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Author Topic: Higher Iterations  (Read 12809 times)
Description: How the interior of a minibulb is filled on higher iterations.
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Pauldelbrot
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pderbyshire2
« Reply #30 on: January 23, 2010, 02:13:06 AM »

It already does that; watch the first 20 seconds carefully. It jumps a bit because it wasn't logmapped.

The principle is similar to my own convergent smoothed iterations algorithm, or to the smooth coloring for the Mandelbrot exterior that you sometimes see.

P.S. hooray for whoever got the site restored after some wacko deleted pretty much the entire database (to judge from all my saved URLs giving 404 errors). Looks like you had backups; when there was no change for over a full day I'd feared the worst.

Please do try to keep the forum software up to date from now on -- unpatched forum software is a very common port of entry for site-blankers and various other vandals and miscreants. PhPBB is the worst, but this SMF 1.1.11 is clearly not invulnerable. (I'd also like to know why it took over 24 hours to get the data restored. Oh, yeah, because it took more than 12 just for anyone to get notified that there was anything in need of restoration, because both webmaster and postmaster bounce, in flagrant violation of standard Internet RFCs. Please, someone, fix that too, preferably before the next instance of vandalism so as to be able to get a heads-up about it immediately if and when there is another incident.)
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gussetCrimp
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« Reply #31 on: January 23, 2010, 02:55:38 AM »

Twinbee: did you notice (I'm sure you did) that the whipped cream reaches maximum levels at every iteration count that is a multiple of 8 (i.e. your shots of the 16th, 24th, and 32nd iteration)? There is probably a nice mathematical reason for this. My question is: does whipped cream reach its maxima at these values everywhere on the mandelbulb (this is the power 8 one, right?). And for the bulbs of other powers, is it the same story or do the whipped cream maxima appear at iteration counts related to the bulb exponent?
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twinbee
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« Reply #32 on: January 23, 2010, 07:12:40 PM »

Can't the same technique that's used for colouring the 2D mandelbrot with smooth colours (instead of jumps), be applied?
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Timeroot
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« Reply #33 on: January 23, 2010, 09:36:58 PM »

Well, it works with iterations, but not really with colors. Here's the code I used:
VaryingBailout {
;For use with the "Pixel" formula.
final:
 complex x = 0
 complex c = #pixel
 int iter = 0
 repeat
  x=x^2 + c
  iter=iter+1
 until (iter==@Iterations) || |x|>=@HighBailout
 If |x|>=@HighBailout
  #index=0
 Else
  #index=1/(|x|-4)
 Endif
default:
 int param Iterations
  default=8
 endparam
 float param HighBailout
  caption="Large Bailout"
  default=1E50
 endparam
}

Even with such a low iteration counts and such high bailouts... nada. Maybe #index=1/log(|x|) would work better, but I don't have time right now. Just one last note: remove the -4, so that it reads #index=1/|x|, and apply it to the inside for Mandelbrot formula with a iteration count on the order of 100. Produces very nice inside coloring!  grin
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Someday, man will understand primary theory; how every aspect of our universe has come about. Then we will describe all of physics, build a complete understanding of genetic engineering, catalog all planets, and find intelligent life. And then we'll just puzzle over fractals for eternity.
twinbee
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« Reply #34 on: January 23, 2010, 10:02:20 PM »

gussetCrimp: Before I answer, one thing to bear in mind is that the smooth effect you see in those pics isn't the "whipped cream" effect I usually refer to. The 'real' insidious whipped cream that plagues the Mandelbulb never sprouts any more detail after further iterations (at least not along all axis the way we'd like). It looks like stretched taffy. However, the smooth bulbous surfaces you notice in my pics will sprout more detail after further iterations as you can see. The real 3D Mandelbrot (if it exists) will also have these.

Even the standard 2D Mandelbrot has these bulbous surfaces with a low enough iteration. See the top right part of this picture for instance:



But anyway to answer your question, I think the answer is a generic 'no', because it depends on zoom level and location of the camera. Explore the 2D Mandelbrot with a finite number of iterations to explore this effect.

For evidence see my original 'Spine' pic which has some sections very smooth, but other sections very detailed:
http://www.skytopia.com/project/fractal/new/full/q85/Mandelbrot-Crustacean-med.jpg

However, there is a chance that given any particular part, the smooth bulbous effect may return according to the power of the fractal (yes it is exponent 8 for my iteration collection).
« Last Edit: January 23, 2010, 10:29:34 PM by twinbee » Logged
Timeroot
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« Reply #35 on: January 25, 2010, 05:36:52 AM »

Okay, I do realise this is a bit Off Topic, but I've found a pretty good coloring algorithim based on the idea of increasing th Bailout. It creates a perfectly smooth gradient for any formula of the type z->z^n+c, and can easily be adapted to any other fractal Escape Time fractal. It also creates a nice, smooth interior coloring that varies a good deal depending on iteration, if you use the Pixel formula; if you want a different interior coloring, use the Mset formula.
Code:
VaryingBailout {
;For use with the "Pixel" formula.
;Copyright Alex Meiburg 2010
final:
If @Type == 0
 complex x = @Perturbation
 complex c = #pixel
Elseif @Type == 1
 complex c = @C
 complex x = #pixel
Endif
 int iter = 0
 repeat
  x=x^@Power + c
  iter=iter+1
 until (iter==#maxiter) || |x|>=@LargeBailout
 If |x|>=@LargeBailout
  #index=cabs(sqrt(log(@Power)*(#maxiter-iter) + log(log(|x|-4))))
 Else
  #index=sqrt(log(log(|x|-4)))
 Endif
default:
 float param LargeBailout
  caption="Large Bailout"
  hint="This does NOT need to be very large; Only large enough for the value of \
  c to be negligible. This means a value of 100 should be more than sufficient."
  default=100
 endparam
 complex param Power
  default=(2,0)
 endparam
 param Type
  enum="Mandelbrot" "Julia"
  default=0
 endparam
 complex param C
  visible = (@Type==1)
  default=(-0.2,-0.79)
 endparam
 complex param Perturbation
  visible = (@Type==0)
 endparam
}
« Last Edit: January 25, 2010, 06:14:17 AM by Timeroot, Reason: Improved formula » Logged

Someday, man will understand primary theory; how every aspect of our universe has come about. Then we will describe all of physics, build a complete understanding of genetic engineering, catalog all planets, and find intelligent life. And then we'll just puzzle over fractals for eternity.
trafassel
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Posts: 531


trafassel
« Reply #36 on: January 25, 2010, 10:06:32 AM »

Smooth iterations:

<a href="http://www.youtube.com/v/yYz3gQ108Mo&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/yYz3gQ108Mo&rel=1&fs=1&hd=1</a>
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cortexman
Guest
« Reply #37 on: January 25, 2010, 10:03:11 PM »

Smooth iterations:

<a href="http://www.youtube.com/v/yYz3gQ108Mo&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/yYz3gQ108Mo&rel=1&fs=1&hd=1</a>


How do you know that the increased "detail" is not due to the amplification of errors due to rounding? That's what it looks like to me.
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LesPaul
Guest
« Reply #38 on: January 25, 2010, 11:36:47 PM »

Smooth iterations:

Cooooooool.  I hope that the title "Part 1/3" means that there is a "Part 2/3" and "Part 3/3" in the works.  It's like eagerly anticipating the next Star Wars sequel.  smiley
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twinbee
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Posts: 383



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« Reply #39 on: January 27, 2010, 05:30:07 PM »

Great! So smooth iteration building is possible. I'm guessing it uses the same kind of idea that the 'smooth colouring' does for the normal 2D Mandelbrot?
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trafassel
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Posts: 531


trafassel
« Reply #40 on: January 28, 2010, 07:03:39 AM »

<a href="http://www.youtube.com/v/qM9fAaZ0_SA&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/qM9fAaZ0_SA&rel=1&fs=1&hd=1</a>
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LesPaul
Guest
« Reply #41 on: January 28, 2010, 08:10:09 AM »

 the wave
 the wave
Spectacular!

Why does it appear that higher iterations are adding to the set?  Doesn't iteration normally remove from the set?
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makc
Strange Attractor
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Posts: 272



« Reply #42 on: January 28, 2010, 08:16:39 AM »

Doesn't iteration normally remove from the set?
Should be removing when sequence diverges, and adding when converges... no?
« Last Edit: January 28, 2010, 08:18:17 AM by makc » Logged
cKleinhuis
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Posts: 7044


formerly known as 'Trifox'


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« Reply #43 on: January 28, 2010, 12:53:00 PM »

they simply are considered as belonging to the set, because they diverge after the maximum iteration reached
 evil

when then increasing the iteration they will diverge and thus wont belong to the set anymore...  police


 afro
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divide and conquer - iterate and rule - chaos is No random!
Pauldelbrot
Fractal Senior
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Posts: 2592



pderbyshire2
« Reply #44 on: January 28, 2010, 03:21:59 PM »

the wave
 the wave
Spectacular!

Why does it appear that higher iterations are adding to the set?  Doesn't iteration normally remove from the set?

These were views from inside the set looking out, i.e. the outside was solid and the inside hollow.
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