119532611555692231864126608560335395830677937445526989723059565598206590367891898467394911556358386179234972500593142081869841790362238266553240041814213356337388982751141354993190142472710950391187727081323027505479136452308448064386283517972781015524209855681335788673744269374677859162070197587477940738704947651087173844617863477560652949841356272962069000764292715737566241003234711235604702236267464602650182979303592890434112792457622396987721882609742859931177341818273187666138340519774842720206982492134896277509574980239897474381333076472746958026712349438349815843426360562006556338835568676503377969930912294441203580507236635231202356397041368243232643242096171465697636427613416621899589621369247195520694822931509253743782828991308508271169923822030731210535571166710661634286964536828249166255244271393901158417890480842237451269264760879763337428453836904776814955516384029161979047340525822848503691702878388529457727649610800474540097130947938914768777069390653907386626598219197537707103472852773928487954022576173090300830083242555816576865669179621054197697073306371497138708514154639978238654075859456762116645947307361805048300920909051144054863714541677404962777535046378220013445059535852715247670869344676733074159242947217492726089180421013247557742926237274643330802114405643723680982693493678065080321054992095840747165747342651598907576468756405957676973239464394898303249003035346570430635955169214125771039539353732135890401126564177895335144578423771521555862079611886138494457318930595157328431568295871786565347458710924343342091618717199655736488581664771241781759853956839288876770557346040681486625539604836220182498710582505322335248
zooms: 4478
zoom: 1.05E1348
size: 3.815E-1348

Iterations from 327205 of which 324577 (99.2%) can be skipped with 40 terms (1m11sec in KF)
Location parameters calculated with claude's methods in 55 minutes
Period of minibrot: 164864
Netwon-Rhapson iterations needed: 8
Zooms of minibrot: 5986.7

[stardust4ever]

Each pixel has a "period" like the big cardioid and its bulbs and all the copies/minibrots with their bulbs and cardioids have. It is the iteration number at which the iterated value has its lowest absolute value. If all the pixels in the image have the same "period" as the cardioid of a minibrot in the region this minibrot is dominant in that image/region. The pixels either have the period of a dominant minibrot further down or further up or they have different periods because several (mini)brots have their own regions of dominance in the image. There can be thousands and thousands such as in some spirals spiraling towards a Misiurewicz point.

Ah yes, that would explain why the middle sections of the "X" chromosomes do not resolve in the initial seed orbit, even if they have a lower bailout than the first seed value. I consider it good practice to zoom into the centroid at least a dozen or so zoom levels before backing out to the target formation. This ensures the seed pixel will have higher bailout than the rest of the image in most cases. Then I framing it by setting the exact zoom depth, aspect ratio, rotation, and color cycling. Then I save the fractal, increase the image resolution to high quality / large size, sit back and render, then save again the fractal iteration data plus the PNG output.

[hapf]

Skippable due to Series Approximation I assume?

Yes.

I loaded the location in KF in 640x360:
Max-iteration: 165163
Min-iteration: 163292
With 10 terms 154561 iteration can be skipped (94.7%), image rendered in 37 seconds
With 20 terms 159713 iteration can be skipped (97.8%), image rendered in 19 seconds
With 40 terms 162289 iteration can be skipped (99.4%), image rendered in 14 seconds
Using 100 terms unfortunately gives incorrect result and only gives a small number of additional skipped iterations.

100 terms give incorrect results? Then you try to skip too much. You can't skip much more since you have exhausted the period rule. 

[stardust4ever]

Skippable due to Series Approximation I assume?

I loaded the location in KF in 640x360:
Max-iteration: 165163
Min-iteration: 163292
With 10 terms 154561 iteration can be skipped (94.7%), image rendered in 37 seconds
With 20 terms 159713 iteration can be skipped (97.8%), image rendered in 19 seconds
With 40 terms 162289 iteration can be skipped (99.4%), image rendered in 14 seconds
Using 100 terms unfortunately gives incorrect result and only gives a small number of additional skipped iterations.

Next doubling of "chromosomes" is this location, following your method and going just left of the leftmost chromosome:

Code:

re:-1.76973927107570278411520061419065946542347799372595152472101192040064094711336613121195702934273509531110920827719735988557827595999806330371323751859876701146210458806064544177069996999828391655185561042469808511402300223535883114455472484511423365855441119164893414414188906330197301018546484636691096364711626721208178249174072183033484198989852428081400589644144574714652215363016893192108533799955241566248299427809467589643090726924845245039740319491540054239489718441137692571568233952550429974345579671201229362878862729943368276039880766440708454565350593901104555321239593383546867095019659471571166180242494183400421258098231638606655519421918832801505565036377462175161679479913432266064848634617218812509508674693057633239911906027116361934291810218686634514232406561410364816866722072165101789409275784262461387890519359482022010390695157846380903429796008812574713407054196678588644723528860910520590905196670451139094295823495172577538778175845923533617587491427006033016602930900278404641562369796189524200550352924161304993887520131641880565034516078770708206943090725951473272863101719316646550470305393493596164656491513320724401548205844175371960034916997591335500423936357608167240915110205662919250126158696276192747025878258149810015608721712554623391357094690983844241174709909015252279070297968544176041518336849095481262345439207029158860533966756384084999639478493365075861692547428353717288319345008285956733788737950087166990901738963854816259215373861305911053377248373347327533386764341081400371266836934831405817979283935531422790742412459811321804473839844435089769762155990155250178014931357778215347716254080455258628723421585799192229526300512890358421750654418464175339338700097626083812582737428156995792247813443649353878917473590509904660167197369608670966587464420205758779389875106
im:0.00479575593386521785056156632204869711543048868866739207812561117523981061072803666759892986917797971367373649618307585237412544802866649204604791951282119532611555692231864126608560335395830677937445526989723059565598206590367891898467394911556358386179234972500593142081869841790362238266553240041814213356337388982751141354993190142472710950391187727081323027505479136452308448064386283517972781015524209855681335788673744269374677859162070197587477940738704947651087173844617863477560652949841356272962069000764292715737566241003234711235604702236267464602650182979303592890434112792457622396987721882609742859931177341818273187666138340519774842720206982492134896277509574980239897474381333076472746958026712349438349815843426360562006556338835568676503377969930912294441203580507236635231202356397041368243232643242096171465697636427613416621899589621369247195520694822931509253743782828991308508271169923822030731210535571166710661634286964536828249166255244271393901158417890480842237451269264760879763337428453836904776814955516384029161979047340525822848503691702878388529457727649610800474540097130947938914768777069390653907386626598219197537707103472852773928487954022576173090300830083242555816576865669179621054197697073306371497138708514154639978238654075859456762116645947307361805048300920909051144054863714541677404962777535046378220013445059535852715247670869344676733074159242947217492726089180421013247557742926237274643330802114405643723680982693493678065080321054992095840747165747342651598907576468756405957676973239464394898303249003035346570430635955169214125771039539353732135890401126564177895335144578423771521555862079611886138494457318930595157328431568295871786565347458710924343342091618717199655736488581664771241781759853956839288876770557346040681486625539604836220182498710582505322335248
zooms: 4478
zoom: 1.05E1348
size: 3.815E-1348

Iterations from 327205 of which 324577 (99.2%) can be skipped with 40 terms (1m11sec in KF)
Location parameters calculated with claude's methods in 55 minutes
Period of minibrot: 164864
Netwon-Rhapson iterations needed: 8
Zooms of minibrot: 5986.7

[hapf]

Next doubling of "chromosomes" is this location, following your method and going just left of the leftmost chromosome:
...
Location parameters calculated with claude's methods in 55 minutes
Period of minibrot: 164864
Netwon-Rhapson iterations needed: 8
Zooms of minibrot: 5986.7

What exactly did you do? You go left of the leftmost chromosome and then what?

[hapf]

Ah yes, that would explain why the middle sections of the "X" chromosomes do not resolve in the initial seed orbit, even if they have a lower bailout than the first seed value.

How so? And if you use the minibrot as reference they should work like the rest for some time zooming further in.

[Kalles Fraktaler]

What exactly did you do? You go left of the leftmost chromosome and then what?

Then I followed the magic that claude described here
http://www.fractalforums.com/index.php?topic=18289.msg90972#msg90972

[hapf]

Then I followed the magic that claude described here
http://www.fractalforums.com/index.php?topic=18289.msg90972#msg90972

Ah, I did not see that posting before. Looks promising.

[hapf]

Next doubling of "chromosomes" is this location, following your method and going just left of the leftmost chromosome:
Location parameters calculated with claude's methods in 55 minutes
Period of minibrot: 164864
Netwon-Rhapson iterations needed: 8
Zooms of minibrot: 5986.7

So you ran the polygon algorithm left of the leftmost chromosome and after 55 minutes it spat out period
164864?

[stardust4ever]

What exactly did you do? You go left of the leftmost chromosome and then what?

Zoom between the outside arms of the outermost chromosome. More specifically, there are seven sideways facing dendrites along the edge of each chromosomal arm counting from center to outer edge. To continue the pattern such that all chromosomes are equidistant to each other, zoom into the diamond-shaped julia midway between the endpoints of the third dendrites on the outer arms of the outer chromosome. There are several of these shapes one could zoom into but the distance from the center of the chromosome will determine the exact spacing between the center two chromosomes in the next target formation.

It's easier to take a screengrab but I'm away from the PC right now and don't feel like booting it as it is late.

EDIT: I appear to have misinterpreted the post. Hapf was referring to the automated process by which one finds the central minibrot (which also reveal the location of the next formation without manual progression)

[stardust4ever]

So you ran the polygon algorithm left of the leftmost chromosome and after 55 minutes it spat out period
164864?

If so, I could really use this tool.

[Kalles Fraktaler]

So you ran the polygon algorithm left of the leftmost chromosome and after 55 minutes it spat out period
164864?

Finding the period is just the first step of 3 steps, which in all took 55 minutes.
But it is fascinating that the period got exactly twice of the value that you calculated for the 1024 image 

[stardust4ever]

Finding the period is just the first step of 3 steps, which in all took 55 minutes.
But it is fascinating that the period got exactly twice of the value that you calculated for the 1024 image  

Not too surprising though. Each "feature" in the progression has twice the complexity of the last. The 1024 X formation has 4096 arms, and the 2048X formation will have 8192 arms. Each arm also has a specific number of dendrites coming off it. The length of said arms, and period of the spirals therin are determined early on in the zoom sequence.

In fact, your period number, 164864, is divisible by 1024 x 161. And 161 = 7 x 23. No coincidence each "arm" have 7 segments. So I will postulize that if you zoomed all the way out to the original little "X" formation I zoomed sideways from, the period of said minibrot contained within would be 161. Or perhaps the 161 refers to the minibrot that lies in the center of the initial 2X formation. Every chromosome in the series has a pair of little chromosomes inside it.