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Author Topic: Comparing the morpologies of Mandelbrot and Julia sets at different scales  (Read 8770 times)
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Duncan C
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« on: July 18, 2007, 05:10:28 AM »

Has anybody studied the relationship between the topology of Mandelbrot and Julia sets (neighborhoods), and in particular the scales at which sets from the same area have similar structures?

If you take a plot from deep in the Mandelbrot set, and plot the corresponding Julia set, the Julia set's structure resembles the topology of the Mandelbrot set, at several different scales.

For example, if I plot the largest baby mandelbrot on the real axis, it has many "tendrils" coming off of it. All the Julia sets from near this baby Mandelbrot also have tendrils, throughout the entire plot. These tendrils are visible at all scales.

Then, if you zoom in to the cleft of the main cardioid ("Elephant valley") on this baby Mandelbrot, the structure starts to look like "elephants," but with tendrils. Then if you zoom into one of the small "crystal" structures on a tendril, you get rings of elephant shapes. Julia set plots taken from these areas also show rings of elephants, if you zoom in at the proper scale.

A zoom into a Julia set plot will echo the source Mandelbrot plot both at the origin of the Julia set, and if you zoom into the "c" point of the Julia plot, but at different magnifications. In general, it seems to take more magnification to find a Julia plot that echos the source Mandelbrot plot when you zoom in on the "c" point than when you zoom in on the origin of the Julia plot. Julia set plots that zoom in on the origin of a Julia set at high magnification break down due to floating point rounding errors, however, making study of these zooms difficult.

How much you need to zoom varies depending on what part of the Mandelbrot set you are exploring, and how deep you zoom in. For a given area of the Mandelbrot set, you need some scaling factor between the magnification of the Mandelbrot set and the equivalent Julia set zoom. As you zoom further into a Mandelbrot plot, that scaling factor seems to increase.

Here is a small gallery of images that illustrate what I am talking about, using two sample sequences of Mandelbrot and Julia plots:

http://www.pbase.com/duncanc/mandelbrotjulia_morphology_study
« Last Edit: July 18, 2007, 05:14:53 AM by Duncan C » Logged

Regards,

Duncan C
alister
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« Reply #1 on: August 14, 2007, 06:10:48 AM »

>>If you take a plot from deep in the Mandelbrot set, and plot the corresponding Julia set, the Julia set's structure resembles the topology of the Mandelbrot set, at several different scales.<<

I read in a book about fractal geometry that there was a correlation between the Mandelbrot set and the Julia set. In the example I was reading about they showed how using different points from within the Mandelbrot would yield different shaped Julia sets.

Some time before reading this I tried an animation experiment.  Basically, I rendered Julia sets for each point that existed on a unit circle about the origin. What I did find was very interesting. The points in the Julia set began to orbit, and returned to their original locations after completing the trip around the circle.

In conclusion, I decided that although there was some correlation between the two sets the Julia set exhibits its own behavior and it is independent of the Mandelbrot.
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lkmitch
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« Reply #2 on: August 14, 2007, 11:05:48 PM »


In conclusion, I decided that although there was some correlation between the two sets the Julia set exhibits its own behavior and it is independent of the Mandelbrot.


The Mandelbrot and Julia sets for the same formula (e.g., z = z*z + c) are deeply related.  The Mandelbrot set acts like a catalog of the Julia sets.  If the Julia seed is taken from the interior of the Mandelbrot set, then the Julia will have a filled-in interior.  If it is taken from outside, then the Julia set will be a disconnected dust.  A tip point gives a dendritic Julia.  Further, if the seed is taken from a point near the boundary (like from a zoom into the edge of the Mandelbrot set), then the structure in the Julia set will be very similar to that in the Mandelbrot zoom.  So, while one can certainly investigate one set independently of the other, the two sets are not independent.

Kerry
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Duncan C
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« Reply #3 on: August 15, 2007, 01:46:27 AM »


In conclusion, I decided that although there was some correlation between the two sets the Julia set exhibits its own behavior and it is independent of the Mandelbrot.


The Mandelbrot and Julia sets for the same formula (e.g., z = z*z + c) are deeply related.  The Mandelbrot set acts like a catalog of the Julia sets.  If the Julia seed is taken from the interior of the Mandelbrot set, then the Julia will have a filled-in interior.  If it is taken from outside, then the Julia set will be a disconnected dust.  A tip point gives a dendritic Julia.  Further, if the seed is taken from a point near the boundary (like from a zoom into the edge of the Mandelbrot set), then the structure in the Julia set will be very similar to that in the Mandelbrot zoom.  So, while one can certainly investigate one set independently of the other, the two sets are not independent.

Kerry


Kerry,

I agree completely. See the link I posted showing sample Mandelbrot and Julia images. If you pick the right magnification, it can be hard to tell whether a given image is a Mandelbrot or Julia set plot.

I show a couple of examples of a Mandelbrot plot, and a couple of corrisponding Julia set plots from the same area that have strikingly similar structures.


Duncan C


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Regards,

Duncan C
alister
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« Reply #4 on: August 15, 2007, 05:41:56 AM »

 :-[
Yeah, you all are right.
*eats words*

We all say stupid stuff sometimes.

I made this image today http://chaos5.deviantart.com/art/WTF-62309433 and in doing so I started to realize just how wrong my conclusion was.


« Last Edit: August 15, 2007, 05:47:22 AM by alister » Logged
FractalMonster
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« Reply #5 on: August 15, 2007, 05:12:15 PM »

Thank you Duncan for a very interesting post and also a very nice an instructive site smiley
Regarding the relation between the M set and the Julia sets, the phenomena embedded
Julia sets,
partly dealt with here would be interesting. This is the subject in the following links:

http://www.mrob.com/pub/muency/embeddedjuliaset.html
http://www.ibiblio.org/e-notes/MSet/Embed.htm

I, myself, use the term Julia-like barriers and my article 12 in the Chaotic series

http://klippan.seths.se/fractals/articles/

is devoted for this. Regarding the Julia sets for higher degree polynomials there are more
properties o than being disc-like or Cantor dust. For Cubics there is a a third case, disconnected
Julia sets made up of enclosed regions. This is the case when one critical point (belonging to
such a region) has a bounded orbit and the other one escapes to infinity. A quick illustrated
site for all this is, is my old Cubic Tutorial,

http://klippan.seths.se/ik/CubTut/cubictut.html
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Duncan C
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Posts: 348



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« Reply #6 on: August 16, 2007, 01:09:01 AM »

Thank you Duncan for a very interesting post and also a very nice an instructive site smiley
Regarding the relation between the M set and the Julia sets, the phenomena embedded
Julia sets,
partly dealt with here would be interesting. This is the subject in the following links:

http://www.mrob.com/pub/muency/embeddedjuliaset.html
http://www.ibiblio.org/e-notes/MSet/Embed.htm

I, myself, use the term Julia-like barriers and my article 12 in the Chaotic series

http://klippan.seths.se/fractals/articles/

is devoted for this. Regarding the Julia sets for higher degree polynomials there are more
properties o than being disc-like or Cantor dust. For Cubics there is a a third case, disconnected
Julia sets made up of enclosed regions. This is the case when one critical point (belonging to
such a region) has a bounded orbit and the other one escapes to infinity. A quick illustrated
site for all this is, is my old Cubic Tutorial,

http://klippan.seths.se/ik/CubTut/cubictut.html

FractalMonster,

Thanks for your post.

I read your chaotic series with interest. One suggestion I have: could you include the coordinates of the plots in your article? It would help readers to follow along. I've had mixed luck trying to find the regions you reference in your articles.



Duncan C
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Regards,

Duncan C
FractalMonster
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« Reply #7 on: August 16, 2007, 01:43:20 AM »

Duncan

See you have been working very hard with my articles wink Well of cause, if you are lucky to
have UF you only have to run my parameter files included in the zip,

http://klippan.seths.se/fractals/articles/params.zip

(also link on my index page). If you don't have UF, you can easily open the paramaeter files
with notepad. As non of my parameter files are compressed the coordinates, magnification, etc
can be read off in clear text wink below an example,

MysteryZoom1 {
fractal:
  title="MysteryZoom1" width=640 height=480 layers=1
  credits="Ingvar Kullberg;5/1/2006"
layer:
  method=multipass caption="Background" opacity=100
mapping:
  center=-0.26875/1.275 magn=18.8235294117647059
formula:
  maxiter=5000 filename="sp3.ufm" entry="CubicParameterspace3"
  p_PlottedPlane="6.(b-real,b-imag)" p_M=M+ p_SetBorders=no p_hide=yes
  p_areal=0.0 p_aimag=0.0 p_breal=0.0 p_bimag=0.0 p_xrot=0.0
  p_yrot=0.0 p_xrott=0.0 p_yrott=0.0 p_zrot=0.0 p_LocalRot=no
  p_diff=no p_bailout=100.0 p_dbailout=1E-6
inside:
  transfer=none
outside:
  transfer=linear
gradient:
  smooth=yes index=2 color=17749 index=42 color=62720 index=73 color=0
  index=97 color=65530 index=153 color=329215 index=211 color=15335546
  index=280 color=32
opacity:
  smooth=no index=0 opacity=255
}

You find it under "mapping". If there will be any problems, please let me now smiley

Regards,
Ingvar
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Duncan C
Fractal Fanatic
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Posts: 348



WWW
« Reply #8 on: August 16, 2007, 02:04:01 AM »

Duncan

See you have been working very hard with my articles wink Well of cause, if you are lucky to
have UF you only have to run my parameter files included in the zip,

http://klippan.seths.se/fractals/articles/params.zip

(also link on my index page). If you don't have UF, you can easily open the paramaeter files
with notepad. As non of my parameter files are compressed the coordinates, magnification, etc
can be read off in clear text wink below an example,

MysteryZoom1 {
fractal:
  title="MysteryZoom1" width=640 height=480 layers=1
  credits="Ingvar Kullberg;5/1/2006"
layer:
  method=multipass caption="Background" opacity=100
mapping:
  center=-0.26875/1.275 magn=18.8235294117647059
formula:
  maxiter=5000 filename="sp3.ufm" entry="CubicParameterspace3"
  p_PlottedPlane="6.(b-real,b-imag)" p_M=M+ p_SetBorders=no p_hide=yes
  p_areal=0.0 p_aimag=0.0 p_breal=0.0 p_bimag=0.0 p_xrot=0.0
  p_yrot=0.0 p_xrott=0.0 p_yrott=0.0 p_zrot=0.0 p_LocalRot=no
  p_diff=no p_bailout=100.0 p_dbailout=1E-6
inside:
  transfer=none
outside:
  transfer=linear
gradient:
  smooth=yes index=2 color=17749 index=42 color=62720 index=73 color=0
  index=97 color=65530 index=153 color=329215 index=211 color=15335546
  index=280 color=32
opacity:
  smooth=no index=0 opacity=255
}

You find it under "mapping". If there will be any problems, please let me now smiley

Regards,
Ingvar

Ingvar,

Thanks for pointing me to the parameter files.

I'm a Mac user/developer, so I don't have access to UltraFractal. Most of the paremeter files make sense. However, I'm not sure what the magnification parameter means. My program, and several others I've seen, use the width of the plot (the value of max real - min real). What does the magnification value mean in UltraFractal?



Duncan C
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Regards,

Duncan C
FractalMonster
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Posts: 21



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« Reply #9 on: August 16, 2007, 02:24:15 AM »

Well, "what the magnification parameter means" may be a good questions as there seems to
be different systems. However, I don't think they will differ so much. As non of the examples are
any extreme deep magnifications (often only one or two levels) I think there will be no problems
if you fix the coordinates on the parent fractals and perform the zooms yourself wink Eager to hear
the results of your experiments..

And congrats for using as Mac smiley Under the ninities i ran MandelZot on a Mac IIcx. That prog has
a color editor that I miss in other softare, including UF..

Regards,
/Ingvar
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lkmitch
Fractal Lover
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Posts: 238



« Reply #10 on: August 17, 2007, 12:41:53 AM »

I'm a Mac user/developer, so I don't have access to UltraFractal. Most of the paremeter files make sense. However, I'm not sure what the magnification parameter means. My program, and several others I've seen, use the width of the plot (the value of max real - min real). What does the magnification value mean in UltraFractal?

In UltraFractal, the magnification is just a way of specifying how "zoomed in" an image is; higher magnifications lead to smaller sections of the complex plane.  There are various ways to define it, depending on the aspect ratio of the window, but they are generally some multiple of 1/(max real - min real) or 1/(max imaginary - min imaginary).

Kerry
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lkmitch
Fractal Lover
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Posts: 238



« Reply #11 on: August 17, 2007, 12:43:13 AM »

:-[
Yeah, you all are right.
*eats words*

We all say stupid stuff sometimes.

Don't sweat it--it's the way we all learn.  I've made my share of mistakes, too.

Kerry
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Duncan C
Fractal Fanatic
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Posts: 348



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« Reply #12 on: August 17, 2007, 01:10:12 AM »

I'm a Mac user/developer, so I don't have access to UltraFractal. Most of the paremeter files make sense. However, I'm not sure what the magnification parameter means. My program, and several others I've seen, use the width of the plot (the value of max real - min real). What does the magnification value mean in UltraFractal?

In UltraFractal, the magnification is just a way of specifying how "zoomed in" an image is; higher magnifications lead to smaller sections of the complex plane.  There are various ways to define it, depending on the aspect ratio of the window, but they are generally some multiple of 1/(max real - min real) or 1/(max imaginary - min imaginary).

Kerry

Kerry,

I played with 1/(max_real - min_real) but couldn't get the scale right. Now I tried again, and decided that UltraFractal uses 4/(max_real - min_real). My app just uses real width (max_real - min_real). So I have to use 4X the reciprocal of the UF magnfication value. Seems to be just about perfect.

Thanks for the input. I'm not sure why I couldn't work it out before.


Duncan C
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Regards,

Duncan C
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