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 Author Topic: *Continued* SuperFractalThing: Arbitrary precision mandelbrot set rendering in Java.  (Read 28200 times) Description: 0 Members and 1 Guest are viewing this topic.
stardust4ever
Fractal Bachius

Posts: 513

 « Reply #15 on: February 27, 2016, 11:00:43 PM »

In my experimental mandelbrot-perturbator I cache the series approximation iterations of the centroid, so zooming further in only needs a few more iterations.  Calculating it fresh for a brand new view (eg coordinates from a file) or when zooming off-center does take a long time.  It's also sometimes possible to zoom directly to the next off-center departure lounge - if you know approximately how deep it will be (eg zoom from 1e-128 to 1e-192 to 1e-288 when julia morphing, the exponent is multiplied by 1.5 each time), and know that the central reference is high enough precision in a minibrot (meaning, it really is central - you can use Newton's method to find it once you know the period, and finding the period of the lowest period minibrot in a region is possible too).  An example: http://mathr.co.uk/mandelbrot/2015-06-12_perturbator_deep_zoom_stress_test/  I definitely didn't do the zooming only 16x each frame - that would take insanely heroic amounts of boring manual time and effort!
I have actually attempted to use automatic zooming feature in say Kalles Fraktaler or other software. The way I do my zoom sequence, I often set up patterns with a large amount of identical objects lined together in a row. I also have what I like to call an even and odd method of building these formations. The even method produces 2^N number of similar objects lined in a row, with the centroid between the said objects, and an odd method consisting of 2^N-1 number of similar objects lined in a row, with a half-object on either side at the extremeties, and the centroid within the centermost object.

It is rather difficult to determine the true centroid of such a formation without first rendering at fairly high resolution and meticulously measuring the distance between outer extremeties, then carefully selecting a point equidistant to those two extremeties, and zooming in from there. I cannot "eyeball" it without holding a ruler to my HD monitor display, and oftentimes the software will mistakenly zoom into a "featured" object near the center when it needs to zoom between two such objects.

I imagine the auto brot finding software works by locating hotspots within the fractal image based on iteration depth, and choosing the location of the hotspot nearest the centroid, when the true centroid may not appear as a hotspot at all. So I have my trademark "X of Xs" formation, and it has been duplicated sixteen times. The software sees this:

XXXXXXXXXXXXXXXX

And the true centroid is located here:

XXXXXXXX·XXXXXXXX

But the software notices the location where the bars of the X intersect appears much hotter, so it instead zooms into one of the X patterns ajacent to the centroid, say here, denoted by capitalization:

xxxxxxxX·xxxxxxxx

And the software continues down the wrong path, until it inevitably intersects the same pattern again, and again takes the wrong path. I come back hours later to find the engine has failed to locate the minibrot, and probably never will. I use some algebra to estimate the location of the target formation, back up to this general area, and sure enough just as I suspected, the software failed to properly identify the cerntroid of a formation.

I haven't played with fractals in a while, but I have found Mandel Machine to be much, much faster than Kalles Fraktaler, and the "locate minibrot" function in KF to be less than perfect in certain situations. Software may have improved somewhat than the last time I played with it; IDK.
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stardust4ever
Fractal Bachius

Posts: 513

 « Reply #16 on: February 28, 2016, 12:22:49 AM »

And the software continues down the wrong path, until it inevitably intersects the same pattern again, and again takes the wrong path. I come back hours later to find the engine has failed to locate the minibrot, and probably never will. I use some algebra to estimate the location of the target formation, back up to this general area, and sure enough just as I suspected, the software failed to properly identify the cerntroid of a formation.
Here is an example location of where the algorithm failed, in my Pterodactyl Canyon Render. I made this in summer of 2014, just realized I failed to ever upload it. Full download available at 3600x2700 resolution on deviantart page.
http://stardust4ever.deviantart.com/art/Pterodactyl-Canyon-593472412

Kales Fraktaler failed to find the centroid of the image. In the middle bone span where the 7 "X of Xs" formations reside, Kalles Fraktaller selected the point in between the central "X of Xs" formation and the one adjacent to it, failing to locate the true centroid. I forget how far the simulation ran before I killed it. Here are the coordinates. In fact, the exact position where the centriod deviation occurred is preserved in the coordinate data. See attached.

Real= -1.769797003221398115912725130438998327994233694990687460403123213691394762798997343276853841064249384314392735766803307337049665460755808389013248912202462392189032875057823197659362732380873696894875347373595161248407157606303961329755736109322011630746286872455033371782761711152485963814840985495119858112247809563217001440012335481392958891277404641915770292234769570579423526083615869119473397655144269230554048451408287129839729482745812536821304009849356175786421926754317166054095017677737478909629824101459411484678651540446085496579356154087444768864107144068903495747107840142587494964830790373105466387017637804940200093226948331098336564024101191304782846009251093956024054859850114380942506295799272703040122491695848188554900910110348500660088142142935996917999415780413409072318505658318370986389714499389359946017922054389605549307239863818771223517117958828030858448235437369940778504548655809414086286410278094103602829312453365743012069479897322687170061953674357190866700112517607208995688167519085493168568587128984804788006359593471007812934992508284738813218401067186129216920419813413598507086914378451166514659356530201296859316650641129911816376644360695899121978646876258352313348564609725007303215079702633145899631663504174247063662618357201794491755664334581161063251718266469929996804838236903448728496690668143319600874089515125291764268345534981174976291977855698805746925229399729615225109605245345830722655517606147744507997235610446150765888279849316729036292301646101698262415387848655551453813389172582295590171380746790465457505657035692901532708877919123668700059980974391493025
Imag= 0.004503808149118977453591027370762118116191847489651632102771075493630536031121753213019458488948070234821894347490919 75232128719902266967792409275276218671134664739202538733880630147980377066457243173553858784184258065626405478713476529 94375685863015511904074453632654407731289619946868720085884280405841386804671414034982833768121999000401733388984737998 50835523341852444210373993799979274072458522457971439601401283190488219977380751679864657632594486990141780409069050808 53533679083210095437351400022620788443700681865056074859184889623921225508741770547501475133877301147491846294015630493 19594413147950329230917914373568299313895801070552430312839787385413077643433921434686758800882730741386718858427487804 80173527152642383437688144097648231731279522222357988455250353865370120443546331395472996006556618614941953429666058354 64910451202485512530423175907298924572677884684325102852936015719933302605823099958630951988450410491580664701963842251 46135190645341340161891884063141465638742680614101092435645795624718302058131414609501281021540435472453888874524109018 14702121578711328524425442226752168664749086242203613749999027884515745350840633982861734634138141253642303937961493945 45838176191438823739844915158113285022936463789829746280707055929391192625872076997627990447836359937976951672647199177 81872517689037585583899463944250055017306480718807197254236743510423432718914191161718864625412816080818679138546319519 75989748541205329675986737013154577653006827691952880225127567357459621316524513472420563020300861878311519895655738526 548297377841163569759373958805028572872157804020781688771768375820124065
Zoom depth: 2.00000000000E1596

Sorry if this is OT. I haven't logged in in a while and this thread got a bump notification in my email.
 False Centroid.jpg (49.75 KB, 660x444 - viewed 512 times.) « Last Edit: February 28, 2016, 05:09:38 AM by stardust4ever » Logged
quaz0r
Fractal Molossus

Posts: 652

 « Reply #17 on: February 28, 2016, 02:41:18 AM »

double can not be used for perturbation beyond 10^-308. Quite limiting. So either extended double with separate exponent or extended long double with separate exponent

my only point was that long double is not very relevant in an HPC sort of context, since CPUs can crunch multiple doubles at a time and GPUs dont even have long doubles.

it also seems worth mentioning that there are no hard limits like e308 when using series approximation with perturbation, since the series approximation gets you part of the way there and you dont start iterating from the very beginning.  the soonest ive seen double fall off the cliff is maybe e400, and ive seen it hold up for quite a ways e.g. e2000 or so.  it depends on the location of course.
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hapf
Fractal Lover

Posts: 219

 « Reply #18 on: February 28, 2016, 10:47:00 AM »

my only point was that long double is not very relevant in an HPC sort of context, since CPUs can crunch multiple doubles at a time and GPUs dont even have long doubles.
it also seems worth mentioning that there are no hard limits like e308 when using series approximation with perturbation, since the series approximation gets you part of the way there and you dont start iterating from the very beginning.  the soonest ive seen double fall off the cliff is maybe e400, and ive seen it hold up for quite a ways e.g. e2000 or so.  it depends on the location of course.
Interesting. I did not look at this since I use long double anway (why bother with double if you have long double?). Double is certainly attractive when you can use the GPU with it. The moment the deltas become too small long double is better. And once deltas are too small for long double hardware floats with bigger exponents become very useful.
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hapf
Fractal Lover

Posts: 219

 « Reply #19 on: February 28, 2016, 11:34:56 AM »

Here is an example location of where the algorithm failed, in my Pterodactyl Canyon Render. I made this in summer of 2014, just realized I failed to ever upload it. Full download available at 3600x2700 resolution on deviantart page.
http://stardust4ever.deviantart.com/art/Pterodactyl-Canyon-593472412
Kalle's program uses some kind of pattern matching, I think, trying to find the center of the symmetric structures. If all you want is going in direction of the central minibrot there is a much simpler and more reliable solution. At this location the central minibrot is already dominant, meaning its period is dominating the region entirely (at realistic image sizes) and it's simple to compute the exact location of the minibrot and where it is in the image (see Claude's posting). If the central minibrot is not yet dominant it might be be possible to compute what its period is nonetheless and proceed the same way. If you want to deviate a specific way there are likely again rules how to do it automatically. If the deviation happens at a juncture where the central minibrot becomes dominant there often are multiple regions with different dominant periods in the image and the periods can be computed and their minibrots reached automtically. I have not looked into this but IMHO it is entirely or at least partially possible to do it automatically and reliably.
 « Last Edit: February 28, 2016, 12:55:32 PM by hapf » Logged
stardust4ever
Fractal Bachius

Posts: 513

 « Reply #20 on: February 28, 2016, 02:10:23 PM »

Kalle's program uses some kind of pattern matching, I think, trying to find the center of the symmetric structures. If all you want is going in direction of the central minibrot there is a much simpler and more reliable solution. At this location the central minibrot is already dominant, meaning its period is dominating the region entirely (at realistic image sizes) and it's simple to compute the exact location of the minibrot and where it is in the image (see Claude's posting). If the central minibrot is not yet dominant it might be be possible to compute what its period is nonetheless and proceed the same way. If you want to deviate a specific way there are likely again rules how to do it automatically. If the deviation happens at a juncture where the central minibrot becomes dominant there often are multiple regions with different dominant periods in the image and the periods can be computed and their minibrots reached automtically. I have not looked into this but IMHO it is entirely or at least partially possible to do it automatically and reliably.
I don't know the details regarding periodicity but generally only points within the set (Minibrots, etc) have periodicty in which the orbit path converges into a stable pattern. Some points along the edge of the set also never escape and never converge into a repeating path. Locations like -1.75 + 0i and 0 + 1i are examples.

Given the immense complexity of the image location here, it is no surprise the software algorithm failed to find the centroid. I forget offhand how many arms this formation has (edit: 8 "Magnum Opus Ex Lite" formations, 8192 individual "X" chromosomes, and 32768 arms - wow) but it is huge. My technique basically find a feature at low zoom level (mini julia), distort it (zoom into the edge of shape), then creates a barbell and later an X pattern. Zoom offcenter to duplicate the X pattern, rinse and repeat.

Zooming off center of a formation produces a new formation twice as complex at roughly 1.5x the zoom depth (and 2x the iterations), rotated about the pint of the original deviation. The exact zoom depth is sometimes less than that if the sub feature zoomed into has high complexity. Also at 1.75x the depth (and 3x the iterations) of the deviation point, the original feature is quadrupled. And obviously, and deviation from the centroid, when zoomed into, will produce a minibrot at twice the depth.

But by repeated deviations at select intervals, the complexity of a formation grows by a factor of 2 every time, one can achieve some truly remarkably complex formations at insane depths. It is not surprising the minibrot finding algorithm can sometimes fail in such instances.

Regarding dominance of the central minibrot, I am not sure how that is calculated, but every sub-feature of said formation has a mini somewhere, albeit at deeper depth compared to the central one. It may well be the exact location at which the central minibrot becomes "dominant" as you say, given it is one of the "fork in the road" points when a new level of complex shape is achieved. I can either continue through the intersection towards the destination (minibrot) or take another detour.
 « Last Edit: February 28, 2016, 02:35:47 PM by stardust4ever » Logged
Kalles Fraktaler
Fractal Senior

Posts: 1458

 « Reply #21 on: February 28, 2016, 11:23:49 PM »

By the way, how much slower are extended long doubles with separately stored exponent compared to regular long doubles when performing standard iterations and no assembly language coding is used?
In Kalles Fraktaler you can force which datatype to use (Actions->Special->Use long double always/Use floatexp always)
I compared stardust's location in depth 1.5E294 (no Auto glitch solving)
* floatexp (a double as mantissa and an integer as exponent) is about 4 times slower than long double.
* long double is about 4 times slower than double.

Here is an example location of where the algorithm failed, in my Pterodactyl Canyon Render. I made this in summer of 2014, just realized I failed to ever upload it. Full download available at 3600x2700 resolution on deviantart page.
http://stardust4ever.deviantart.com/art/Pterodactyl-Canyon-593472412
<Quoted Image Removed>
Kales Fraktaler failed to find the centroid of the image. In the middle bone span where the 7 "X of Xs" formations reside, Kalles Fraktaller selected the point in between the central "X of Xs" formation and the one adjacent to it, failing to locate the true centroid. I forget how far the simulation ran before I killed it. Here are the coordinates. In fact, the exact position where the centriod deviation occurred is preserved in the coordinate data. See attached.
...
Sorry if this is OT. I haven't logged in in a while and this thread got a bump notification in my email.
Previously the pattern center was searched for using the iteration data, however that can be very chaotic. So I change that to search on the image data (RGB values) which is much better. But indeed, not perfect...
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hapf
Fractal Lover

Posts: 219

 « Reply #22 on: February 28, 2016, 11:56:29 PM »

In Kalles Fraktaler you can force which datatype to use (Actions->Special->Use long double always/Use floatexp always)
I compared stardust's location in depth 1.5E294 (no Auto glitch solving)
* floatexp (a double as mantissa and an integer as exponent) is about 4 times slower than long double.
* long double is about 4 times slower than double.
Previously the pattern center was searched for using the iteration data, however that can be very chaotic. So I change that to search on the image data (RGB values) which is much better. But indeed, not perfect...
So you have no long double mantissa with an int exponent but only double with an int exponent and this is 16 times slower than a regular double?
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stardust4ever
Fractal Bachius

Posts: 513

 « Reply #23 on: February 29, 2016, 02:23:05 AM »

Previously the pattern center was searched for using the iteration data, however that can be very chaotic. So I change that to search on the image data (RGB values) which is much better. But indeed, not perfect...
I wouldn't sweat it. The date stamp to that file was from June 2014, a year and a half ago so I would imagine the software had been updated since.

I noticed Mandel Machine now has a "find centroid" option, but only zooms 2 at a time and waits for the entire image to render. I typically zoom using tiny rectangles about 5 or so zoom levels at once, and typically I only need to wait for the first orbit to complete to advance unless I'm off center. But i wish it had an "auto" function like Kalles Fraktaler so i could wak away and do other things besides spending hours clicking through thousands of zoom levels at the PC. I typically listen to music to pass the time...
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Kalles Fraktaler
Fractal Senior

Posts: 1458

 « Reply #24 on: February 29, 2016, 09:38:33 AM »

So you have no long double mantissa with an int exponent but only double with an int exponent and this is 16 times slower than a regular double?
Yes. And addition/subtraction is slower than multiplication/division since then the exponents needs to be adjusted to each other.

The floatexp data type is also useful when using many terms in series approximation
Because when using for example 5 terms, the last one often go outside the boundaries of double.
However they come often back again and and can go to magnitudes that have influence.
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hapf
Fractal Lover

Posts: 219

 « Reply #25 on: February 29, 2016, 10:26:16 AM »

Yes. And addition/subtraction is slower than multiplication/division since then the exponents needs to be adjusted to each other.

The floatexp data type is also useful when using many terms in series approximation
Because when using for example 5 terms, the last one often go outside the boundaries of double.
However they come often back again and and can go to magnitudes that have influence.
When using long double the extended type's (double or long double) main use is series approximation indeed (normal perturbation iterations
work without extended types and long double to ~10e-4900 and more and that is is most of the time enough).
The slowdown of 16 is for multiplications or additions or a mix of them?
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hapf
Fractal Lover

Posts: 219

 « Reply #26 on: February 29, 2016, 10:35:41 AM »

Regarding dominance of the central minibrot, I am not sure how that is calculated, but every sub-feature of said formation has a mini somewhere, albeit at deeper depth compared to the central one. It may well be the exact location at which the central minibrot becomes "dominant" as you say, given it is one of the "fork in the road" points when a new level of complex shape is achieved. I can either continue through the intersection towards the destination (minibrot) or take another detour.

In this location the central dominant minibrot has period 418864. Computing the zero crossing of the corresponding polynomial locates the minibrot at

-1.76979700322139811591272513043899832799423369499068746040312321369139476279899734327685384106424938431439273576680330733704966546075580838901324891220246239218903287505782319765936273238087369689487534737359516124840715760630396132975573610932201163074628687245503337178276171115248596381484098549511985811224780956321700144001233548139295889127740464191577029223476957057942352608361586911947339765514426923055404845140828712983972948274581253682130400984935617578642192675431716605409501767773747890962982410145941148467865154044608549657935615408744476886410714406890349574710784014258749496483079037310546638701763780494020009322694833109833656402410119130478284600925109395602405485985011438094250629579927270304012249169584818855490091011034850066008814214293599691799941578041340907231850565831837098638971449938935994601792205438960554930723986381877122351711795882803085844823543736994077850454865580941408628641027809410360282931245336574301206947989732268717006195367435719086670011251760720899568816751908549316856858712898480478800635959347100781293499250828473881321840106718612921692041981341359850708691437845116651465935653020129685931665064112991181637664436069589912197864687625835231334856460972500730321507970263314589963166350417424706366261835720179449175566433458116106325171826646992999680483823690344872849669066814331960087408951512529176426834553498117497629197785569880574692522939972961522510960524534583072265551760614774450799723561044615076588827984931672903629230164610169826241538784865555145381338917258229559017138074679046545750565703569290153270887791912366870238890702486377674493961627842425415072641536223340784982438486048756109238181153075391103742999718461989487988255182749425809658290851105686957800331487046619356847741786931568734133797812990312933679468689355633257241932332586807751783991361005487951858068862626827875513314445086552403572026135269341415265563914895613317095945080129111249617399947471951570374601941026322586575889678534000148411715548247602090863178460885536238487047026969052782268862081294620522011538188275677933094565746782844741895263212598E+00
4.503808149118977453591027370762118116191847489651632102771075493630536031121753213019458488948070234821894347490919752 32128719902266967792409275276218671134664739202538733880630147980377066457243173553858784184258065626405478713476529943 75685863015511904074453632654407731289619946868720085884280405841386804671414034982833768121999000401733388984737998508 35523341852444210373993799979274072458522457971439601401283190488219977380751679864657632594486990141780409069050808535 33679083210095437351400022620788443700681865056074859184889623921225508741770547501475133877301147491846294015630493195 94413147950329230917914373568299313895801070552430312839787385413077643433921434686758800882730741386718858427487804801 73527152642383437688144097648231731279522222357988455250353865370120443546331395472996006556618614941953429666058354649 10451202485512530423175907298924572677884684325102852936015719933302605823099958630951988450410491580664701963842251461 35190645341340161891884063141465638742680614101092435645795624718302058131414609501281021540435472453888874524109018147 02121578711328524425442226752168664749086242203613749999027884515745350840633982861734634138141253642303937961493945458 38176191438823739844915158113285022936463789829746280707055929391192625872076997627990447836359937976951672647199177818 72517689037585583899463944250055017306480718807197254236743510423432718914191161718864625412816080818679138546319519759 89748541205329675986737013154577653006827691952880225127567357459621316524513472420563020300861878311519895655738526548 29737784116356975937395880502857287215780402078167418834602295096014173047038182390355477059048628119343002217338189674 84428900612407421285966391654470156922336601567981570299684787648714514350236588685564191491795576963451396365624203611 89623693814216660262167258794137460777065623334381376669587093792227710384619914833779522355034279775231366236846879929 65077410226071259699613708732240144706025226046260403350230398624904436384826525344982234790191805054228954439652523570 302168603714064304338213704267058855190821114715226120756650420403035424042014899248446596447E-03
2.0E-2105
There is another juncture at 1.7E-1851 .
Now a question for Kalle and other fractal champions: How much can you skip near the minibrot?
A program option suited best for your needs would allow you to click on a pixel where you want to use a minibrot deep down in the neighbourhood of that pixel, locate the minibrot, compute one or more fork locations in between and generate images of the fork locations for you so you can click again on other pixels to deviate from the central minibrot. And so on. Pretty straightforward to implement.
 « Last Edit: February 29, 2016, 01:25:11 PM by hapf » Logged
Kalles Fraktaler
Fractal Senior

Posts: 1458

 « Reply #27 on: February 29, 2016, 06:19:35 PM »

Now a question for Kalle and other fractal champions: How much can you skip near the minibrot?
Between 2095304 and 2513168 of the minimum 4196216 at 1e2105 depending on the number of terms in approximation.

A program option suited best for your needs would allow you to click on a pixel where you want to use a minibrot deep down in the neighbourhood of that pixel, locate the minibrot, compute one or more fork locations in between and generate images of the fork locations for you so you can click again on other pixels to deviate from the central minibrot. And so on. Pretty straightforward to implement.
That's an interesting suggestion indeed!
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Want to create DEEP Mandelbrot fractals 100 times faster than the commercial programs, for FREE? One hour or one minute? Three months or one day? Try Kalles Fraktaler http://www.chillheimer.de/kallesfraktaler
stardust4ever
Fractal Bachius

Posts: 513

 « Reply #28 on: February 29, 2016, 09:08:54 PM »

In this location the central dominant minibrot has period 418864. Computing the zero crossing of the corresponding polynomial locates the minibrot at

<snip>
Now a question for Kalle and other fractal champions: How much can you skip near the minibrot?

A program option suited best for your needs would allow you to click on a pixel where you want to use a minibrot deep down in the neighbourhood of that pixel, locate the minibrot, compute one or more fork locations in between and generate images of the fork locations for you so you can click again on other pixels to deviate from the central minibrot. And so on. Pretty straightforward to implement.
Hey that would be awesome!   Did you manually zoom to the center or use computer algorythm to find it? This is exactly what I want. Gone are the days of waiting days or weeks or months [!] for a single image to render, but the main hangup with exploring these formations is the mind-numbing constant zooming into the centroid of features. Everytime I select a fork point, I zoom through the center of every single fork-in-the-road formation I've seen thus far, as well as tons of other junk formations, before arriving at the target. I can always tell when the target is close because the iteration bands become very dense approaching it. Overall this process requires thousands of mouse clicks over may hours even with the speedup associated with perturbation. I have to wait for that first orbit to complete before eyeballing it and zooming into that black circle in the center. If there was a way to automate it, walk away from the PC, software finds minibrot, then backup to the general area where the location exists.

I attempted to join the voyage to minibrot last night using the "locate centriod" option in Mandel Machine. This requires clicking the button or pressing spacebar with each pass, and I fabricated a system to automate this feature. I plugged in a game controller with turbo function into the PC, and used the joytokey app to map a button to the spacebar. Tie the button down with a rubber band to simulate repeated spacebar presses. I watched for a few dozen zoom levels and it appeared to be working. I check it today and to my horror the software has at some point locked onto an infinite spiral where it remained until hitting the zoom limit of 2^-10000. Well it worked for a few hundred zoom levels at least before going into left field.

Another idea I have tossed around in my mind on multiple occasions: once an interesting formation is found at a reasonably deep zoom depth, an inflection point is selected to be zoomed into, at a point approximately 50% deeper and exactly twice the iteration count, the original formation is wrapped twice around the inflection point. Zoom depth is compressed, so that features approaching the inflection point wrap around it twice. I imagined a new zoom movie format using a tubular image of arbitrary length with polar coordinates and log scale on the axis. Essentially a zoom is like travelling through a tube. And after the feature has gone through a periodic doubling, the patterns on the unrolled tube would be exactly the same, just doubled. Which leads me to another point, if all we are doing is creating a more complex formation by duplicating parts of a simpler formation, this image could be extrapolated without zooming thousands of zoom levels down the rabbit hole. Lock on the inflection point about which you want to distort and duplicate the image. Convert the fractal from rectangular to polar coordinates, then duplicate the cylindrical image. Convert back to rectangular and you've got a mapping that looks exactly or very close to what you will find by zooming down the rabbit hole. You've now got a rectangular grid full of pixels that are scattered about within the source formation. Render the actual locations these pixels within the grid and you have a preview of your deep zoom formation without the extra precision of zooming deeper within the set. You could also create patterns that don't exist within the Mandebrot set. For instance you could extrapolate an formation with period three symmetry which do not exist within second order fractals such as the mandelbrot. Simulated periodicity rather than old fashioned very deep zooming could create some interesting results. At the very least, you could at least preview your target formation before deciding if it is worth the effort to zoom there.
 « Last Edit: February 29, 2016, 09:16:13 PM by stardust4ever » Logged
hapf
Fractal Lover

Posts: 219

 « Reply #29 on: March 01, 2016, 12:26:56 AM »

Hey that would be awesome!   Did you manually zoom to the center or use computer algorythm to find it? This is exactly what I want.
I did it automatically since the minibrot was already dominant (e.g. the period is the same for all pixels) so I knew which polynom's zero crossing to compute. Takes some time because the Newton steps must be done at full arbitrary precision.
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