stardust4ever
Fractal Bachius
Posts: 513


« on: August 11, 2014, 06:53:37 PM » 

Triple Ex Magnum. From the "Because I can" files. Based loosely on my previous "Magnum Opus Ex Lite", showing that the pattern (X of X of Xs) can be manipulated once again to be displayed in neat and orderly rows. In principal by zooming deeper, this pattern can also be replicated and made into a larger X shape, although it begins to get impractical and the finer details will eventually become lost at some point. Due to precision limits of Mandel Machine (<3700), I have rendered this in the slower (but still way faster than FX) Kalles Fraktaler. 9600x5400 pixels, ~14 hours 9 min on AMD 8core Bulldozer, OC @4.2Ghz. Scaled to 4800x2700 for antialiased effect. Deviation link with high quality download: http://stardust4ever.deviantart.com/art/TripleExMagnum474835044Coordinates: R= 1.76979700322139811591272513043899832799423369499068746040312321369139476279899734327685384106424938431439273576680330733704966546075580838901324891220246239218903287505782319765936273238087369689487534737359516124840715760630396132975573610932201163074628687245503337178276171115248596381484098549511985811224780956321700144001233548139295889127740464191577029223476957057942352608361586911947339765514426923055404845140828712983972948274581253679340285067513225744679727645014517715145157480044089144978284962881930308393383563600827339564330522185034725374619497019929115833977390077112303761221265610058340090753971438170787533739611614131156138422855338469152910054086557858305898515384328090016617655263354932120681762555309514571526404505498122822098362730171825178384626908572014703813031309705105698041519194516388496462605601928402688013629674942450923550210343670307022708303866893035581819682694527014372892559065335911904707042805899685563238612975380327884255249372893829066064617478954419453422863255598536592549111492083121989408856893678720891760821661558024969415260515468015019966116359554523196078066571947778324897320768234979487998550641402 I= 0.004503808149118977453591027370762118116191847489651632102771075493630536031121753213019458488948070234821894347490919 75232128719902266967792409275276218671134664739202538733880630147980377066457243173553858784184258065626405478713476529 94375685863015511904074453632654407731289619946868720085884280405841386804671414034982833768121999000401733388984737998 50835523341852444210373993799979274072458522457971439601401283190488219977380751679864657617472858638199063899094311129 50986435475903376689569262854294399348492523985561245444569066562659921792821152021327760735636420754994007141389380134 66704394368255578502070054387656746578639639290386737709456296278667259455305485343757327711865075145113189988740144295 55122965237303553837733913609891540919091150461342497169450966250952174058994785039455189094187291999062801262967852650 44148117178635874792212241305374452915299749705021896223247978466142304421015571809942011237220999458330382486349695968 16286916049558100625775300967063155730103782914902071612493213005093101824179621747948185575806415057520533852725243570 52457881498772523144963478648096824846727758977865709771458262834198797485210250644081564585 Magnification: 8E+1142 (~3796 zooms) Thanks for viewing...


« Last Edit: August 15, 2014, 06:41:43 AM by stardust4ever »

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stardust4ever
Fractal Bachius
Posts: 513


« Reply #1 on: August 15, 2014, 06:36:47 AM » 

New version: Magnum Opus Lite replicated seven times. Quite possibly the deepest object ever rendered. DeviantArt link for high quality view/download: http://stardust4ever.deviantart.com/art/SevenExMagnum475692041Coordinates: R= 1.769797003221398115912725130438998327994233694990687460403123213691394762798997343276853841064249384314392735766803307337049665460755808389013248912202462392189032875057823197659362732380873696894875347373595161248407157606303961329755736109322011630746286872455033371782761711152485963814840985495119858112247809563217001440012335481392958891277404641915770292234769570579423526083615869119473397655144269230554048451408287129839729482745812536793402850675132257446797276450145177151451574800440891449782849628819303083933835636008273395643305221850347253746194970199291158339773900771123037612212656100583400907539714381707875337396116141311561384228553384691529100540865578583058985153843280900166176552633549321206817625553095145715264045054981228220983627301718251783846269085720147038130313097051056980415191945163884964626056019284026880136296749424509235502103436703070227083038668930355818196826945270143728925590653359119047070428058996855632386129753803278842552493728938290660646174789544194534228632555985365925491114920831219894088568936787208917608216615580249694152605154680150199661163595545231960780665719477783248973207682346305276793006733385678682275558964973019802873368327805940339952780871574753576960555246151315945009015725815402278171890609727120630443723092842219679756541447187939991155400642199559184586399549354134182627615264156877718091968276884865541064978778224354147614688835049735000521212773122397546311880875608932796060856226179789067503078902940099692082470670710499310402401659270079295236253981213764340177848778692348010492116951654 I= 0.004503808149118977453591027370762118116191847489651632102771075493630536031121753213019458488948070234821894347490919 75232128719902266967792409275276218671134664739202538733880630147980377066457243173553858784184258065626405478713476529 94375685863015511904074453632654407731289619946868720085884280405841386804671414034982833768121999000401733388984737998 50835523341852444210373993799979274072458522457971439601401283190488219977380751679864657617472858638199063899094311129 50986435475903376689569262854294399348492523985561245444569066562659921792821152021327760735636420754994007141389380134 66704394368255578502070054387656746578639639290386737709456296278667259455305485343757327711865075145113189988740144295 55122965237303553837733913609891540919091150461342497169450966250952174058994785039455189094187291999062801262967852650 44148117178635874792212241305374452915299749705021896223247978466142304421015571809942011237220999458330382486349695968 16286916049558100625775300967063155730103782914902071612493213005093101824179621747948185575806415057520533852725243570 52457881498772523144963478648096824846727758977865709771458262834198797485648944137394527290658436594530285963921685867 55058891805887375505271389751306197860999577248229907581503440232871115350839910002739224253661221918689910780752617548 59806512107020659462065485551577194075067421647607845799981921380106458967684438370585173924305136466397555189969443539 90250598705715691676572054402334626474775012385454729312713211133798242248343097225928743950743497735490596041850892689 906033832484516337373719285424 Magnification: 5210 Zooms Average Iteration Depth: 893,220 Rendered with Mandel Machine v1.2.3 beta Original render resolution: 9600x3600, 2x2 AA Luckily I didn't quite hit the 5300 zoom limit... Rendering this file took only a tiny fraction as long on my quad core laptop as Triple Ex did on my 8 core desktop using Kalles Fraktaler. Here's something I don't normally do; I'm attaching the parameter file. Enjoy!


« Last Edit: August 15, 2014, 06:41:07 AM by stardust4ever »

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Dinkydau


« Reply #3 on: August 16, 2014, 01:09:05 AM » 

Amazing, so many X of X of X... If you continue, the julia morphing itself would approach to be a fractal where every X is made of X. The limit must be raised further! I still want to find another occurence of this shape with different patterns. Something like that could easily surpass 10000 zooms. Maybe it's possible with a future mandel machine. I tried it with kalles fraktaler, but it was too extreme for my patience.



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stardust4ever
Fractal Bachius
Posts: 513


« Reply #4 on: August 16, 2014, 01:52:07 AM » 

Amazing, so many X of X of X... If you continue, the julia morphing itself would approach to be a fractal where every X is made of X. The limit must be raised further! I still want to find another occurence of this shape with different patterns. Something like that could easily surpass 10000 zooms. Maybe it's possible with a future mandel machine. I tried it with kalles fraktaler, but it was too extreme for my patience. Starting with the first render, I could potentially create an "X of Xs of Xs of Xs" formation by zooming into the edge of the bent fragment shape on the side instead of the middle. This would create another XXX/XXX formation as opposed to XXXXXXX. Zooming into the centroid of the formation, the slash will become an X in the center rather than an 8sided starfish. To do this, I'll need about 7200 zooms. The central "X of Xs of Xs" formation will have a large "X of Xs" in the center with 128 Xs (32 per arm), followed by three standard "X of Xs" on each arm (each containing 64 Xs, or 16 per arm), followed by a bent row of 64 Xs, followed by thre more standard "X of Xs" shapes congruent to the first three, followed by a bent row of 32 Xs. Each arm of the "X of Xs of Xs of Xs" formation would then have three formations congruent to "Magnum Opus Ex Lite" followed by a half fragment of Magnum Opus formation consisting of 'XXX/XXX' (32X bent fragment followed by three standard X or Xs formations followed by a long row of 64 Xs followed by three more X of Xs formations, followed by a bent fragment). Each Magnum Opus "X of Xs of Xs" formation contains 1024 X chromosomes with 128 in the central large "X of Xs" (32 per arm), 64 Xs in each of 12 medium sized "X of Xs" (16 Xs per arm of each X of Xs; 3 "X of Xs per arm of the massive "X of Xs of Xs" formation, and a smaller bent fragment of 32 Xs at the tip of each arm of the "X of Xs of Xs" formation. All in all, there is 128 + 12*64 + 4*32 = 1024 Xs in each Magnum Opus Lite formation, and each X has 4 arms for a total of 4096 arms. If a massive "X of Xs of Xs of Xs" formation is realized as planned, it will have a grand total of 16,384 Xs with 65,536 arms, somewhere close to a depth of between 71007200 zooms. EDIT: I miscalculated the zoom depth. The final formation should be around ~6650 zoom levels. The above Magnum feature (5210 zooms) has 7 Mangum Opus Ex formations with 1024 Xs in each formation, and a bent fragment on either end with 512 Xs each. That's 8192 X chromosomes in total. Because the locations of the inflection points chosen within the formation can have an effect on the zoom depth of the final object (the more developed the formation, the fewer zoom levels have to be iterated), the alternate zoom path will result in a deeper zoom depth. Zooming into an inflection point off the centroid will result in double the iteration depth and at most 50% deeper zoom depth when the periodic doubling is reached. As mentioned earlier, the zoom depth can be slightly less depending how advanced the feature the new inflection point is centered in. Zooming into the centroid of a twofold symmetry julia will result in a fourfold symmetry julia with 50% deeper iteration depth and 16.67% deeper zoom depth. Counting iterations is more accurate method of estimating when a feature will appear than counting zoom depth. Zooming into an inflection point will always result in a minibrot at double the zoom depth, or slightly less so if the inflection point is sufficiently advanced. Yeah, I've basically made a science/art out of picking inflection points. The entire image will eventually become double wrapped around the inflection point. That's the science part. Picking an inflection point with which to zoom into is the art part. These formations can easily be replicated in different areas of the MSet with different styling of the dendrites and tendrils, but the overall shapes of the formations can be crafted to be similar. Wow, that was a mouthful...



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Kalles Fraktaler


« Reply #5 on: August 18, 2014, 06:09:49 PM » 

Can you do other letters than X?



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laser blaster
Iterator
Posts: 178


« Reply #7 on: August 19, 2014, 03:13:41 AM » 

Wow, I had no idea you could find shapes like those y's and w's in a thirdorder minibrot. Higher power mandelbrots may be far more interesting than anyone gave them credit for.



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stardust4ever
Fractal Bachius
Posts: 513


« Reply #8 on: August 19, 2014, 05:28:32 PM » 

Yeah those shapes exist because they have three arms. The arms are produced by zooming the edge of a Julia shape in any order. The 2nd order equivalent is actually the "I" shape but you can double it to create "X"s. 4th order also can make Xs but they behave differently. 5th order produces starfish shapes and so on. 6th order and beyond start producing more bulbs than anything.



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plynch27


« Reply #9 on: August 21, 2014, 08:01:26 AM » 

Amazing, so many X of X of X... If you continue, the julia morphing itself would approach to be a fractal where every X is made of X. The limit must be raised further! I still want to find another occurence of this shape with different patterns. Something like that could easily surpass 10000 zooms. Maybe it's possible with a future mandel machine. I tried it with kalles fraktaler, but it was too extreme for my patience. Actually, after I finished that zoom sequence I made, I got a squirrel up my butt that was making me want to make an indirect extension of this as a longterm project. I'm planning on calling it The Human Genome Project. The attached image is a very basic rendering of my last detour.¹ Its magnification is at 1.71890907366E4445. The uneven spacing between the X's is entirely intentional ─ because I'm not making another X for this project ─ and I still have one more detour to make. The current detour is doubling the row of X's in the attached image one more time and my next ─ and last ─ detour will create two straight, parallel rows of X's with the intended effect of resembling an entire replicated diploid genome entering anaphase. My current projections put the target magnification of this project at 5.964E9994 (2^33201.92567) which would put its corresponding minibrot ─ I'm definitely not going there ─ at 3.55695E19989 (2^66403.85). Right now, I'm at 1.0756E4941, so it's gonna be awhile. Plus, the project's temporarily on hold at the moment. I'm rendering a zoom video of an old double Julia morph I wrote back in 2010. ¹As implied, it's still a work in progress, thus I still haven't drawn a color palette for it, so this image's color palette is still the same as the one stardust4ever drew.



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((π10^{45,111,908,392}) mod 10)^{πi} + 1 ≈ 0



stardust4ever
Fractal Bachius
Posts: 513


« Reply #10 on: August 21, 2014, 10:16:43 AM » 

Wow, I've really passed the torch on this one. Kudos to you for making this. Hard to tell based on the low res image but I assume you did this based on my Magnum Opus Ex coordinates and not my Magnum Opus Ex Lite? Not sure where you are going with the uneven spacing but mistake or intentional, I guess we can still call it artistic license... Another thing to consider is odd multireplication versus even multireplication (is that even a word? ) You can zoom to a point adjacent to any julia to create a pair of julias deeper in the set, then zoom to a point adjacent to the side of that pair for 4, 8, 16, and so on. This is even multireplication. The odd method is simple enough to do but it only works with shapes that have at least fourfold symmetry. Zooming into a twofold julia will result in a fourfold julia at a point deeper in the MSet. Zooming into a point adjacent to a twofold julia will result in a pair of twofold julias deeper in the MSet. Zoom into the centroid of either twofold julia, and deeper in the set you will find a fourfold julia with a twofold julia on either side. Now it's a nobrainer to zoom into one of the distorted twofold julias on the edge, to create a pattern with three fourfold julias in the center, with two distorted twofold julias on either edge. I call this the odd method, because there is always (2^n)1 fully formed julias (an odd number) with 2 partially formed julias on either extremity. I used a combination of even and odd multireplication to generate the Magnum Opus Ex fractal (both the original and lite versions are essentially the same pattern but in different areas of the set, with the full version being the deeper of the two). Also your zoom depth estimates seem to be a bit off. Zooming into a point located outside the centroid of the julia pattern will generally produce a minibrot at a zoom distance a maximum of twice the zoom depth of the original detour. This distance can be shorter if the detour julia zoomed into is reasonably complex. As a result, the odd method of multireplication will often generate patterns at a slightly shallower zoom depth than even method will. Half the distance to the minibrot from the detour location (typically 3/4 of the total zoom path), the patterns surrounding the detour location will be doubled. Iteration bands will also be twice as dense, and the iteration depth will be twice the detour point. Three quarters of the distance from the detour location to the minibrot (7/8 of the total zoom path), the pattern will be quadrupled and iteration bands will be 4x as dense. Iteration depth will be triple that of the detour point. This continues with each periodic doubling being half the distance to the minibrot and increase in iteration depth being equal to the depth of the original detour location. To borrow a term from astrophysics, the iteration density reaches the event horizon at the location of the minibrot, which can be viewed as a black hole. Iteration count increases by an amount equal to the location of the detour, for every iteration band of the minibrot. Overall, iteration counts yeild a more accurate prediction of when a desired object will materialize, rather than counting zoom levels. Your estimate that the final minibrot will occur at nearly E20,000 implies that your last detour occured at nearly E10,000, realizing a "feature" with twofold symmetry at nearly E15,000 and a feature with fourfold symmetry at nearly E17,500. Assuming you compounded your calculation error several times, you may find to your liking that the feature you are searching for is at a shallower zoom depth than you anticipated. Hope this helps... AJ One more thing, I mentioned the how to measure when certain patterns are likely to occur in the base2 Mandelbrot. If anyone is brave enough to attempt multireplication in the higher order sets, you might find that one does not have to zoom as far to find complex shapes. In a second order Mandelbrot, a minibrot will always be found a maximum of twice the depth of the detour, in other words, the final detour takes place at 1/2 the depth of the minibrot. In the third order Mandelbrot, the minibrot is 1.5 times the depth of the detour. This results in the detour location being 2/3 the depth of the minibrot, with a periodic tripling at 8/9 the depth of the minibrot, with 9fold symmetry of the detour pattern at 26/27 the depth of the mini, so each successive periodic tripling occurs at onethird the distance to the minibrot. In a fourth order Mandelbrot, the detour location will be 3/4 the distance to a minibrot with a periodic quadrupling of the pattern at 15/16 of the minibrot depth. So, for a Mandelbrot fractal of integer order N, taking a scenic detour at zoom depth D will result in a minibrot at a approximate depth DN/(N1). Patterns will always be replicated by a factor of N^I where I is an integer, starting with I=0 (remember anything to the zero power equals 1) at the location of the detour. Starting around N=8 and above, the fractals tend to get extremely blobby, requiring to zoom moderately deep just to render an image that's not littered with minibrots. Maybe thats why N=8 appears to be a sweet spot for 3D Mandelbulbers?


« Last Edit: August 21, 2014, 11:11:44 AM by stardust4ever »

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stardust4ever
Fractal Bachius
Posts: 513


« Reply #11 on: August 21, 2014, 10:47:16 AM » 



« Last Edit: August 21, 2014, 10:49:25 AM by stardust4ever »

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Kalles Fraktaler


« Reply #12 on: August 21, 2014, 03:48:17 PM » 

I made a series of zoom movies into higher order Mandelbot, from 3 to 10. Check YouTube Kalles Fraktaler 77 to 84 Each of them were rendered over the night, one night each. On a 4 year old quad core 64bit laptop


« Last Edit: August 21, 2014, 04:29:01 PM by Kalles Fraktaler »

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youhn
Fractal Molossus
Posts: 696
Shapes only exists in our heads.


« Reply #13 on: August 21, 2014, 04:46:36 PM » 

Those small things in MetaphaseHexapod6thOrder274054207 begin to look like torches or icecreams.



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plynch27


« Reply #14 on: August 21, 2014, 09:01:24 PM » 

Half the distance to the minibrot from the detour location (typically 3/4 of the total zoom path), the patterns surrounding the detour location will be doubled. ... Yeah, I know all that, actually.¹ I copied the projection for the final minibrot from an old text file and I'd forgotten that when I calculated that projection, I'd lost track of the algebra. The final minibrot would actually be at a magnification of 5.72655997E13335 or 2^44300.429 . My divergence from your coordinates actually² occurs here:


« Last Edit: August 22, 2014, 12:29:21 AM by plynch27 »

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If you'd like to leave me a text message, my 11digit phone number can be found in π starting at digit 224,801,520,878
((π10^{45,111,908,392}) mod 10)^{πi} + 1 ≈ 0



