Title: Calcyman's Idea (2D)
Post by: LesPaul on January 23, 2010, 12:35:16 AM
Hi all,
Calcyman had an interesting idea regarding the 3D Mandelbulb. His original post is here: http://www.fractalforums.com/index.php?topic=2354.msg10836#msg10836 (http://www.fractalforums.com/index.php?topic=2354.msg10836#msg10836)
His idea was to include all the "higher orders" when calculating the Mandelbrot (Mandelbulb) set, and he came up with a clever way to do it that can be calculated efficiently. His example started with:
... which is the basic form of the Mandelbulb. Then he added in higher orders, like so:
The idea is for each term to contribute less as the exponent gets higher. So, each term is divided by a factorial, which gets large very quickly, and the really high-order terms become negligible. But the clever thing about this is that this series is actually just the MacLaurin series expansion for})
. So, his formula can be calculated just like the original with one additional multiplication by a cosine term.
When I saw his post, the question that came to my mind was whether this had been studied in the 2D realm of the original Mandelbrot. I started some very basic research and found some interesting things lurking within that magical complex plane. :)
I decided to generalize his idea to be "z raised to any power, multiplied by any function of z, plus z0."
First, let me paste the UltraFractal formula that I used, in case anyone else is interested in experimenting further. The formula is so simple that I'm nearly certain this has been done before, to some extent.
That will allow you to choose any function you like (as long as it's supported by UltraFractal) and multiply it by z raised to any power, in other words:
} \: + \: z_0)
As the "hint" suggests, you can also set the power to zero, reducing the calculation to simply:
} \: + \: z_0)
So, I began by looking at just sin(z) and cos(z), without multiplication by a power of z.
sin(z) + z0 looks like this:
Calcyman had an interesting idea regarding the 3D Mandelbulb. His original post is here: http://www.fractalforums.com/index.php?topic=2354.msg10836#msg10836 (http://www.fractalforums.com/index.php?topic=2354.msg10836#msg10836)
His idea was to include all the "higher orders" when calculating the Mandelbrot (Mandelbulb) set, and he came up with a clever way to do it that can be calculated efficiently. His example started with:
... which is the basic form of the Mandelbulb. Then he added in higher orders, like so:
The idea is for each term to contribute less as the exponent gets higher. So, each term is divided by a factorial, which gets large very quickly, and the really high-order terms become negligible. But the clever thing about this is that this series is actually just the MacLaurin series expansion for
When I saw his post, the question that came to my mind was whether this had been studied in the 2D realm of the original Mandelbrot. I started some very basic research and found some interesting things lurking within that magical complex plane. :)
I decided to generalize his idea to be "z raised to any power, multiplied by any function of z, plus z0."
First, let me paste the UltraFractal formula that I used, in case anyone else is interested in experimenting further. The formula is so simple that I'm nearly certain this has been done before, to some extent.
Code:
MandelTimesFunc {
init:
z = 0
loop:
z = z^@power * @Function(z) + #pixel
bailout:
|z| < @bail
default:
title = "Complex Power Times Complex Function"
float param power
caption = "Power"
hint = "The power to which Z is raised. Set to zero to reduce calculation to just Function(Z)."
default = 2.0
endparam
float param bail
caption = "Bailout"
default = 1e6
endparam
}
That will allow you to choose any function you like (as long as it's supported by UltraFractal) and multiply it by z raised to any power, in other words:
As the "hint" suggests, you can also set the power to zero, reducing the calculation to simply:
So, I began by looking at just sin(z) and cos(z), without multiplication by a power of z.
sin(z) + z0 looks like this:
Title: Re: Calcyman's Idea (2D)
Post by: LesPaul on January 23, 2010, 12:36:39 AM
and cos(z) + z0 looks like this:
Title: Re: Calcyman's Idea (2D)
Post by: LesPaul on January 23, 2010, 12:38:38 AM
These images are interesting and definitely have some fractal-like qualities. In fact, if you zoom in on the cosine version, you can find some familiar things!
Title: Re: Calcyman's Idea (2D)
Post by: LesPaul on January 23, 2010, 12:47:55 AM
I continued by getting back to the original idea and including a power of z in the calculation. Things become interesting even with a power of one, giving
} \: + \: z_0)
This produces the attached images. "z_1_sin.png" shows the bulk of the set, although it appears to continue quite far in both directions. "z_1_sin_zoom_1.png" is a zoom into the flat-ish area between the two main "bulbs" shown in the original. Surprise! Another minibrot is hiding there, along with some other shapes that are distinctly not similar to the Mandelbrot. "z_1_sin_zoom_2.png" is a further zoom into the same area showing some really surprising features.
This produces the attached images. "z_1_sin.png" shows the bulk of the set, although it appears to continue quite far in both directions. "z_1_sin_zoom_1.png" is a zoom into the flat-ish area between the two main "bulbs" shown in the original. Surprise! Another minibrot is hiding there, along with some other shapes that are distinctly not similar to the Mandelbrot. "z_1_sin_zoom_2.png" is a further zoom into the same area showing some really surprising features.
Title: Re: Calcyman's Idea (2D)
Post by: LesPaul on January 23, 2010, 12:56:19 AM
The next experiment was to increase the power of z to 2:
} \: + \: z_0)
This gives the attached images and zooms. This set is interesting because it has a lot of characteristics of the basic Mandelbrot. There are many, many instances of smaller, rotated/skewed versions of the original (what we would call "minibrots" or "midgets" in the the Mandelbrot). These midgets appear to be far more dense in this set than in the Mandelbrot. But interestingly, there are also some parts of the set that are not self-similar. I don't believe I've ever seen a part of the Mandelbrot set that was actually inside the set that didn't look similar to the original set.
The "stem" area of this set looks rather non-interesting at first, but there are some really bizarre shapes hiding there.
This gives the attached images and zooms. This set is interesting because it has a lot of characteristics of the basic Mandelbrot. There are many, many instances of smaller, rotated/skewed versions of the original (what we would call "minibrots" or "midgets" in the the Mandelbrot). These midgets appear to be far more dense in this set than in the Mandelbrot. But interestingly, there are also some parts of the set that are not self-similar. I don't believe I've ever seen a part of the Mandelbrot set that was actually inside the set that didn't look similar to the original set.
The "stem" area of this set looks rather non-interesting at first, but there are some really bizarre shapes hiding there.
Title: Re: Calcyman's Idea (2D)
Post by: LesPaul on January 23, 2010, 01:01:26 AM
There is a lot more experimentation that I'd like to do, but there's only so much time in a day. :) One last thing I tried was Calcyman's original suggestion, which was the eigth-order cosine version:
} \: + \: z_0)
That gives the attached image (and zoom), which really does have the "feel" of the Mandelbulb. Here again, many parts are self-similar and many parts are not. It's all interesting, to me, anyway!
That gives the attached image (and zoom), which really does have the "feel" of the Mandelbulb. Here again, many parts are self-similar and many parts are not. It's all interesting, to me, anyway!
Title: Re: Calcyman's Idea (2D)
Post by: Timeroot on January 23, 2010, 03:32:54 AM
I think that, yes, much experimentation has been done with other functions such as sin(z) or cos(z), but I'd never seen z^n * fn(z). Nice images! I'm not sure if you know, but the shape featured in z_2_sin.png is a multibrot; the shape that appears from iterating z=z^3 + z0, instead of z^2. I find it interesting that this appears in your z^2*sin(z) version. The object to the right in z_2_sin_zoom_2.png is Julia set. Nothing new, really, to see that hiding there, but the interesting thing is that it's filled in - as far as I know, that never happens in the regular M-Set. For the record, it can be easily shown that any formula of the form z=z^n * sin(z) + z0 or z=z^n * cos(z) + z0 extends out to infinity in both directions. :P I really like your picture z_1_sin_zoom_2.png, by the way!
Title: Re: Calcyman's Idea (2D)
Post by: Paolo Bonzini on January 23, 2010, 09:34:34 AM
I'm not sure if you know, but the shape featured in z_2_sin.png is a multibrot; the shape that appears from iterating z=z^3 + z0, instead of z^2. I find it interesting that this appears in your z^2*sin(z) version.
That's because z^2*sin(z) = z^3+O(z^5).
The object to the right in z_2_sin_zoom_2.png is Julia set. Nothing new, really, to see that hiding there, but the interesting thing is that it's filled in - as far as I know, that never happens in the regular M-Set.
Yes, that's very interesting!Title: Re: Calcyman's Idea (2D)
Post by: matsoljare on January 23, 2010, 12:39:54 PM
Don't forget to try the "julibrot" projections as well!
Title: Re: Calcyman's Idea (2D)
Post by: kram1032 on January 23, 2010, 01:51:46 PM
very nice experiments :D
They look great :)
Just an other thought:
what's about doing+i*z^k*sin(z)+z0)
?
They look great :)
Just an other thought:
what's about doing
Title: Re: Calcyman's Idea (2D)
Post by: bib on January 23, 2010, 02:13:22 PM
here is a test using z*tan(z) :)
Title: Re: Calcyman's Idea (2D)
Post by: kram1032 on January 23, 2010, 02:19:01 PM
O.o How's that coloured?
Really nice!
Really nice!
Title: Re: Calcyman's Idea (2D)
Post by: bib on January 23, 2010, 02:40:38 PM
There are 2 layers : one with smooth iterations in yellow and blue, with a steep variation in gradient to make the patterns inside the big bud appear, and another with orbit traps to highlight the minibrots.
Title: Re: Calcyman's Idea (2D)
Post by: Nahee_Enterprises on January 23, 2010, 03:58:09 PM
I continued by getting back to the original idea and including a power of z in the calculation.
Things become interesting even with a power of one, giving
<Quoted Image Removed>
This produces the attached images.
Things become interesting even with a power of one, giving
<Quoted Image Removed>
This produces the attached images.
These "z_1_sin" images are quite interesting. :)
Title: Re: Calcyman's Idea (2D)
Post by: LesPaul on January 23, 2010, 06:17:21 PM
here is a test using z*tan(z) :)
WOW!! I will definitely experiment with tan(z) having seen that image!
The MacLaurin series for tan(x) is interesting. It is identical to sin(x) except for two things:
1) the tan(x) series has no negative terms, and
2) the coefficients of the tan(x) series are larger.
This makes me think that inverse-tangent is probably also interesting:
Title: Re: Calcyman's Idea (2D)
Post by: LesPaul on January 23, 2010, 06:52:27 PM
Tangent is a strange beast!
The first image here is what appears to be the whole set. I had to zoom out quite a bit to see it -- the top-left is (-1000, 750) and the bottom right is (1000, -750). This image is so unusual that it almost seems like there is some calculation error happening. I did see some strange behavior (like the image changing when zooming) until I turned off "guessing" and "periodicity checking."
The two zooms here are not deep at all. There is no layering or any other post-processing of the images. "Function" is set to "tan" and "Power" is set to one. It's just a really strange set! The different areas show almost no resemblance to each other...
The first image here is what appears to be the whole set. I had to zoom out quite a bit to see it -- the top-left is (-1000, 750) and the bottom right is (1000, -750). This image is so unusual that it almost seems like there is some calculation error happening. I did see some strange behavior (like the image changing when zooming) until I turned off "guessing" and "periodicity checking."
The two zooms here are not deep at all. There is no layering or any other post-processing of the images. "Function" is set to "tan" and "Power" is set to one. It's just a really strange set! The different areas show almost no resemblance to each other...
Title: Re: Calcyman's Idea (2D)
Post by: kram1032 on January 23, 2010, 07:21:23 PM
nice O.o
Did you also try different bailout values?
Did you also try different bailout values?
Title: Re: Calcyman's Idea (2D)
Post by: LesPaul on January 23, 2010, 07:52:07 PM
I'm not sure if you know, but the shape featured in z_2_sin.png is a multibrot; the shape that appears from iterating z=z^3 + z0, instead of z^2. I find it interesting that this appears in your z^2*sin(z) version.
It is very similar to the z3 multibrot, but different in a lot of ways. The standard multibrot doesn't have that "stem" along the real axis. The overall shape is slightly different, too.
You can toggle between the two by just switching the formula from "sin" to "ident." The identity formula just returns z, so you get
zn+1 = zn2 * zn + z0 = zn3 + z0.
That would suggest that the zn2 term dominates the shape on the large scale, so that multiplying by sin(zn) isn't much different from multiplying by just zn. But they are quite different when you zoom in.
Title: Re: Calcyman's Idea (2D)
Post by: LesPaul on January 23, 2010, 08:02:46 PM
nice O.o
Thanks :)
Did you also try different bailout values?
Good idea... It looks like changing the bailout does have a dramatic effect. The black semicircle on the +real half just gets larger and larger if you increase the bailout. I guess that suggests that the entire +real half of the graph is in the set? Or that it isn't in the set but tends to infinity very slowly? I'm not sure. Are there coloring algorithms that don't rely on a bailout value? It would probably make rendering unacceptably slow if there was no bailout.
Title: Re: Calcyman's Idea (2D)
Post by: kram1032 on January 24, 2010, 12:48:53 AM
that's what I thought...
With functions like the tan, going to infinity in a cyclic pattern, I guess you can't define a true bailout value.
You could have a value close to infinity which then returns a value close to zero, if you plot it in again.
However it might help a bit (for bringing slightly more order in this) to use)
, rather than )
, as then the intervals will be at lenght 2, having infinity at odd and zero at even numbers...
With functions like the tan, going to infinity in a cyclic pattern, I guess you can't define a true bailout value.
You could have a value close to infinity which then returns a value close to zero, if you plot it in again.
However it might help a bit (for bringing slightly more order in this) to use
Title: Re: Calcyman's Idea (2D)
Post by: Timeroot on January 24, 2010, 04:04:23 AM
It seemed to me it would be interesting to find the locus of points (I believe they would be Cantor Dust) that do escape to infinity. Although that isn't very easy to do, I created an image of the locus of points that monotonically increase. Here's the code I used:
It created a Cantor Set along the positive real axis (with the exception of an narrow, boring, error-ridden shape from 0 to 0.23) with no other points in the positive half-plane. In the other half plane, some interesting figures were produced. I'll eventually post some pics, but I'm doing some other stuff right now... :-\
Code:
TangentEscape {
init:
c=#pixel
z=@Perturbation
loop:
zOld=z
z=z*tan(z) + c
bailout:
|z| >= |zOld|
}Title: Re: Calcyman's Idea (2D)
Post by: bib on January 24, 2010, 12:24:01 PM
a video showing the effect of bailout on z*tan(z)+c
http://www.youtube.com/watch?v=YviGjZaQGLQ
http://www.youtube.com/watch?v=YviGjZaQGLQ
Title: Re: Calcyman's Idea (2D)
Post by: bib on January 24, 2010, 02:48:01 PM
Another one that dezooms from near the origin and zooms far from the origin:
http://www.youtube.com/watch?v=-wK6tMPR_Nc
http://www.youtube.com/watch?v=-wK6tMPR_Nc
Title: Re: Calcyman's Idea (2D)
Post by: Calcyman on January 24, 2010, 03:49:01 PM
Wow! Some truly stunning images from such simple formulae. The z*tan(z) fractal looks totally pathological until you zoom into its core, where it is dominated with minibrots and multiply connected lacy structures. By comparison, ordinary z˛ Mandelbrot-Julia fractals have simply connected dendritic structures.
The diversity of phenomena in the tangent fractal is exciting. The fractal detail seems to lie on radial lines emanating from the origin, and the Manhattan-style lines that bridge them. It's quite unexpected to see Mandelbrot Sets hiding in the details.
The Maclaurin Series of the natural logarithm function also rapidly converges. z*ln(z) should be interesting -- the real line terminates at the base of the logarithm (e in this case).
My 'idea' was to investigate fractals with Maclaurin series (integer powers of z). I wonder whether Fourier series conceal the same detail, considering the fractal complexity of the sinusoidal fractals. This would yeild a new family of fractals, all composed of sinusoidal components.
In fact, the Weierstrass function, a particular Fourier series, is a fractal on the first iteration!
The diversity of phenomena in the tangent fractal is exciting. The fractal detail seems to lie on radial lines emanating from the origin, and the Manhattan-style lines that bridge them. It's quite unexpected to see Mandelbrot Sets hiding in the details.
The Maclaurin Series of the natural logarithm function also rapidly converges. z*ln(z) should be interesting -- the real line terminates at the base of the logarithm (e in this case).
My 'idea' was to investigate fractals with Maclaurin series (integer powers of z). I wonder whether Fourier series conceal the same detail, considering the fractal complexity of the sinusoidal fractals. This would yeild a new family of fractals, all composed of sinusoidal components.
In fact, the Weierstrass function, a particular Fourier series, is a fractal on the first iteration!
Title: Re: Calcyman's Idea (2D)
Post by: David Makin on January 24, 2010, 05:56:11 PM
I think you'll find formulas already built for Fractint, Ultra Fractal and ChaosPro investigating formulas along the lines you're describing :)
There are also some where a continued series is extended with iteration.
There are also some where a continued series is extended with iteration.
Title: Re: Calcyman's Idea (2D)
Post by: LesPaul on January 25, 2010, 08:48:56 PM
Another one that dezooms from near the origin and zooms far from the origin:
Wow, I watched this one multiple times! :)
Did you find that the calculation time becomes unbearably slow as soon as you zoom to a level where arbitrary precision is required? In UltraFractal, the "pixels per second" drops from several thousand down to like 9! The algorithm they use for tan(z) must not be very friendly to arbitrary-precision math.
I took a shot at animating what happens as the formula varies smoothly from z2*tan(z) to just z*tan(z) and the result is interesting. The set kind of "folds over" on itself and all kinds of strange details emerge right before the power reaches 1. It looks as if the set "explodes" in the -real direction.
I'd post it but I used an evaluation version of UltraFractal and the whole AVI is stamped with "Evaluation Copy." I guess it's time to shell out the hundred dollars. :)
Those videos are really great, bib.
Title: Re: Calcyman's Idea (2D)
Post by: LesPaul on January 25, 2010, 09:25:17 PM
The Maclaurin Series of the natural logarithm function also rapidly converges. z*ln(z) should be interesting ...
I'm not able to get z*ln(z) to do anything interesting, so far. The entire complex plane appears to be in the set. I'll have to check a few points by hand and figure out what's going on there.
Title: Re: Calcyman's Idea (2D)
Post by: Timeroot on January 26, 2010, 01:21:46 AM
I'd post it but I used an evaluation version of UltraFractal and the whole AVI is stamped with "Evaluation Copy." I guess it's time to shell out the hundred dollars. :)
Those videos are really great, bib.
Those videos are really great, bib.
Just a tip: While UltraFractal is generally more user-friendly and what not and has a better GUI and stuff, ChaosPro can load UF formulas, animate them, and it's free. I just use it when I have a good fractal form UF and want to render it. :-P
Can't wait to see vid!
Title: Re: Calcyman's Idea (2D)
Post by: LesPaul on January 26, 2010, 07:50:03 AM
Just a tip: While UltraFractal is generally more user-friendly and what not and has a better GUI and stuff, ChaosPro can load UF formulas, animate them, and it's free. I just use it when I have a good fractal form UF and want to render it. :-P
Can't wait to see vid!
Can't wait to see vid!
Nice, "free" is my favorite price. :)
Is it also open source? I'd be more than willing to contribute to the project, if so.
Title: Re: Calcyman's Idea (2D)
Post by: LesPaul on January 26, 2010, 09:18:46 AM
Can't wait to see vid!
Here it is:
http://www.youtube.com/watch?v=dwHUkbhI4CA
Sorry the compression is a bit severe. I'm just getting the hang of uploading to YouTube.
Many of the small black islands that you see swirling around are minibrots.
Title: Re: Calcyman's Idea (2D)
Post by: stigomaster on January 26, 2010, 10:55:01 AM
Nice, "free" is my favorite price. :)
Is it also open source? I'd be more than willing to contribute to the project, if so.
Is it also open source? I'd be more than willing to contribute to the project, if so.
ChaosPro is unfortunately not open source, just freeware. And it's Windows-only, but it runs well in Wine. :dink:
Title: Re: Calcyman's Idea (2D)
Post by: kram1032 on January 26, 2010, 03:34:14 PM
LesPauls: Which powers are included? It looks nice :)
Title: Re: Calcyman's Idea (2D)
Post by: LesPaul on January 27, 2010, 12:00:42 AM
LesPauls: Which powers are included? It looks nice :)
Thanks! I want to experiment more, both with the formula and the video codec settings to try to make it cleaner. I did a version with no compression but it came out to about 400MB and YouTube probably wouldn't be happy about that. :)
The power goes from 2.0 to 1.0, so the first frame is exactly
zn+1 = zn2 * tan(zn) + z0
and the last frame is exactly
zn+1 = zn * tan(zn) + z0
The power-2 version behaves very nicely but the power-1 version does not. I'll make a video that zooms out as the power decreases and it will show how the whole thing basically blows up when the power becomes 1.
Title: Re: Calcyman's Idea (2D)
Post by: LesPaul on January 27, 2010, 08:17:45 AM
A related question - Is there any particular software that you guys use for your video editing? I'd really like to be able to just render the animation to individual frames (PNG) and then play around with different levels of compression.
Title: Re: Calcyman's Idea (2D)
Post by: LesPaul on January 27, 2010, 09:18:22 AM
The power-2 version behaves very nicely but the power-1 version does not. I'll make a video that zooms out as the power decreases and it will show how the whole thing basically blows up when the power becomes 1.
Here it is.
http://www.youtube.com/watch?v=HHDOqBfdO_Y&feature=player_embedded (http://www.youtube.com/watch?v=HHDOqBfdO_Y&feature=player_embedded)
Title: Re: Calcyman's Idea (2D)
Post by: Nahee_Enterprises on January 27, 2010, 12:46:13 PM
A related question - Is there any particular software that you guys use for your video editing?
I'd really like to be able to just render the animation to individual frames....
I'd really like to be able to just render the animation to individual frames....
Have you tried VirtualDub (http://www.virtualdub.org/) ?? It is FREE !!!! :)
You might also be interested in the FREE K-Lite Codec Pack (http://www.codecguide.com/download_kl.htm), which comes in various downloads:
- Basic
- Standard
- Full
- Mega
- Corporate
- 64-bit
Title: Re: Calcyman's Idea (2D)
Post by: David Makin on January 27, 2010, 03:32:22 PM
A related question - Is there any particular software that you guys use for your video editing?
I'd really like to be able to just render the animation to individual frames....
I'd really like to be able to just render the animation to individual frames....
Have you tried VirtualDub ?? It is FREE !!!! :)
I use VirtualDub and the Xvid codec - *both* free :)
Title: Re: Calcyman's Idea (2D)
Post by: bib on January 28, 2010, 11:43:31 PM
Hi
Another vid using the formula z=z^n*tan(z)+c
http://www.youtube.com/watch?v=ToWAOKeNKXw
Another vid using the formula z=z^n*tan(z)+c
http://www.youtube.com/watch?v=ToWAOKeNKXw
Title: Re: Calcyman's Idea (2D)
Post by: kram1032 on January 30, 2010, 11:50:49 AM
whoa, that's a great zoom and value-shift :D
Title: Re: Calcyman's Idea (2D)
Post by: LesPaul on February 01, 2010, 01:10:43 AM
Wow, fantastic, bib!
I thought for sure that there was going to be a minibrot at the end of the zoom. :)
Several places in z*tan(z) that I've roamed around and zoomed have revealed minibrots.
I thought for sure that there was going to be a minibrot at the end of the zoom. :)
Several places in z*tan(z) that I've roamed around and zoomed have revealed minibrots.
Title: Re: Calcyman's Idea (2D)
Post by: LesPaul on February 01, 2010, 01:17:35 AM
I just noticed that at 1:17 in bib's video, two minibrots "float" across the chaos from right to left... It's very unusual, because they aren't classical minibrots in the sense that they don't appear to actually be inside the set. The chaotic background noise appears to shape itself into minibrots. I've definitely never seen that before. z*tan(z) is such an unusual set.
Title: Re: Calcyman's Idea (2D)
Post by: Timeroot on February 01, 2010, 05:53:09 AM
I had actually noticed that too. Due to the high level of noise present and the unusual inside coloring, it's hard to say whether or not these minibrots are inside the set. But I'm not the one to know things here... ??? It is a strange beast indeed.
Title: Re: Calcyman's Idea (2D)
Post by: Calcyman on February 02, 2010, 08:18:22 PM
On one of the first images of this fractal set, it is possible to see Mandelbrot-shaped formations with internal substructure. In other words, those formations are not actually part of the Fatou set of the function. Rather, it appears that crossing the border into these curiosities is like passing through an asymptote -- something which seems quite possible given the nature of the tangent function.
Indeed, the first picture (courtesy of Bib) displays this artefact really clearly. The inside of this bubble-like anomaly is populated with fractal swirls, which contrasts with the globular clusters outside the anomaly. Minibrots bridge the gap between these worlds, as if unperturbed by the asymptote.
(http://www.fractalforums.com/index.php?action=dlattach;topic=2483.0;attach=1607;image)
This can also be seen in LesPaul's second image, albeit less clearly.
Maybe we should translate this series back into triplex, and see if it yields the elusive "True 3D Mandelbrot".
What about z = tan(z)^2 + c? That might be even more remarkable than the hybrid form (z*tan(z) + c). It's certainly simpler and more elegant in definition.
Indeed, the first picture (courtesy of Bib) displays this artefact really clearly. The inside of this bubble-like anomaly is populated with fractal swirls, which contrasts with the globular clusters outside the anomaly. Minibrots bridge the gap between these worlds, as if unperturbed by the asymptote.
(http://www.fractalforums.com/index.php?action=dlattach;topic=2483.0;attach=1607;image)
This can also be seen in LesPaul's second image, albeit less clearly.
Maybe we should translate this series back into triplex, and see if it yields the elusive "True 3D Mandelbrot".
What about z = tan(z)^2 + c? That might be even more remarkable than the hybrid form (z*tan(z) + c). It's certainly simpler and more elegant in definition.
Title: Re: Calcyman's Idea (2D)
Post by: bib on February 02, 2010, 08:56:05 PM
These kinds of shadow buds is something I have been looking for in other fractals, like in these pics. I don't know the details of the maths behind unfortunately, but I guess that they have in common to diverge for some real values (e.g 1/(1-x) or tan(x))
(http://www.fractalforums.com/gallery/0/492_07_06_09_8_28_40.jpg)
(http://www.fractalforums.com/gallery/0/492_06_06_09_4_01_03.jpg)
(http://www.fractalforums.com/gallery/0/492_06_06_09_9_46_23.JPG)
(http://www.fractalforums.com/gallery/0/492_08_06_09_11_18_34.JPG)
(http://www.fractalforums.com/gallery/0/492_07_06_09_8_28_40.jpg)
(http://www.fractalforums.com/gallery/0/492_06_06_09_4_01_03.jpg)
(http://www.fractalforums.com/gallery/0/492_06_06_09_9_46_23.JPG)
(http://www.fractalforums.com/gallery/0/492_08_06_09_11_18_34.JPG)
Title: Re: Calcyman's Idea (2D)
Post by: kram1032 on February 02, 2010, 10:16:50 PM
tan delivers some of the most beautiful sets, really O.o
Title: Re: Calcyman's Idea (2D)
Post by: stigomaster on February 02, 2010, 10:23:30 PM
Warning: Horrible pun ahead!
I guess the Mandelbrot set got a little makeover... It's such a nice tan.
I guess the Mandelbrot set got a little makeover... It's such a nice tan.
Title: Re: Calcyman's Idea (2D)
Post by: kram1032 on February 02, 2010, 10:25:09 PM
First fractal joke being funny so far :D
Title: Re: Calcyman's Idea (2D)
Post by: LesPaul on February 02, 2010, 10:30:32 PM
Haha, stigoman. :)
Bib, where did you find those images? Are you saying that they're not z*tan(z)?
Bib, where did you find those images? Are you saying that they're not z*tan(z)?
Title: Re: Calcyman's Idea (2D)
Post by: bib on February 02, 2010, 10:41:39 PM
lol
I did these images last year, with different formulae: Supernova, Herman Ring, Phoenix-Halley-Nova, so not z*tan(z)
I did these images last year, with different formulae: Supernova, Herman Ring, Phoenix-Halley-Nova, so not z*tan(z)
Title: Re: Calcyman's Idea (2D)
Post by: stigomaster on February 09, 2010, 10:27:53 PM
This is the bifurcation diagram of xn+1 = xn * tan(xn) + x0. It features a moderately sized logistic map-ish thing in the middle along with many others of different sizes spread around the entire set. View size (-5, -3.75)-(5, 3.75)
Title: Re: Calcyman's Idea (2D)
Post by: kram1032 on February 10, 2010, 03:57:24 PM
strange bifurcation diagram :)
How wide is that range that is clearly not chaotic, in the middle?
How wide is that range that is clearly not chaotic, in the middle?
Title: Re: Calcyman's Idea (2D)
Post by: stigomaster on February 10, 2010, 04:58:55 PM
It goes from about -1.11 to about 0.24, a width of a tiny bit less than 1.35.
Title: Re: Calcyman's Idea (2D)
Post by: Timeroot on February 10, 2010, 07:39:08 PM
Interesting diagram; rather than "bands" of chaos or periodic "windows", it looks more like chaotic "curtains" and periodic "brdiges". :)
For those of you on the UF mailing list, someone recently posted that he had seen the "mandelbfields" - those weird, flying Mandelbrots we saw at 1:16 in Reply #37 - in the phoenixDoubleNova fractal. Oh dear god, fractals are much, much too weird for us mortals... :surrender:
For those of you on the UF mailing list, someone recently posted that he had seen the "mandelbfields" - those weird, flying Mandelbrots we saw at 1:16 in Reply #37 - in the phoenixDoubleNova fractal. Oh dear god, fractals are much, much too weird for us mortals... :surrender:
Title: Re: Calcyman's Idea (2D)
Post by: kram1032 on February 10, 2010, 08:56:39 PM
interesting that the biggest "island of order" lies on the lower site of the diagram rather than on the higher one :)
Title: Re: Calcyman's Idea (2D)
Post by: Nahee_Enterprises on February 11, 2010, 01:52:45 AM
For those of you on the UF mailing list, someone recently posted that
he had seen the "mandelbfields" - those weird, flying Mandelbrots....
he had seen the "mandelbfields" - those weird, flying Mandelbrots....
Actually, I believe that Daniel J. Wills (aka. inkling, enkling, enk, gdanzo) stated the following:
- Turn up the 'Relaxation' (eg > 2.5), a noisy lake will form in the 'inside' area.. this is interesting to explore in itself, it's full of 'mandelfields' (those fuzzy mandelbrot-shaped things - 'mandelfield' is my name for them. They also contain the 'micro-lakes' - areas where the detail flattens out.)
- Play with the number of iterations - very low numbers will reveal the large structures more, and hide the mandelfields.
Title: Re: Calcyman's Idea (2D)
Post by: lkmitch on February 12, 2010, 08:34:03 AM
Here's another view of the same bifurcation diagram--I find it easier to see as black on white.
Title: Re: Calcyman's Idea (2D)
Post by: bib on March 16, 2010, 08:57:01 PM
Hi
I post here as an answer to Fractex, in case it might interest anybody.
Fractex asked if I could provide the UF file for the animations I did. I have to apologize because I lost my parameters. As I was playing with the script (which I don't do often) I messed up with the versions and names of the formulae and lost my work :(. So I did another one quickly. It's not exactly the same and you don't have the animations, but it's something to start again with.
There are 2 layers.
Some suggestions to play and get more familiar with UF (if needed)
- Change the layering mode
- Change and play with the parameters of the the inside/outside coloring of each layer
- change the bailout (NB : to change a parameter value at once for multiple layers, simply multi-select the layers with the shift key before editing the value)
- change the powers (there are two parameters to do z^p1*tan(z)^p2, so for example one could do an animated transition from z˛+c to z*tan(z)+c... ;))
- Pan, zoom, Explore !!!
I probably said it in a previous post, but there are 2 particular things to me in this z*tan(z)+c fractal (by particular I mean very unusual compared to the Mandelbrot set despite the graphical similarities)
1- increasing the bailout has a very original effect, it create new shapes (due to the periodically divergent nature of f(x)=tan(x) I guess)
2- The inside is really full of surprises (these so-called Mandelfields in particular, I have no idea what they are)
And Calcymansidea formula:
I post here as an answer to Fractex, in case it might interest anybody.
Fractex asked if I could provide the UF file for the animations I did. I have to apologize because I lost my parameters. As I was playing with the script (which I don't do often) I messed up with the versions and names of the formulae and lost my work :(. So I did another one quickly. It's not exactly the same and you don't have the animations, but it's something to start again with.
There are 2 layers.
Some suggestions to play and get more familiar with UF (if needed)
- Change the layering mode
- Change and play with the parameters of the the inside/outside coloring of each layer
- change the bailout (NB : to change a parameter value at once for multiple layers, simply multi-select the layers with the shift key before editing the value)
- change the powers (there are two parameters to do z^p1*tan(z)^p2, so for example one could do an animated transition from z˛+c to z*tan(z)+c... ;))
- Pan, zoom, Explore !!!
I probably said it in a previous post, but there are 2 particular things to me in this z*tan(z)+c fractal (by particular I mean very unusual compared to the Mandelbrot set despite the graphical similarities)
1- increasing the bailout has a very original effect, it create new shapes (due to the periodically divergent nature of f(x)=tan(x) I guess)
2- The inside is really full of surprises (these so-called Mandelfields in particular, I have no idea what they are)
Code:
ztanzForFractex {
fractal:
title="ztanz for fractex" width=1024 height=768 layers=2
credits="arsene;3/16/2010"
layer:
caption="Layer 1" opacity=100 mergemode=screen transparent=yes
mapping:
center=0/0 magn=.1
formula:
maxiter=100 filename="Standard.ufm" entry="Calcymansidea"
p_bailout=400000 f_function=tan p_power=1/0 p_power2=1/0 p_start=0/0
inside:
density=10 transfer=sqrt filename="Standard.ucl"
entry="GenericDirectColoring"
p_coloringClass="Standard.ulb:Standard_Smooth"
p_coloringClass.v_generic=100 p_coloringClass.v_coloring=100
p_coloringClass.v_gradientcoloring=100 p_coloringClass.power=2/0
p_coloringClass.bailout=128.0
outside:
density=2 transfer=cuberoot filename="Standard.ucl"
entry="OrbitTraps" p_trapshape=point p_diameter=1.0 p_traporder=4.0
p_trapfreq=1.0 p_trapcolor=distance p_traptype=closest
p_threshold=0.25 p_trapcenter=0/0 p_aspect=1.0 p_angle=0.0
p_solidcolor=no
gradient:
smooth=yes rotation=-28 index=54 color=16711713 index=213 color=0
index=215 color=16777215 index=222 color=16313190 index=238
color=11075600 index=361 color=13403696
opacity:
smooth=no index=0 opacity=255
layer:
caption="Background" opacity=100
mapping:
center=0/0 magn=.1
formula:
maxiter=100 filename="Standard.ufm" entry="Calcymansidea"
p_bailout=400000 f_function=tan p_power=1/0 p_power2=1/0 p_start=0/0
inside:
transfer=cuberoot filename="Standard.ucl" entry="Decomposition"
outside:
transfer=cuberoot filename="Standard.ucl" entry="Smooth" p_power=2/0
p_bailout=128.0
gradient:
smooth=yes rotation=-7 index=29 color=6211054 index=148 color=0
index=225 color=943060
opacity:
smooth=no index=0 opacity=255
}
And Calcymansidea formula:
Code:
Calcymansidea {
init:
z = @start
loop:
z=z^@power*(@function(z))^@power2 + #pixel
bailout:
|z| <= @bailout
default:
title = "Calcyman's Idea"
center = (-0.5, 0)
$IFDEF VER50
rating = recommended
$ENDIF
param start
caption = "Starting point"
default = (0,0)
hint = "The starting point parameter can be used to distort the Mandelbrot \
set. Use (0, 0) for the standard Mandelbrot set."
endparam
param power
caption = "Power"
default = (1,0)
endparam
param power2
caption = "Power2"
default = (1,0)
hint = "z=(z^powerbizarre*f(z))^power2"
endparam
float param bailout
caption = "Bailout value"
default = 4.0
min = 1.0
$IFDEF VER40
exponential = true
$ENDIF
hint = "Try very large values"
endparam
switch:
type = "Julia"
seed = #pixel
power = power
bailout = bailout
}
Title: Re: Calcyman's Idea (2D)
Post by: kram1032 on March 20, 2010, 08:17:49 PM
I just did an Antibuddhabrot style render of the set. It doesn't really do it justice - I'd love to see a highquality one - but it looks great and has a lot of interesting details :D
If you go directly to dA, you can find a slightly higher resolution version :)
(http://www.deviantart.com/download/157868619/A_nice_tan_by_kram1032.png)
http://kram1032.deviantart.com/art/A-nice-tan-157868619
If you go directly to dA, you can find a slightly higher resolution version :)
(http://www.deviantart.com/download/157868619/A_nice_tan_by_kram1032.png)
http://kram1032.deviantart.com/art/A-nice-tan-157868619
Title: Re: Calcyman's Idea (2D)
Post by: Fractex on March 21, 2010, 01:30:08 AM
Interesting! Did you try the regular Buddhabrot style render of z*tan(z)+c?
Title: Re: Calcyman's Idea (2D)
Post by: stardust4ever on March 21, 2010, 06:33:43 AM
I'd post it but I used an evaluation version of UltraFractal and the whole AVI is stamped with "Evaluation Copy." I guess it's time to shell out the hundred dollars. :)
Those videos are really great, bib.
Those videos are really great, bib.
Just a tip: While UltraFractal is generally more user-friendly and what not and has a better GUI and stuff, ChaosPro can load UF formulas, animate them, and it's free. I just use it when I have a good fractal form UF and want to render it. :-P
Can't wait to see vid!
Title: Re: Calcyman's Idea (2D)
Post by: kram1032 on March 21, 2010, 03:42:22 PM
There you have your escaping orbits:
(http://www.deviantart.com/download/157962594/A_Nice_Tan_II_by_kram1032.png)
http://kram1032.deviantart.com/art/A-Nice-Tan-II-157962594
and a try on combining the two:
(http://www.deviantart.com/download/157962821/A_Nice_Tan_III_by_kram1032.png)
http://kram1032.deviantart.com/art/A-Nice-Tan-III-157962821
in the combination, blue is rather from the escaping point, green from the non-escaping ones.
I personally think, the non-escaping points are more interesting in case of tan. :)
Note, that the two have different bailout values.
The radius is 2*pi for both of them but as the results wheren't clear enough on the escaping one, I set the bailout to 1000 while it was 200 for the inner one.
My guess is, that you'd need WAY higher bailouts for clear images ^^
(http://www.deviantart.com/download/157962594/A_Nice_Tan_II_by_kram1032.png)
http://kram1032.deviantart.com/art/A-Nice-Tan-II-157962594
and a try on combining the two:
(http://www.deviantart.com/download/157962821/A_Nice_Tan_III_by_kram1032.png)
http://kram1032.deviantart.com/art/A-Nice-Tan-III-157962821
in the combination, blue is rather from the escaping point, green from the non-escaping ones.
I personally think, the non-escaping points are more interesting in case of tan. :)
Note, that the two have different bailout values.
The radius is 2*pi for both of them but as the results wheren't clear enough on the escaping one, I set the bailout to 1000 while it was 200 for the inner one.
My guess is, that you'd need WAY higher bailouts for clear images ^^
Title: Re: Calcyman's Idea (2D)
Post by: Timeroot on March 21, 2010, 06:25:09 PM
Woah! The Buddha has a bunch of other monks with him!
In the third picture, I like the two little globs towards the bottom that are so twisted... and now I see that the corresponding monks have two "Jewels" on their heads, instead of three! They look like ears! I think that, undoubtedly, one of the coolest features of the tan function is how it can behave locally like x^2, x^3, x^4, etc, which results in the various multibrots. I would be curious to see what happens for atanh, instead of tan, because they have somewhat similar shapes and expansions. If that doesn't look interesting, then maybe atanh(pi*x/2). O0
In the third picture, I like the two little globs towards the bottom that are so twisted... and now I see that the corresponding monks have two "Jewels" on their heads, instead of three! They look like ears! I think that, undoubtedly, one of the coolest features of the tan function is how it can behave locally like x^2, x^3, x^4, etc, which results in the various multibrots. I would be curious to see what happens for atanh, instead of tan, because they have somewhat similar shapes and expansions. If that doesn't look interesting, then maybe atanh(pi*x/2). O0
Title: Re: Calcyman's Idea (2D)
Post by: kram1032 on March 21, 2010, 08:35:00 PM
I can try but I don't really think, it will be that interesting^^
(Interesting, yes. THAT interesting, no...)
Also worth a try might be 1/tan (cot) :)
(Interesting, yes. THAT interesting, no...)
Also worth a try might be 1/tan (cot) :)
Title: Re: Calcyman's Idea (2D)
Post by: Calcyman on March 22, 2010, 04:57:12 PM
Whoa! Impressive nebula-esque renderings!
As for the co-tangent idea, the set will have similar features to the z*tan(z) set, since, after all, cot(z) = tan((pi/2)-z).
The main difference is that tan(z) resembles z for small values of z, whereas cot(z) resembles 1/z. This means that the linear term will actually cancel out the dominant term in cot(z), resulting in small terms being mapped towards 1.
As for the co-tangent idea, the set will have similar features to the z*tan(z) set, since, after all, cot(z) = tan((pi/2)-z).
The main difference is that tan(z) resembles z for small values of z, whereas cot(z) resembles 1/z. This means that the linear term will actually cancel out the dominant term in cot(z), resulting in small terms being mapped towards 1.
Title: Re: Calcyman's Idea (2D)
Post by: kram1032 on March 22, 2010, 06:49:16 PM
Well, I see an important difference:
While tan(0) is 0, so tan(tan(0)+0)+ .... is also 0, cot 0 might be totally different. cot 0 = infinity...
While tan(0) is 0, so tan(tan(0)+0)+ .... is also 0, cot 0 might be totally different. cot 0 = infinity...
Title: Re: Calcyman's Idea (2D)
Post by: bib on March 22, 2010, 08:16:28 PM
z*cotan(z)+c gives interesting results (if zstart<>0) with a lot of similarities and also a lot of differences compared to z*tan(z)+c
Title: Re: Calcyman's Idea (2D)
Post by: kram1032 on March 22, 2010, 08:57:42 PM
tried already?
I noticed that z=0 seemingly doesn't converge or diverge at all.
All the points seem to get undefined or so. The screen stays black :)
(Thinking about it, they probably get undefined.)
So I'll use c0=z0 as starting condition... :)
EDIT: Or actually, I'll first try random conditions between +/-2*pi - probably not the most interesting way to do it but the shape 'till now looks interesting anyway. Just not overly fractal ^^
I noticed that z=0 seemingly doesn't converge or diverge at all.
All the points seem to get undefined or so. The screen stays black :)
(Thinking about it, they probably get undefined.)
So I'll use c0=z0 as starting condition... :)
EDIT: Or actually, I'll first try random conditions between +/-2*pi - probably not the most interesting way to do it but the shape 'till now looks interesting anyway. Just not overly fractal ^^
Title: Re: Calcyman's Idea (2D)
Post by: kram1032 on March 23, 2010, 12:24:19 AM
Silly me:
I forgot the multiplication part and just did cot(z), rather than cot(z)*z
cot(z)*z actually also gives results for x=0 and y=0 :)
It's 1 in the limit.
So, the "correct" starting conditions could either be or, skipping the first step
[1,0]
it seems to be nice but it's late now and I'll have to start the render over tomorrow....
It looks promissing, though.
Bib: Would you bother reendering propper high-iteration and high-bail-radius renders of tan and cot?
In case of such a complicated function, it'd probably be neccesary to skip bailout alltogeher.. Too bad, that this makes things so extremely slow :(
I forgot the multiplication part and just did cot(z), rather than cot(z)*z
cot(z)*z actually also gives results for x=0 and y=0 :)
It's 1 in the limit.
So, the "correct" starting conditions could either be or, skipping the first step
[1,0]
it seems to be nice but it's late now and I'll have to start the render over tomorrow....
It looks promissing, though.
Bib: Would you bother reendering propper high-iteration and high-bail-radius renders of tan and cot?
In case of such a complicated function, it'd probably be neccesary to skip bailout alltogeher.. Too bad, that this makes things so extremely slow :(
Title: Dragon Galaxy
Post by: bib on April 17, 2010, 08:36:31 PM
A zoom into an outer region of z=z*tan(z)+C
(http://www.fractalforums.com/gallery/2/492_17_04_10_8_28_55.jpeg)
http://www.fractalforums.com/index.php?action=gallery;sa=view;id=2102
I will try to produce some more images soon, but to answer kram's request for a global kind of "template" view of these fractals, calculation is extremely long with high iteration count AND high bailout, and the result is not very beautiful, lots of chaotic places (/artifacts?)
(http://www.fractalforums.com/gallery/2/492_17_04_10_8_28_55.jpeg)
http://www.fractalforums.com/index.php?action=gallery;sa=view;id=2102
I will try to produce some more images soon, but to answer kram's request for a global kind of "template" view of these fractals, calculation is extremely long with high iteration count AND high bailout, and the result is not very beautiful, lots of chaotic places (/artifacts?)
Title: Re: Calcyman's Idea (2D)
Post by: kram1032 on April 18, 2010, 10:51:26 AM
yup...
Due to the way, tan works, mapping + and - infinity to every pi/2 steps periodically, to REALLY know the shape of this, you'd have to do a no-bailout render with insanely high iterations...
Beautiful render :D
I looked at cot and it looked nice but rather boring, at least to me.
Maybe, an actual Mbrot render of this would work better...
Due to the way, tan works, mapping + and - infinity to every pi/2 steps periodically, to REALLY know the shape of this, you'd have to do a no-bailout render with insanely high iterations...
Beautiful render :D
I looked at cot and it looked nice but rather boring, at least to me.
Maybe, an actual Mbrot render of this would work better...
Title: Re: Calcyman's Idea (2D)
Post by: kram1032 on April 28, 2010, 08:07:37 PM
I tried something different now. While it's not as beautiful as ztanz, it's not too far away of that in my opinion:
First, I tried to do a logistic growth Mset. Not too interesting. The graph already made that to be expected.
Then I found out, that certain logistic growths simplify to 1+tanh(z) - still not interesting, but the connection to tanh(z) made me even more curious about the results. - It now seems like tanh(z) is in a way the base of any logistic growth.
Then I thought: Hmmm: this thing actually looks like an integral. It's how many individuals exist per timestep.
So, my next idea was, to take the differential of tanh(z) which is sech²(z). - It's basically how many individuals become added in that certain time-step.
So that's what I'm rendering right now. - The anti-buddhabrot version for now.
It looks very nice and promising. Still quite noisy though. 1/sin(iz)² - I just renormed it a little bit so that the integral over the total set gives 1 but that shouldn't change too much. In theory, that curve now could be used as a probabillity function, at least over the reals :)
So, the actual formula I'm using is
with a being any positive real - they will always produce 1 as an integral. The specific value of a, I chose, was 
, as that made it more symmetric... (2/sqrt(2) = sqrt(2))
Probably, you could directly use sech²(z) for nice results, too. The image shouldn't vary too much...
+cos(2 a y))^2<br />c=\sin(a y) \sinh(a x)<br />d=\cos(a y) \cosh(a x)<br /><br />xnew=2 a (d^2-c^2)/b+x0<br />ynew=-{4 a c d\over b}+y0)
Oh, btw, I also tried the exiting orbit version. It looked nice as well but cleared up very slowly...
There surely are good reasons for using the gaussean normal distribution but the sech(x)² shares the property of the simple integral and on the imaginaryaxis it's pretty much like a cyclic version of the normal distribution... Looking at the general normal distrubution with sigma and µ, I think it would be very trivial to add that to the sech distribution aswell...
EDIT: Ok, found. It's actually called Logistic distribution (d'uh)
http://en.wikipedia.org/wiki/Logistic_distribution
EDIT:
Argh! As usual: Errors make beauty. I missed an h, turning sinh to sin. Now I have to check, wether it still looks as nice, without the error...
EDIT:
Lucky: It changed shape quite a bit, but it still looks interesting :) (Hyper short rendertime 'till now - just a few secs... But it already looks reasonable) - certain features still are more or less unaltered, actually...
If you want to look at the error, just use the above mentioned formulae but change the line
 \sinh(a x))
to
 \sin(a x))
First, I tried to do a logistic growth Mset. Not too interesting. The graph already made that to be expected.
Then I found out, that certain logistic growths simplify to 1+tanh(z) - still not interesting, but the connection to tanh(z) made me even more curious about the results. - It now seems like tanh(z) is in a way the base of any logistic growth.
Then I thought: Hmmm: this thing actually looks like an integral. It's how many individuals exist per timestep.
So, my next idea was, to take the differential of tanh(z) which is sech²(z). - It's basically how many individuals become added in that certain time-step.
So that's what I'm rendering right now. - The anti-buddhabrot version for now.
It looks very nice and promising. Still quite noisy though. 1/sin(iz)² - I just renormed it a little bit so that the integral over the total set gives 1 but that shouldn't change too much. In theory, that curve now could be used as a probabillity function, at least over the reals :)
So, the actual formula I'm using is
Probably, you could directly use sech²(z) for nice results, too. The image shouldn't vary too much...
Oh, btw, I also tried the exiting orbit version. It looked nice as well but cleared up very slowly...
There surely are good reasons for using the gaussean normal distribution but the sech(x)² shares the property of the simple integral and on the imaginaryaxis it's pretty much like a cyclic version of the normal distribution... Looking at the general normal distrubution with sigma and µ, I think it would be very trivial to add that to the sech distribution aswell...
EDIT: Ok, found. It's actually called Logistic distribution (d'uh)
http://en.wikipedia.org/wiki/Logistic_distribution
EDIT:
Argh! As usual: Errors make beauty. I missed an h, turning sinh to sin. Now I have to check, wether it still looks as nice, without the error...
EDIT:
Lucky: It changed shape quite a bit, but it still looks interesting :) (Hyper short rendertime 'till now - just a few secs... But it already looks reasonable) - certain features still are more or less unaltered, actually...
If you want to look at the error, just use the above mentioned formulae but change the line
to
Title: Re: Calcyman's Idea (2D)
Post by: kram1032 on April 29, 2010, 01:03:48 AM
Ok, here you go. It's still very noisy but you get the idea...
(http://www.deviantart.com/download/162296856/Sech_Mask_by_kram1032.png)
Here, I used a bailout value of 200 and a bailout radius of sqrt(20) (so x²+y²=20)
However for this it's probably similar to ztanz: It would need a bailout-free render with insanely high iterations to truely know how this one looks.
You can clearly see a lot of places where details only begin to show up. This would seriously need a way longer render.
If the image isn't there, try it here:
http://kram1032.deviantart.com/art/Sech-Mask-162296856
There you can also click the download button to see a higher resolution but due to all the noise, it's not quite worth it.
(http://www.deviantart.com/download/162296856/Sech_Mask_by_kram1032.png)
Here, I used a bailout value of 200 and a bailout radius of sqrt(20) (so x²+y²=20)
However for this it's probably similar to ztanz: It would need a bailout-free render with insanely high iterations to truely know how this one looks.
You can clearly see a lot of places where details only begin to show up. This would seriously need a way longer render.
If the image isn't there, try it here:
http://kram1032.deviantart.com/art/Sech-Mask-162296856
There you can also click the download button to see a higher resolution but due to all the noise, it's not quite worth it.
Title: Re: Calcyman's Idea (2D)
Post by: kram1032 on May 01, 2010, 12:40:02 PM
Ok, I did an overnight render with roughly 18h in total...
(http://www.deviantart.com/download/162544028/Mask_of_Sechsqua_by_kram1032.png)
http://kram1032.deviantart.com/art/Mask-of-Sechsqua-162544028
Certain details still aren't there, most likely. It litterally takes for ever, especially for the smaller details, to show up. for instance, inside all the holes of the bright area, structures emerged but they converged so slowly, I didn't get the chance to ever see, how they actually looked...
(http://www.deviantart.com/download/162544028/Mask_of_Sechsqua_by_kram1032.png)
http://kram1032.deviantart.com/art/Mask-of-Sechsqua-162544028
Certain details still aren't there, most likely. It litterally takes for ever, especially for the smaller details, to show up. for instance, inside all the holes of the bright area, structures emerged but they converged so slowly, I didn't get the chance to ever see, how they actually looked...
Title: Re: Calcyman's Idea (2D)
Post by: stigomaster on May 01, 2010, 01:05:02 PM
I was going to complain about your inappropriate use of the term “Literally”, but then I thought that because fractals have detail on an arbitrarily small level, the smallest details do literally take forever to show up. ;D
How does the orbit density map of the escaped orbits look?
How does the orbit density map of the escaped orbits look?
Title: Re: Calcyman's Idea (2D)
Post by: kram1032 on May 01, 2010, 05:22:17 PM
To show you that, I'll have to render for ever yet again xD
I did a short test on that and what I saw by that time was nice :)
I'd love to see an Mset of it, too, like the last two ztanz renders :)
Yup, I actually DID mean it litterally :P
Oh, I forgot to mention: My renders where with bailout at xxyy=20 and iteration=2000
To fasten things up, I now do 200 iterations
I did a short test on that and what I saw by that time was nice :)
I'd love to see an Mset of it, too, like the last two ztanz renders :)
Yup, I actually DID mean it litterally :P
Oh, I forgot to mention: My renders where with bailout at xxyy=20 and iteration=2000
To fasten things up, I now do 200 iterations
Title: Re: Calcyman's Idea (2D)
Post by: kram1032 on May 02, 2010, 02:16:09 AM
Ok, here you go:
(http://www.deviantart.com/download/162620564/Firey_Kapoera_by_kram1032.png)
http://kram1032.deviantart.com/art/Firey-Kapoera-162620564
While it's still a bit noisy, I'm very happy with how it looks :)
I guess, higher bailout values would have made it more clear. for such complex-behaving functions which cycle between + and - infinity in a rather high frequency, you most likely need WAY more iterations to get "close" results and definitely no bailout radius, as even a point close to infinity could suddenly jump back to near zero...
(http://www.deviantart.com/download/162620564/Firey_Kapoera_by_kram1032.png)
http://kram1032.deviantart.com/art/Firey-Kapoera-162620564
While it's still a bit noisy, I'm very happy with how it looks :)
I guess, higher bailout values would have made it more clear. for such complex-behaving functions which cycle between + and - infinity in a rather high frequency, you most likely need WAY more iterations to get "close" results and definitely no bailout radius, as even a point close to infinity could suddenly jump back to near zero...
Title: Re: Calcyman's Idea (2D)
Post by: stigomaster on May 02, 2010, 12:24:30 PM
Really a dramatic gradient, I would not like to encounter that in a dark alley! I mean, it has eyes, right? It's not just me? :-\
Title: Re: Calcyman's Idea (2D)
Post by: kram1032 on May 02, 2010, 12:45:13 PM
Well... Way before you encounter it, you'll see its glow, so it should be rather easy to avoid. - Except it only glows while attacking^^
There are many ways to interpret those images. I actually thought about it was two humanoids, doing breakdance or kapoera or something.
While... on fire....
Btw, you said dramatic gradient. Probably it is... The noise level was already less prominent but I pushed the contrast a bit to see fainter details better. And as a down-side, noise too^^
I actually wonder, what could be done with the image data, if you interpret the orbit map as an HDR-image... It should be possible to get way more detail out of it, without cutting off the upper end. I probably slightly did so here in the very bright areas but for the details, visible at this resolution, the losses should be rather minimal.
(Of course the smaller details, which need forever to show up :P should be cut off that way^^)
If you look at it on dA and download it, you can see it at a resolution of 1680*1018, btw...
There are many ways to interpret those images. I actually thought about it was two humanoids, doing breakdance or kapoera or something.
While... on fire....
Btw, you said dramatic gradient. Probably it is... The noise level was already less prominent but I pushed the contrast a bit to see fainter details better. And as a down-side, noise too^^
I actually wonder, what could be done with the image data, if you interpret the orbit map as an HDR-image... It should be possible to get way more detail out of it, without cutting off the upper end. I probably slightly did so here in the very bright areas but for the details, visible at this resolution, the losses should be rather minimal.
(Of course the smaller details, which need forever to show up :P should be cut off that way^^)
If you look at it on dA and download it, you can see it at a resolution of 1680*1018, btw...