Title: A new look into escapetime fractals using abs & inner coloring
Post by: Kali on May 16, 2011, 07:36:14 AM
First I must say that this thread derives from this one (http://www.fractalforums.com/new-theories-and-research/very-simple-formula-for-fractal-patterns/), in which I presented a simple formula that produces what Sam Monnier's, creator of "Ducks", calls "space-filling patterns", or sometimes "Ducky-Thalis" patterns (I prefer "kaleidoscopic patterns"). That was after reading what Sam wrote on another thread (http://www.fractalforums.com/general-discussion-b77/fractal-expressionism/) about how this patterns are created, and as I like a lot the patterns from Ducks formula, I wanted to try another ways for getting this kind of shapes. I saw there was a relationship with Mandelbox so I wanted to try with abs function and the mandelbox fold. But first I tried the most simpler version of that... just taking the absolute real values of x and y, then dividing by the squared modulus. No 'if' conditions, just abs+inversion, and it turned to be the simplest method for this kaleidoscopic patterns yet, that also has a 3D version.
The complex number analog of my 2D formula turned to be z=abs(z)^-1+c - Syntopia explains well the analogy in my previous thread, and called it a "power -1 burning ship". So I decided to try with other powers, and I discovered that even the well-known "burning ship" fractal, has this patterns on it's insides... and it seems that nobody have noticed that before... why? I think the coloring method is the answer... as the patterns doesn't diverge, an inner coloring must be used... and the patterns doesn't show up or become evident with any coloring.
I found that exponential smoothing (http://formulas.ultrafractal.com/reference/Standard/Standard_ExponentialSmoothing.html) works perfectly on this. I know of this coloring technique as it's included in the standard set of formulas on Ultrafractal, but I'm using my own version... it's similar but it computes the magnitude only of Z, and it's pretty simple:
At each iteration, do this: (|z| means the squared module of z in UF)
Then the color index is picked depending on the value of 'sum'.
I really like how this coloring works. It gives good effects with low iterations and with certains patterns (glows, blur, sometimes like a DOF effect), and with the correct coloring settings, produces a cool 3D feel on the images. The color density and number of iterations must be fine-tuned for this.
Well, the fact is that after this encounter with the burning ship, I begin to explore another fractals using this method of adding an abs function and coloring with exp.smoothing, and I will be posting some of my findings in this thread. I will use always the default gradient from UF5 and no postprocessing, so don't expect too much artistic work here because the idea is just to show the great variety of patterns and structures I found. (Even when there are some similarities that most of this formulas shares, each one have unique kind of shapes and patterns somewhere)
I also invite you to post here your own findings and artworks using this formulas ;)
One thing I forgot to mention, is that in Mandelbrot mode you get the variety of patterns but most are streched/distorted and very chaotic... you can find cool images, though, but the nicer ones are Julias, and you can use the Mandelbrot as a map for a preview of what kind of patterns you will get with the Julia values corresponding at that neighborhood of points.
The regular burning ship fractal will be the first series of images, with three variations... abs(z)^2+c, abs(z^2)+c and abs(z^2+c).
Enough talking, this are the first images:
(http://img69.imageshack.us/img69/9853/burningship2.jpg)
(http://img135.imageshack.us/img135/6978/burningship3.jpg)
(http://img812.imageshack.us/img812/5638/burningship4.jpg)
(http://img508.imageshack.us/img508/5231/burningship6.jpg)
(http://img651.imageshack.us/img651/6540/miniships.jpg)
The last is in Mandelbrot mode, and it shows a "miniship" inside the patterns area. I raised iterations and coloring density to show that inside the miniship there is a mini-set of patterns too. I have looked at this closely; patterns are not the same of the main fractal, and there's also "mini-mini-ships" inside... this goes on and on as more iterations are made.
I'll post more images later...
The complex number analog of my 2D formula turned to be z=abs(z)^-1+c - Syntopia explains well the analogy in my previous thread, and called it a "power -1 burning ship". So I decided to try with other powers, and I discovered that even the well-known "burning ship" fractal, has this patterns on it's insides... and it seems that nobody have noticed that before... why? I think the coloring method is the answer... as the patterns doesn't diverge, an inner coloring must be used... and the patterns doesn't show up or become evident with any coloring.
I found that exponential smoothing (http://formulas.ultrafractal.com/reference/Standard/Standard_ExponentialSmoothing.html) works perfectly on this. I know of this coloring technique as it's included in the standard set of formulas on Ultrafractal, but I'm using my own version... it's similar but it computes the magnitude only of Z, and it's pretty simple:
At each iteration, do this: (|z| means the squared module of z in UF)
Code:
z=abs(z)^2+c (or whatever formula you want to use)
mold=m
m=sqrt(|z|)
sum=sum+exp(-1/abs(mold-m))
Then the color index is picked depending on the value of 'sum'.
I really like how this coloring works. It gives good effects with low iterations and with certains patterns (glows, blur, sometimes like a DOF effect), and with the correct coloring settings, produces a cool 3D feel on the images. The color density and number of iterations must be fine-tuned for this.
Well, the fact is that after this encounter with the burning ship, I begin to explore another fractals using this method of adding an abs function and coloring with exp.smoothing, and I will be posting some of my findings in this thread. I will use always the default gradient from UF5 and no postprocessing, so don't expect too much artistic work here because the idea is just to show the great variety of patterns and structures I found. (Even when there are some similarities that most of this formulas shares, each one have unique kind of shapes and patterns somewhere)
I also invite you to post here your own findings and artworks using this formulas ;)
One thing I forgot to mention, is that in Mandelbrot mode you get the variety of patterns but most are streched/distorted and very chaotic... you can find cool images, though, but the nicer ones are Julias, and you can use the Mandelbrot as a map for a preview of what kind of patterns you will get with the Julia values corresponding at that neighborhood of points.
The regular burning ship fractal will be the first series of images, with three variations... abs(z)^2+c, abs(z^2)+c and abs(z^2+c).
Enough talking, this are the first images:
(http://img69.imageshack.us/img69/9853/burningship2.jpg)
(http://img135.imageshack.us/img135/6978/burningship3.jpg)
(http://img812.imageshack.us/img812/5638/burningship4.jpg)
(http://img508.imageshack.us/img508/5231/burningship6.jpg)
(http://img651.imageshack.us/img651/6540/miniships.jpg)
The last is in Mandelbrot mode, and it shows a "miniship" inside the patterns area. I raised iterations and coloring density to show that inside the miniship there is a mini-set of patterns too. I have looked at this closely; patterns are not the same of the main fractal, and there's also "mini-mini-ships" inside... this goes on and on as more iterations are made.
I'll post more images later...
Title: Re: A new look into escapetime fractals using abs & inner coloring
Post by: Kali on May 16, 2011, 02:16:24 PM
More images of formula z=abs(z^2+c). All are Julias except one, try to guess which is :)
(http://img194.imageshack.us/img194/3083/burningship5.jpg)
(http://img405.imageshack.us/img405/9519/burningship7.jpg)
(http://img607.imageshack.us/img607/183/burningship8.jpg)
Next post: Negative powers
(http://img194.imageshack.us/img194/3083/burningship5.jpg)
(http://img405.imageshack.us/img405/9519/burningship7.jpg)
(http://img607.imageshack.us/img607/183/burningship8.jpg)
Next post: Negative powers
Title: Re: A new look into escapetime fractals using abs & inner coloring
Post by: Kali on May 17, 2011, 02:58:01 AM
Instead of moving to higher powers, let's go to the negative ones... the patterns area is bigger and more interesting results.
I start with z=abs(z^-1+c), which is related to the real number "kaliset" formula:
(http://img829.imageshack.us/img829/207/power1w.jpg)
Next is power -2
(http://img638.imageshack.us/img638/2290/power2f.jpg)
Power -3
(http://img818.imageshack.us/img818/6775/power3.jpg)
Power -4
(http://img560.imageshack.us/img560/3048/power4a.jpg)
(http://img856.imageshack.us/img856/8550/power4b.jpg)
Power -8
(http://img339.imageshack.us/img339/7467/power8.jpg)
Next post: Playing with fractional powers
I start with z=abs(z^-1+c), which is related to the real number "kaliset" formula:
(http://img829.imageshack.us/img829/207/power1w.jpg)
Next is power -2
(http://img638.imageshack.us/img638/2290/power2f.jpg)
Power -3
(http://img818.imageshack.us/img818/6775/power3.jpg)
Power -4
(http://img560.imageshack.us/img560/3048/power4a.jpg)
(http://img856.imageshack.us/img856/8550/power4b.jpg)
Power -8
(http://img339.imageshack.us/img339/7467/power8.jpg)
Next post: Playing with fractional powers
Title: Re: A new look into escapetime fractals using abs & inner coloring
Post by: Kali on May 19, 2011, 03:13:59 AM
Just choosing random fractional powers gives more variety of similar patterns, and sometimes not so similar... still something to be explored more deeply.
But in the following image I show how the patterns can be altered and "fine-tuned" by changing power slightly, using small fractional steps.
The base formula is z=abs(z)^2+c with Julia values -0.635-0.635i
(http://img534.imageshack.us/img534/8839/fract1.jpg)
This is another example, with Julia values -0.25+0.35i
(http://img535.imageshack.us/img535/9241/fracb1.jpg)
The variations can be similar but are not the same of making small adjustments to the Julia values.
In the next post: Complex powers
But in the following image I show how the patterns can be altered and "fine-tuned" by changing power slightly, using small fractional steps.
The base formula is z=abs(z)^2+c with Julia values -0.635-0.635i
(http://img534.imageshack.us/img534/8839/fract1.jpg)
This is another example, with Julia values -0.25+0.35i
(http://img535.imageshack.us/img535/9241/fracb1.jpg)
The variations can be similar but are not the same of making small adjustments to the Julia values.
In the next post: Complex powers
Title: Re: A new look into escapetime fractals using abs & inner coloring
Post by: bib on May 19, 2011, 08:02:04 AM
(http://img535.imageshack.us/img535/9241/fracb1.jpg)
:o ;)
(http://nocache-nocookies.digitalgott.com/gallery/7/492_06_05_11_9_13_16.jpeg)
Title: Re: A new look into escapetime fractals using abs & inner coloring
Post by: KRAFTWERK on May 19, 2011, 11:55:15 AM
Wow, great images Kali, and cool 3D render bib! :beer: :beer:
Title: Re: A new look into escapetime fractals using abs & inner coloring
Post by: Kali on May 19, 2011, 04:18:00 PM
Thanks Johan! and yes, a lovely 3D version of the "entangled trees" you made, bib... if I remember well, you did it using Darkbeam's custom transforms, and it's pretty much the formula I'm using in 2D.
Speaking of Darkbeam, I remember when our friend Luca wrote that "no fractal art can be produced with complex powers", or something like that (I'm too lazy to search for the thread). When I read that, I inmediatly posted some images from my collection, dedicated to him :)
So Luca, you are truly one of the greatest and prolific minds around here, but everybody makes mistakes... the following images are also for you :D
The formulas here are z=abs(z^complexpow+c) (all julias, except one)
The first example is power 2i:
(http://img715.imageshack.us/img715/2026/complex02.jpg)
power -2i:
(http://img6.imageshack.us/img6/8843/complex02m.jpg)
power -2i (mandelbrot):
(http://img812.imageshack.us/img812/8843/complex02m.jpg)
power 2i:
(http://img847.imageshack.us/img847/8382/complex2i.jpg)
power -i:
(http://img822.imageshack.us/img822/4213/complex01.jpg)
power 1+2i:
(http://img853.imageshack.us/img853/7202/complex12.jpg)
power -5i:
(http://img30.imageshack.us/img30/6043/complex518519186.jpg)
And the following are using weird complex-fractional powers, that I searched in Ultrafractal in the same way you search for Julia values:
(The lower cutting line from the left to the center, is an effect of complex powers)
(http://img685.imageshack.us/img685/6959/complexfract1.jpg)
(http://img849.imageshack.us/img849/8310/complexraro.jpg)
(http://img838.imageshack.us/img838/5397/complexfract2.jpg)
So I had this idea of using C as complex power also... in this way: z=abs(z^c+c) and in this another one: z=abs(z^(1/c)+c)
Both worked good, and with the second I found this cute snowmans :D
(http://img716.imageshack.us/img716/8894/snowfg.jpg)
Well, all of this are just a few images I made, there are plenty of different shapes and patterns that can be found, so now with fractional and complex powers I can truly say that "the possibilites are infinite" for this kind of fractals too ;)
But there are even more variants I will show in further posts,
In the next: Nova and Phoenix fractals
Speaking of Darkbeam, I remember when our friend Luca wrote that "no fractal art can be produced with complex powers", or something like that (I'm too lazy to search for the thread). When I read that, I inmediatly posted some images from my collection, dedicated to him :)
So Luca, you are truly one of the greatest and prolific minds around here, but everybody makes mistakes... the following images are also for you :D
The formulas here are z=abs(z^complexpow+c) (all julias, except one)
The first example is power 2i:
(http://img715.imageshack.us/img715/2026/complex02.jpg)
power -2i:
(http://img6.imageshack.us/img6/8843/complex02m.jpg)
power -2i (mandelbrot):
(http://img812.imageshack.us/img812/8843/complex02m.jpg)
power 2i:
(http://img847.imageshack.us/img847/8382/complex2i.jpg)
power -i:
(http://img822.imageshack.us/img822/4213/complex01.jpg)
power 1+2i:
(http://img853.imageshack.us/img853/7202/complex12.jpg)
power -5i:
(http://img30.imageshack.us/img30/6043/complex518519186.jpg)
And the following are using weird complex-fractional powers, that I searched in Ultrafractal in the same way you search for Julia values:
(The lower cutting line from the left to the center, is an effect of complex powers)
(http://img685.imageshack.us/img685/6959/complexfract1.jpg)
(http://img849.imageshack.us/img849/8310/complexraro.jpg)
(http://img838.imageshack.us/img838/5397/complexfract2.jpg)
So I had this idea of using C as complex power also... in this way: z=abs(z^c+c) and in this another one: z=abs(z^(1/c)+c)
Both worked good, and with the second I found this cute snowmans :D
(http://img716.imageshack.us/img716/8894/snowfg.jpg)
Well, all of this are just a few images I made, there are plenty of different shapes and patterns that can be found, so now with fractional and complex powers I can truly say that "the possibilites are infinite" for this kind of fractals too ;)
But there are even more variants I will show in further posts,
In the next: Nova and Phoenix fractals
Title: Re: A new look into escapetime fractals using abs & inner coloring
Post by: Kali on June 05, 2011, 02:39:13 PM
Before Nova & Phoenix, abs(sqrt(z)+c):
(http://img3.imageshack.us/img3/3192/sqrt.jpg)
(http://img3.imageshack.us/img3/3192/sqrt.jpg)
Title: Re: A new look into escapetime fractals using abs & inner coloring
Post by: Kali on June 05, 2011, 04:36:51 PM
This is based on Nova formula:
z = abs(z - (z^pow-1) / (pow * z^(pow-1)) + c)
Some Julias:
pow=2
(http://img233.imageshack.us/img233/33/newton2.jpg)
pow=3
(http://img827.imageshack.us/img827/4242/newton3.jpg)
z = abs(z - (z^pow-1) / (pow * z^(pow-1)) + c)
Some Julias:
pow=2
(http://img233.imageshack.us/img233/33/newton2.jpg)
pow=3
(http://img827.imageshack.us/img827/4242/newton3.jpg)
Title: Re: A new look into escapetime fractals using abs & inner coloring
Post by: bkercso on October 15, 2012, 10:59:37 PM
Hi!
I wrote a fractal generator program just for fun, and when I found this topic I tried Kali's
formulas. I experienced that
(if m>1E-10 then)
sum:=sum+abs(mold-m)/m
formula gives more contoured internal patters, and requires less calculation time.
I played with burning ship and burning bird fractals, classic Mandelbrot didn't show any patters inside.
You can see there are places where more patters meet each other.
Here are some images. (Yes, there are better color palettes... Geometry matters. :) )
Burning ship:
(http://nocache-nocookies.digitalgott.com/gallery/18/4917_25_07_15_1_13_24.png)
(http://nocache-nocookies.digitalgott.com/gallery/18/4917_26_07_15_7_00_33.png)
04_burning_ship_0004b
(https://lh3.googleusercontent.com/xxtigyuEKthHjbUs1tgxb4dE_mdDdM3-FXfAiI-Hrks=w920-h614-no)
05_burning_ship_0004b_1
(http://nocache-nocookies.digitalgott.com/gallery/18/4917_17_08_15_11_42_21.png)
06_burning_ship_0002
(https://lh3.googleusercontent.com/I96S8a0OQVp_DxHyz89lRyZKJaGXwR3m1zNwS9rua6s=w1200-h800-no)
06a_burning_ship_0031
(https://lh3.googleusercontent.com/yflHNznEUMo8-ttOqICrMyfBYTB6Vyg1RY764Amy8qI=w1200-h800-no)
06b_burning_ship_0032
(https://lh3.googleusercontent.com/MKj4Yu6_C9eSSRM-EWp7RkMOfC7XuYP2RPMxiRiYVmI=w1280-h854-no)
07_burning_ship_0003
(https://lh3.googleusercontent.com/0SJnzEZ-ovvlbJpYvkKkN2DA8A0L6i5bMlBX_i8h5cc=w1200-h800-no)
08_burning_ship_0025
(https://lh3.googleusercontent.com/X3IrzbhlmNnyNPfTsnNEHfuZrqHVOdnFRRoHbhCpJvk=w1200-h800-no)
09_burning_ship_0010
(https://lh3.googleusercontent.com/PPnJ-S48grJZ6KotEI5vbOQe9ZRfuIvePU4HDnnzTl8=w1076-h717-no)
10_burning_ship_0024
(https://lh3.googleusercontent.com/-M1M-OoxpoCt-1QEz3GpxcEg_RutVkCB4khJG6AUnxs=w1280-h855-no)
11_burning_ship_0019
(https://lh3.googleusercontent.com/46qFEd12LVkB_BggkiEaaV6JdZXT1mFbhWczWPhmWpQ=w1076-h717-no)
12_burning_ship_0001
(https://lh3.googleusercontent.com/YY9QuYpGWc8FMhzhRuqSpzOIbALbHvP8NbWIS4MLS-I=w1200-h800-no)
I wrote a fractal generator program just for fun, and when I found this topic I tried Kali's
Quote
z=abs(z)^2+c (or whatever formula you want to use)
mold=m
m=sqrt(|z|)
sum=sum+exp(-1/abs(mold-m))
mold=m
m=sqrt(|z|)
sum=sum+exp(-1/abs(mold-m))
formulas. I experienced that
(if m>1E-10 then)
sum:=sum+abs(mold-m)/m
formula gives more contoured internal patters, and requires less calculation time.
I played with burning ship and burning bird fractals, classic Mandelbrot didn't show any patters inside.
You can see there are places where more patters meet each other.
Here are some images. (Yes, there are better color palettes... Geometry matters. :) )
Burning ship:
(http://nocache-nocookies.digitalgott.com/gallery/18/4917_25_07_15_1_13_24.png)
(http://nocache-nocookies.digitalgott.com/gallery/18/4917_26_07_15_7_00_33.png)
04_burning_ship_0004b
(https://lh3.googleusercontent.com/xxtigyuEKthHjbUs1tgxb4dE_mdDdM3-FXfAiI-Hrks=w920-h614-no)
05_burning_ship_0004b_1
(http://nocache-nocookies.digitalgott.com/gallery/18/4917_17_08_15_11_42_21.png)
06_burning_ship_0002
(https://lh3.googleusercontent.com/I96S8a0OQVp_DxHyz89lRyZKJaGXwR3m1zNwS9rua6s=w1200-h800-no)
06a_burning_ship_0031
(https://lh3.googleusercontent.com/yflHNznEUMo8-ttOqICrMyfBYTB6Vyg1RY764Amy8qI=w1200-h800-no)
06b_burning_ship_0032
(https://lh3.googleusercontent.com/MKj4Yu6_C9eSSRM-EWp7RkMOfC7XuYP2RPMxiRiYVmI=w1280-h854-no)
07_burning_ship_0003
(https://lh3.googleusercontent.com/0SJnzEZ-ovvlbJpYvkKkN2DA8A0L6i5bMlBX_i8h5cc=w1200-h800-no)
08_burning_ship_0025
(https://lh3.googleusercontent.com/X3IrzbhlmNnyNPfTsnNEHfuZrqHVOdnFRRoHbhCpJvk=w1200-h800-no)
09_burning_ship_0010
(https://lh3.googleusercontent.com/PPnJ-S48grJZ6KotEI5vbOQe9ZRfuIvePU4HDnnzTl8=w1076-h717-no)
10_burning_ship_0024
(https://lh3.googleusercontent.com/-M1M-OoxpoCt-1QEz3GpxcEg_RutVkCB4khJG6AUnxs=w1280-h855-no)
11_burning_ship_0019
(https://lh3.googleusercontent.com/46qFEd12LVkB_BggkiEaaV6JdZXT1mFbhWczWPhmWpQ=w1076-h717-no)
12_burning_ship_0001
(https://lh3.googleusercontent.com/YY9QuYpGWc8FMhzhRuqSpzOIbALbHvP8NbWIS4MLS-I=w1200-h800-no)
Title: Re: A new look into escapetime fractals using abs & inner coloring
Post by: bkercso on October 15, 2012, 11:10:41 PM
Burning bird:
13_burning_bird_0004
(https://lh3.googleusercontent.com/B35xxMZwGllsgFyxiUraO90emvOHPI7-59HB1lP-Wg=w1200-h800-no)
14_burning_bird_0005
(https://lh3.googleusercontent.com/hn6WkqdmD_2-diPMrwN-JXVwxkfXC4KqSCtOXaV5eQ=w1200-h800-no)
15_burning_bird_0006
(https://lh3.googleusercontent.com/LJ3JbllAu0CgCAJJBr7gxeBSgyxp16vsePMg-69TCA=w1280-h854-no)
16_burning_bird_0021
(http://kepfeltoltes.hu/121015/16_burning_bird_0021_www.kepfeltoltes.hu_.png)
18_burning_bird_0019
(https://lh3.googleusercontent.com/zoQy9s-ZCisOY94pHhPxh7sgBdxxIphEFOpJGpdexaQ=w920-h614-no)
13_burning_bird_0004
(https://lh3.googleusercontent.com/B35xxMZwGllsgFyxiUraO90emvOHPI7-59HB1lP-Wg=w1200-h800-no)
14_burning_bird_0005
(https://lh3.googleusercontent.com/hn6WkqdmD_2-diPMrwN-JXVwxkfXC4KqSCtOXaV5eQ=w1200-h800-no)
15_burning_bird_0006
(https://lh3.googleusercontent.com/LJ3JbllAu0CgCAJJBr7gxeBSgyxp16vsePMg-69TCA=w1280-h854-no)
16_burning_bird_0021
(http://kepfeltoltes.hu/121015/16_burning_bird_0021_www.kepfeltoltes.hu_.png)
18_burning_bird_0019
(https://lh3.googleusercontent.com/zoQy9s-ZCisOY94pHhPxh7sgBdxxIphEFOpJGpdexaQ=w920-h614-no)
Title: Re: A new look into escapetime fractals using abs & inner coloring
Post by: bkercso on October 16, 2012, 03:40:01 PM
Here is a video to show how the patterns get more complex while do more and more iterations (youtube quality not so good...):
http://youtu.be/haQrkjm3R6g (http://youtu.be/haQrkjm3R6g)
It's hard to coloring because maxiter changes frame to frame in every point. I fixed the palette to midpont pixel of the video
(-1.9853441533503659E+0000 ; -3.7603428234884621E-0007 ; Zoom: 1E16).
Some more pics:
Burning ship:
burning_ship_0043
(https://lh3.googleusercontent.com/7uu92ULtK_5BMfLTJhKqLDl5M2J_GlVkviw_59DXqoo=w640-h427-no)
burning_ship_0043a
(http://kepfeltoltes.hu/121016/burning_ship_0043a_www.kepfeltoltes.hu_.png)
Burning bird:
burning_bird_0015
(https://lh3.googleusercontent.com/KoU-Cadc2anhmRb2VJ4TRb_hs-xy1t_WWiz7-IjeRA=w1280-h854-no)
burning_bird_0018
(https://lh3.googleusercontent.com/VtiytiPfoAoxGN9M0yAbg7jhnfVLPv8i8ll5j6gXVvI=w1280-h854-no)
burning_bird_0022
(https://lh3.googleusercontent.com/Tv20AqAiOkQxPvdLYeipgzJ1G_H_MeDerhQFoeeRHPw=w1200-h800-no)
burning_bird_0025
(https://lh3.googleusercontent.com/UAyqSngwludXWxkShB-u3ZFRqKTohOVyGb__79CsV5k=w640-h427-no)
burning_bird_0031
(https://lh3.googleusercontent.com/G_qTpdGV8_yLNAsFNr1woV-nqjn2_5neXROL51ZSGK4=w1280-h854-no)
The same image with coloring algoith from FractView (Android)
burning bird_0031_-1
(https://lh3.googleusercontent.com/S3ogGOVi4U_i7962Gk1ctwpW31cQ4P8FpfWBvAlaiUw=w1280-h854-no)
...and with an other formula from FractView too
burning bird_0031_-2
(https://lh3.googleusercontent.com/b6vtksBmBR9aWqC4MgOEhRXcjxyOcXWgLzYGfu2P0Vc=w1280-h854-no)
http://youtu.be/haQrkjm3R6g (http://youtu.be/haQrkjm3R6g)
It's hard to coloring because maxiter changes frame to frame in every point. I fixed the palette to midpont pixel of the video
(-1.9853441533503659E+0000 ; -3.7603428234884621E-0007 ; Zoom: 1E16).
Some more pics:
Burning ship:
burning_ship_0043
(https://lh3.googleusercontent.com/7uu92ULtK_5BMfLTJhKqLDl5M2J_GlVkviw_59DXqoo=w640-h427-no)
burning_ship_0043a
(http://kepfeltoltes.hu/121016/burning_ship_0043a_www.kepfeltoltes.hu_.png)
Burning bird:
burning_bird_0015
(https://lh3.googleusercontent.com/KoU-Cadc2anhmRb2VJ4TRb_hs-xy1t_WWiz7-IjeRA=w1280-h854-no)
burning_bird_0018
(https://lh3.googleusercontent.com/VtiytiPfoAoxGN9M0yAbg7jhnfVLPv8i8ll5j6gXVvI=w1280-h854-no)
burning_bird_0022
(https://lh3.googleusercontent.com/Tv20AqAiOkQxPvdLYeipgzJ1G_H_MeDerhQFoeeRHPw=w1200-h800-no)
burning_bird_0025
(https://lh3.googleusercontent.com/UAyqSngwludXWxkShB-u3ZFRqKTohOVyGb__79CsV5k=w640-h427-no)
burning_bird_0031
(https://lh3.googleusercontent.com/G_qTpdGV8_yLNAsFNr1woV-nqjn2_5neXROL51ZSGK4=w1280-h854-no)
The same image with coloring algoith from FractView (Android)
burning bird_0031_-1
(https://lh3.googleusercontent.com/S3ogGOVi4U_i7962Gk1ctwpW31cQ4P8FpfWBvAlaiUw=w1280-h854-no)
...and with an other formula from FractView too
burning bird_0031_-2
(https://lh3.googleusercontent.com/b6vtksBmBR9aWqC4MgOEhRXcjxyOcXWgLzYGfu2P0Vc=w1280-h854-no)
Title: Re: A new look into escapetime fractals using abs & inner coloring
Post by: kram1032 on October 17, 2012, 09:03:43 AM
Those are amazing! I really like them all.
Title: Re: A new look into escapetime fractals using abs & inner coloring
Post by: Kali on October 18, 2012, 03:12:03 AM
Nice renders bkercso, what did you use to write your program? I started to write one using Visual Basic more than a year ago, but never finished it. I'm very entertained with GPU programming and 3D stuff, but I think I will be back to finish it at some point. Or maybe now that I know more C language I'll rewrite it.
Title: Re: A new look into escapetime fractals using abs & inner coloring
Post by: bkercso on October 18, 2012, 12:12:32 PM
Thank you Kali and kram1032! I use Pascal under Freepascal. :D Sometimes I have to write small programs for my work and for this pascal is useful. I didn't want to learn a new language... My program hasn't GUI, you can control it via keyboard. It more than 2500 lines and developed during ~10 month with smaller-bigger breaks.
I wrote a high quality image render and video maker for this too, which can handle multiple CPUs if you start it several times parallel.
It calculates 20 digit numbers only, but it is enough for a lot of things.
Rendering 3D is an other interesting area for me, but nowdays I have no time for these things... :) But 2D sections of >=3D fractals are also beauty, I played this too.
I wrote a high quality image render and video maker for this too, which can handle multiple CPUs if you start it several times parallel.
It calculates 20 digit numbers only, but it is enough for a lot of things.
Rendering 3D is an other interesting area for me, but nowdays I have no time for these things... :) But 2D sections of >=3D fractals are also beauty, I played this too.
Title: Re: A new look into escapetime fractals using abs & inner coloring
Post by: subblue on October 18, 2012, 12:38:10 PM
This is fun! Exp. smoothing really is great for revealing interesting features :)
These are -1 power abs julias:
(http://farm9.staticflickr.com/8184/8099499351_86cfdb45f9_z.jpg) (http://www.flickr.com/photos/subblue/8099499351/sizes/o/in/photostream/)
(http://farm9.staticflickr.com/8048/8099499517_f2eea9474e_z.jpg) (http://www.flickr.com/photos/subblue/8099499517/sizes/o/in/photostream/)
For the following two I tweaked the Exp. smoothing to take the iteration max. of the difference between the magnitudes of z:
To reveal a different structure:
(http://farm9.staticflickr.com/8468/8099499665_b72e5c9d8b_z.jpg) (http://www.flickr.com/photos/subblue/8099499665/sizes/o/in/photostream/)
(http://farm9.staticflickr.com/8473/8099507392_aacbc66050_z.jpg) (http://www.flickr.com/photos/subblue/8099507392/sizes/o/in/photostream/)
These are -1 power abs julias:
(http://farm9.staticflickr.com/8184/8099499351_86cfdb45f9_z.jpg) (http://www.flickr.com/photos/subblue/8099499351/sizes/o/in/photostream/)
(http://farm9.staticflickr.com/8048/8099499517_f2eea9474e_z.jpg) (http://www.flickr.com/photos/subblue/8099499517/sizes/o/in/photostream/)
For the following two I tweaked the Exp. smoothing to take the iteration max. of the difference between the magnitudes of z:
Code:
m0 = m;
m = length(z);
float diff = abs(m0 - m);
m1 = max(diff, m1);
idx += exp(-1.0 / m1);
To reveal a different structure:
(http://farm9.staticflickr.com/8468/8099499665_b72e5c9d8b_z.jpg) (http://www.flickr.com/photos/subblue/8099499665/sizes/o/in/photostream/)
(http://farm9.staticflickr.com/8473/8099507392_aacbc66050_z.jpg) (http://www.flickr.com/photos/subblue/8099507392/sizes/o/in/photostream/)
Title: Re: A new look into escapetime fractals using abs & inner coloring
Post by: kram1032 on October 18, 2012, 01:39:15 PM
What is it with fractal checkerboards lately? And why aren't there any pieces on them yet? :D
Title: Re: A new look into escapetime fractals using abs & inner coloring
Post by: Syntopia on October 18, 2012, 10:12:04 PM
Although not very well-documented, Fragmentarium also does exponential smoothening in its Progressive2DJulia.frag. Just set ColoringType=1.
A couple of weeks ago, I explored this little system with exp. smoothening:
(Is Latex broken?)
(http://blog.hvidtfeldts.net/media/sketch-10.jpg)
(http://blog.hvidtfeldts.net/media/sketch-14.jpg)
Code:
A couple of weeks ago, I explored this little system with exp. smoothening:
(Is Latex broken?)
(http://blog.hvidtfeldts.net/media/sketch-10.jpg)
(http://blog.hvidtfeldts.net/media/sketch-14.jpg)
Code:
Code:
#include "Progressive2DJulia.frag"
uniform float Scaling; slider[-5,3,5]
vec2 formula(vec2 z,vec2 c) {
vec2 z2 = vec2(z.x*z.x-z.y*z.y, 2*z.x*z.y);
z =z2/dot(z2,z2)*Scaling;
return z+c;
}
Title: Re: A new look into escapetime fractals using abs & inner coloring
Post by: Alef on October 23, 2012, 05:48:56 PM
If you are interested, I tried to unite all the possible modifications in E++ Exponent smoothing in UF. Say you can use -e or some other exponent base. This one looks like Fractal Explorer colour methods.
Code:
Fractal2 {
fractal:
title="Fractal2" width=640 height=480 layers=1
credits="Asdam;10/22/2012"
layer:
caption="Background" opacity=100
mapping:
center=-0.9299999623/0.5341666628 magn=5.3973016
formula:
maxiter=30 percheck=off filename="em.ufm" entry="Kalisetes_Sefirii"
p_equation="1- kaleido Kaliset j" p_bailout=777000 p_settype=Julia
p_switchsettype=Mandelbrot p_julia=-0.475/0.65 p_D=1
f_func_post=flip p_scale=1/0 p_vectype=None p_vfactor=-0.31/-0.41
inside:
transfer=linear filename="em.ucl" entry="E++ExpSmoothting"
p_posneg=0/0 p_iternum=7770000 p_expdim=no p_darkness=0.9
p_calculation="Mandelbrot type" p_exponentype="Free exponent"
p_base=-2/2 p_prefn="1 None" p_postfn="1- None"
outside:
transfer=linear filename="Standard.ucl" entry="Default"
gradient:
comments="My favorite and my original creation." smooth=yes index=0
color=15202559 index=66 color=4194432 index=133 color=15202559
index=199 color=4618495 index=266 color=1646882 index=332
color=55717
opacity:
smooth=yes index=50 opacity=255 index=100 opacity=255 index=200
opacity=255 index=302 opacity=255 index=376 opacity=255
}