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Author Topic: Mandelbrot Safari  (Read 34786 times)
Description: Elephants, Squid, and Peanuts, Oh My!
0 Members and 1 Guest are viewing this topic.
Kalles Fraktaler
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« Reply #480 on: May 02, 2014, 07:10:48 PM »

Ok like I did with my movie diving to seahorse, dragon and elephant walleys in different minibrots by increasing iteration division.
I think the iteration count would be multiplied with it self for the next minibrot, and since the safari minibrot requires  some 2 million iterations, we need to wait until the computers can have a couple of american trillions iterations in memory, some terabyte...
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Dinkydau
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« Reply #481 on: May 02, 2014, 09:19:03 PM »

Yes, like that video
How did you arrive at the thought that it's the amount of iterations multiplied with itself? It sounds reasonable.

I did research in fractal extreme on how the iteration count is affected by composing two zoom paths. I think it's more complicated than to just multiply them. I can't really give a formula, but I do think depth is probably a factor.
« Last Edit: May 03, 2014, 02:12:23 AM by Dinkydau » Logged

Dinkydau
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« Reply #482 on: May 03, 2014, 03:07:55 AM »

This is fantastic. In my little visual research I just learned something very remarkable about the mandelbrot set that I didn't know before. Minibrots have their own iteration bands like the main mandelbrot has iteration bands. The iteration bands of minibrots are symmetry increases of what happened before the minibrot. I just found this by pure visual exploring. Assuming that to be true, it leads to the following.

We're making a composition of two zoom paths S1 and S2 (for Safari1 and Safari2). I wonder how many iterations are required at the minibrot of this composition. Let's define the amount of iterations required for S1 and S2 to be I1 and I2 respectively.

For performing S1, each main-mandelbrot iteration band adds one iteration. After I1 iteration bands passed, we have I1*(iterations per main-mandelbrot iteration band) = I1*1 = I1 iterations in total.

Now for performing S2, each S1-minibrot iteration band adds a different amount of iterations. There would have been I2 iteration bands passed if S2 were performed on the main mandelbrot, so on the minibrot that's no different. The amount of S1-minibrot iteration bands passed is I2. For each iteration band passed, the amount of required interations increases.

That means doing the composition of S1 and S2 requires
I2 * (iterations per S1-minibrot iteration band).

So what is that "iterations per S1-minibrot iteration band"? It's not I1 itself because that's the iterations required for the minibrot of S1. By there, there have already been symmetry increases. Instead, what is repeating and increasing in symmetry all the time is the part up until half the depth of S1.

That makes the composition of S1 and S2 require
I2 * (iterations at half S1).

I did a practical test and the formula is... somewhat accurate. In the case of the mandelbrot safari, I measured 59512 iterations at half S1. We know the final minibrot took 2 000 000, so the composition S1×S2 would require about 120 000 000 000 iterations. That's probably still too much, right?
« Last Edit: May 03, 2014, 03:09:44 AM by Dinkydau » Logged

Pauldelbrot
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pderbyshire2
« Reply #483 on: May 03, 2014, 07:20:24 AM »

This is fantastic. In my little visual research I just learned something very remarkable about the mandelbrot set that I didn't know before. Minibrots have their own iteration bands like the main mandelbrot has iteration bands. The iteration bands of minibrots are symmetry increases of what happened before the minibrot. I just found this by pure visual exploring. Assuming that to be true, it leads to the following.

We're making a composition of two zoom paths S1 and S2 (for Safari1 and Safari2). I wonder how many iterations are required at the minibrot of this composition. Let's define the amount of iterations required for S1 and S2 to be I1 and I2 respectively.

For performing S1, each main-mandelbrot iteration band adds one iteration. After I1 iteration bands passed, we have I1*(iterations per main-mandelbrot iteration band) = I1*1 = I1 iterations in total.

Now for performing S2, each S1-minibrot iteration band adds a different amount of iterations. There would have been I2 iteration bands passed if S2 were performed on the main mandelbrot, so on the minibrot that's no different. The amount of S1-minibrot iteration bands passed is I2. For each iteration band passed, the amount of required interations increases.

That means doing the composition of S1 and S2 requires
I2 * (iterations per S1-minibrot iteration band).

So what is that "iterations per S1-minibrot iteration band"? It's not I1 itself because that's the iterations required for the minibrot of S1. By there, there have already been symmetry increases. Instead, what is repeating and increasing in symmetry all the time is the part up until half the depth of S1.

That makes the composition of S1 and S2 require
I2 * (iterations at half S1).

I did a practical test and the formula is... somewhat accurate. In the case of the mandelbrot safari, I measured 59512 iterations at half S1. We know the final minibrot took 2 000 000, so the composition S1×S2 would require about 120 000 000 000 iterations. That's probably still too much, right?

You were pretty close. The exact value is 59444. wink
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Adam Majewski
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« Reply #484 on: September 27, 2014, 03:24:26 PM »

Where is filament which should land on the root of cardioid ?
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stardust4ever
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« Reply #485 on: September 27, 2014, 06:47:50 PM »

Where is filament which should land on the root of cardioid ?
Julias appear everywhere surrounding the minibrot. These Julias are not part of  the minibrot set, only the larger mandelbrot. To duplicate an existing mandelbrot zoom within a mini would require not taking these Julia detours. Iteration depth would be ao dense that most of the layers tacked on by the large Mset would be completely lost in the iterations of the minibrot set. It the minibrot set was set sufficiently deep within the main set, and the iteration bands set sufficiently sparse, the iteration bands of the mini would take on the look and feel of a smooth-shaded brot, albeit a much grainier image would apply. Almost no guessing would occur here because even with highest AA settings, pixels would share different values from their neiboors. Rendering of said minibrot track would take exponentially long. My advice to attemp this would be to utilise minibrot at depths of 2^200 or less. This method also may generate subsequent minibrots of minibrots more quickly if the explorer successfully avoids the plethora of Julia formations surrounding the mini. IE, if a minibrot A exists at a depth of 200 zooms in the parent set, this mini would theoretically exist as a child of minibrot B of the parent, at a depth of +200. Remember, Julias contain only minis that are a child of the parent brot, not children of the minibrot. So avoiding Julia formations may create minis of minis at shallower depths (but huger iteration counts) compared to minis of the parent brot.
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Kalles Fraktaler
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« Reply #486 on: May 20, 2016, 10:54:28 PM »

<a href="http://www.youtube.com/v/SpUKjPvyi5U&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/SpUKjPvyi5U&rel=1&fs=1&hd=1</a>
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stardust4ever
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« Reply #487 on: May 21, 2016, 12:18:41 AM »

That's nice. Even the negative space is filled with intricate Julia patterns. I would like to see a high quality Mega download link if possible...
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Fatbear
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« Reply #488 on: June 04, 2017, 12:18:22 AM »

Amazing!! Thanks for showing it
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Chillheimer
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« Reply #489 on: August 14, 2017, 02:02:14 PM »

what a shame that the external images are offline.. It was an epic journey.
if you want your images to stay online to "infinity", please use our gallery or attachments.
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Dinkydau
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« Reply #490 on: August 15, 2017, 02:17:16 PM »

what a shame that the external images are offline.. It was an epic journey.
if you want your images to stay online to "infinity", please use our gallery or attachments.
It's a photobucket problem and it's affecting many websites. I made a userscript to fix this. Maybe you find it useful.
https://github.com/DinkydauSet/Photobucket-3rd-party-fix
Of course it's best if everyone who used photobucket rehosts their images but in most cases it's just not going to happen.
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Tas_mania
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« Reply #491 on: August 17, 2017, 06:06:48 AM »

Kalles That video is amazing. After watching it twice I can see why forums like FF put new comments below old comments. You can take that comment as an iteration  Repeating Zooming Self-Silimilar Thumb Up, by Craig
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claude
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« Reply #492 on: August 17, 2017, 12:24:03 PM »

Here is some information about the location, generated by the m-describe code in my https://code.mathr.co.uk/mandelbrot-numerics repository:


the input point was +2.75337647746737993588667124824627881566714069895426285916274363067437510130230301309671975356653639860582884204637353849973626635844461696577733396177173659502869597622654858047830473369233652610609631007219270037919896108613318635711410655928412269957977397230123742985898239211816931398241903797459102438729408702005271145966616545036e-01 + +6.75964940532785067018170045619492950218975023461430484635726913710673103258247167757358200829449470582619413145077310704967071714678595763311924422571027117886784050420240236249129631789483532106497151867377563025274513529470021667381579073334313498412010852400179935107657764228375162746931512488396245301309385347189831168355578240409e-03 i
the point didn't escape after 60000 iterations
nearby hyperbolic components to the input point:

- a period 1 cardioid
  with nucleus at +0e+00 + +0e+00 i
  the component has size 1.00000e+00 and is pointing west
  the atom domain has size 0.00000e+00
  the nucleus is 2.75421e-01 to the west of the input point
  the input point is exterior to this component at
  radius 1.01030e+00 and angle 0.051481301889481974 (in turns)
  the multiplier is +9.57901e-01 + +3.21128e-01 i
  a point in the attractor is +4.78943e-01 + +1.60561e-01 i

- a period 94 cardioid
  with nucleus at +2.7533764780368214e-01 + +6.7596492803599804e-03 i
  the component has size 7.07849e-12 and is pointing south
  the atom domain has size 1.18454e-07
  the nucleus is 1.37330e-10 to the south-south-east of the input point
  the input point is exterior to this component at
  radius 8.63943e+00 and angle 0.259140356745496070 (in turns)
  the multiplier is -4.95894e-01 + +8.62518e+00 i
  a point in the attractor is +4.6369718780481874e-05 + -5.3911600324196745e-05 i

- a period 292 cardioid
  with nucleus at +2.75337647746737993589450443e-01 + +6.75964940532785066642972551e-03 i
  the component has size 1.33239e-21 and is pointing north-west
  the atom domain has size 1.86167e-15
  the nucleus is 3.83287e-21 to the south-south-east of the input point
  the input point is exterior to this component at
  radius 2.57308e+00 and angle 0.432408664872798276 (in turns)
  the multiplier is -2.34450e+00 + +1.06020e+00 i
  a point in the attractor is -1.69564475530324405445050437e-10 + +2.40199294799935024599726391e-10 i

- a period 2489 cardioid
  with nucleus at +2.7533764774673799358866712482462788156671406989418941304667747156400867e-01 + +6.7596494053278506701817004561949295021897502349756765590698454557430955e-03 i
  the component has size 9.47543e-66 and is pointing south-east
  the atom domain has size 3.44580e-43
  the nucleus is 1.28858e-48 to the west-north-west of the input point

- a period 4099 cardioid
  with nucleus at +2.753376477467379935886671248246278815667140698954262859162743630674375099341391278329e-01 + +6.759649405327850670181700456194929502189750234614304846357269137106730807072789532301e-03 i
  the component has size 4.44948e-79 and is pointing east-north-east
  the atom domain has size 2.79233e-55
  the nucleus is 2.98842e-73 to the south-west of the input point
  the input point is exterior to this component at
  radius 1.63810e+03 and angle 0.544382068892239390 (in turns)
  the multiplier is -1.57482e+03 + -4.50905e+02 i
  a point in the attractor is +5.736936311897576207312683073012650676614467026706682207757961736257834954672699928867e-37 + -3.156955688691252502284958583502381443762749996210058333306072137990928921151874304218e-36 i

- a period 14782 cardioid
  with nucleus at +2.753376477467379935886671248246278815667140698954262859162743630674375101302303013096719753566536398605828842046373538499736266358442511053152383621080583e-01 + +6.759649405327850670181700456194929502189750234614304846357269137106731032582471677573582008294494705826194131450773107049670717146427804564532075148987065e-03 i
  the component has size 4.12681e-149 and is pointing west-south-west
  the atom domain has size 2.87802e-108
  the nucleus is 4.15478e-133 to the south-south-west of the input point
  the input point is exterior to this component at
  radius 2.00677e+08 and angle 0.295619955576930882 (in turns)
  the multiplier is -5.67373e+07 + +1.92489e+08 i
  a point in the attractor is +1.01589285022300801326268381889642621905716620784746689862495330888851160159813669881526266107957297165711347175450146405953589018925006749566518877570748e-66 + -3.646715913906026744021125419729861152443251137995300638705681163542312469192942799571640391988745181049600996957058080383959716199954755595151824794120333e-66 i

- a period 59444 cardioid
  with nucleus at +2.75337647746737993588667124824627881566714069895426285916274363067437510130230301309671975356653639860582884204637353849973626635844461696577733396177173659502869597622654858047830473369233652610609631007219270037919896108613318635711410655928412269957977397230123742985898239211816931398241903797459102438729408702005271146604041e-01 + +6.75964940532785067018170045619492950218975023461430484635726913710673103258247167757358200829449470582619413145077310704967071714678595763311924422571027117886784050420240236249129631789483532106497151867377563025274513529470021667381579073334313498412010852400179935107657764228375162746931512488396245301309385347189831133866673e-03 i
  the component has size 1.06317e-324 and is pointing north-west
  the atom domain has size 6.66873e-221
  the nucleus is 7.24745e-325 to the east-south-east of the input point
  the input point is interior to this component at
  radius 9.35775e-01 and angle 0.541761546433570995 (in turns)
  the multiplier is -9.03744e-01 + -2.42735e-01 i
  a point in the attractor is -2.6025538574296415952332667091128199346211267632708216906442009663141234426147312488425278300035554053468777171243258952047261577623034475153849717684849957653364661711338481367484672303255804446676714005247665149565145445218216807377219405913550928161747542074194897884493104533200856460675330342275057978441273417134673601626392e-162 + +1.12008951668276505060976485788508740261881654301286406200102941504218073528504840309816859782346828879017415162926827002367559002018428414455323029172624912530914605048101267372377557011128767232093377253073998711371596110109288661492571318569611990167238983116030755092055668694551869794767592874969987603636013091634709434749682e-162 i
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