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Author Topic: An escape time algorithm for Kleinian group limit sets  (Read 64241 times)
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bib
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« Reply #75 on: January 05, 2017, 02:48:32 PM »

... mister Yoshiaki Araki...

http://www.fractalforums.com/fractal-news-across-the-world/3d-fractal-meeting-in-tokyo/
cheesy
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DarkBeam
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« Reply #76 on: January 05, 2017, 03:08:33 PM »

But he is still not registered in here cry please Dr. Araki join us A Beer Cup
I am almost sure he rendered some images directly using the .frag script wink
Too bad the language barrier is not surpassable for me!

https://mobile.twitter.com/alytile/status/813727903854653440
« Last Edit: January 05, 2017, 03:32:07 PM by DarkBeam » Logged

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pupukuusikko
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« Reply #77 on: January 05, 2017, 03:34:26 PM »

EDIT: This post is not relevant, Aexion's script on the previous page does what I propose and more.

It would be nice if the spirals spiralled in all three dimensions for cork-screw type of things.
I tried to implement this and it kind of works by adding a variable to transA() to modify z-axis:






* Kleinian_test23_zrot.frag (27.11 KB - downloaded 131 times.)
« Last Edit: January 05, 2017, 04:00:04 PM by pupukuusikko » Logged

eiffie
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« Reply #78 on: January 06, 2017, 02:10:25 AM »

I just wanted to add these are absolutely wonderful, but you all knew that already.
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KRAFTWERK
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« Reply #79 on: January 08, 2017, 10:50:00 AM »

3D print ready to order smiley Thank you guys!!!
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Tim Emit
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« Reply #80 on: January 09, 2017, 07:45:47 PM »

Must say everything is looking sweet folks.. Aexions latest is great fun, will look at yours pupukuusikko : )
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JosLeys
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« Reply #81 on: January 10, 2017, 02:08:09 PM »

For those who want more info on the rapid calculation of Kleinian group limit sets in 2D, here is an article for you to read: http://www.josleys.com/articles/Kleinian%20escape-time_2.pdf

(it was actually working on this 2D algorithm that gave me the idea for the 3D one)


* Kleinian_001b.jpg (112.72 KB, 400x400 - viewed 831 times.)
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trafassel
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« Reply #82 on: January 11, 2017, 11:40:50 PM »

JosLeys Thank you for sharing you ideas. I like you easy to understand style of describing you geometrical and mathematical research. It would be cool to see some of this in a scientific paper.

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pupukuusikko
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« Reply #83 on: January 12, 2017, 09:03:56 AM »

 joy breaking news! joy

It looks like KleinR, I (and J) can be interpreted as Julia coordinates, and there exists a kleinbrot.
All kinds of problems, but I managed to take this terrible shot:



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bib
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« Reply #84 on: January 12, 2017, 10:24:42 AM »

Not surprisingly it looks very much like the scale 1 amazing box M-set
http://bib993.deviantart.com/art/Underwater-amazing-object-208457489
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DarkBeam
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« Reply #85 on: January 12, 2017, 01:01:36 PM »

It is a troble because kleinr must be > kleini and kleinj. Else the skewness becomes too high.
The algo should be redesigned avoiding to use a skew factor into wrap() somehow to make it more flexible... don't know how btw.
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pupukuusikko
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« Reply #86 on: January 12, 2017, 09:19:24 PM »

I think it's going to be fine  Azn

In the previous pic I handled the separation line part wrong, so it became more box-like than it should have.
I still have no idea of the correct way, but the real kleinbrot will probably look more like this:



EDIT: added frag, including brot and Julia mode, as well as 3d boxfold.
There might still be gross misunderstandings as well as minor sign issues,
but it's now quite clean and explorable, although slowish.
 

* kleinbrot.frag (11.86 KB - downloaded 119 times.)
« Last Edit: January 13, 2017, 08:06:38 PM by pupukuusikko, Reason: added frag » Logged

trafassel
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« Reply #87 on: January 13, 2017, 12:04:10 AM »

I was not able to implement the box foldings an line of separation definition, given in Reply #81. So my research is still based on the Fragment code Kleinian_test15-bis.frag, presented in Reply #23.

I did some simplifications and scale the line of separation, but the fractal set seems compatible to the picture in Reply #55 .

The following images should demonstrate the difference in using the line of separation and using classic box folding. The only image without cut is the classical mandelbox julia without using the line of separation.

Default Kleinian Formula:
Code:
public override bool GetBool(double x, double y, double z)
{
  double KleinR=1.94;
  double KleinI=0.033;
  for (int i = 0;i < 44;  i++)
  {
    x = x-Math.Floor(x+0.5);
    z = z-Math.Floor(z+0.5);
    if ( y >= KleinR*(0.5+0.1*Math.Sin(-2.0*Math.PI*x)) )
    {
      x = KleinI-x;
      y = KleinR-y;
      z = -z;
    }
    double r=x*x+y*y+z*z;
    if(r>0)
    {
      x /= r;
      y /= r;
      z /= r;
      if(r<0.2) return false;
    }
    x = KleinI + x;
    y = KleinR - y;
    z = -z;
  }
  return true;
}

Same code without line of separation:
Code:
public override bool GetBool(double x, double y, double z)
{
  double KleinR=1.94;
  double KleinI=0.033;
  for (int i = 0;i < 44;  i++)
  {
    x = x-Math.Floor(x+0.5);
    z = z-Math.Floor(z+0.5);
    double r=x*x+y*y+z*z;
    if(r>0)
    {
      x /= r;
      y /= r;
      z /= r;
      if(r<0.2) return false;
    }
    x = KleinI + x;
    y = KleinR - y;
    z = -z;
  }
  return true;
}

With boxfold and line of Separation:
Code:
public override bool GetBool(double x, double y, double z)
{
  double KleinR=1.94;
  double KleinI=0.033;
  for (int i = 0;i < 44;  i++)
  {
    x=x%2;
    z=z%2;
    while(Math.Abs(x)>0.5)
    {
      if(x>0.5) x = 1-x;
      if(x<-0.5) x = -1-x;
    }
    while(Math.Abs(z)>0.5)
    {
      if(z>0.5) z = 1-z;
      if(z<-0.5) z = -1-z;
    }
    if ( y >= KleinR*(0.5+0.1*Math.Sin(-2.0*Math.PI*x)) )
    {
      x = KleinI-x;
      y = KleinR-y;
      z = -z;
    }
    double r=x*x+y*y+z*z;
    if(r>0)
    {
      x /= r;
      y /= r;
      z /= r;
      if(r<0.2) return false;
    }
    x = KleinI + x;
    y = KleinR - y;
    z = -z;
  }
  return true;
}

With boxfold (Mandelbox Julia with scale=1):
Code:
public override bool GetBool(double x, double y, double z)
{
  double KleinR=1.94;
  double KleinI=0.033;
  for (int i = 0;i < 44;  i++)
  {
    x=x%2;
    z=z%2;
    while(Math.Abs(x)>0.5)
    {
      if(x>0.5) x = 1-x;
      if(x<-0.5) x = -1-x;
    }
    while(Math.Abs(z)>0.5)
    {
      if(z>0.5) z = 1-z;
      if(z<-0.5) z = -1-z;
    }
    double r=x*x+y*y+z*z;
    if(r>0)
    {
      x /= r;
      y /= r;
      z /= r;
      if(r<0.2) return false;
    }
    x = KleinI + x;
    y = KleinR - y;
    z = -z;
  }
  return true;
}


* Data147pic10019s55.jpg (66.24 KB, 900x450 - viewed 154 times.)

* Data147pic10021s50.jpg (55.06 KB, 900x450 - viewed 138 times.)

* Data147pic10022s50.jpg (67.33 KB, 900x450 - viewed 142 times.)

* Data147pic10020s50.jpg (38.24 KB, 900x450 - viewed 126 times.)
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pupukuusikko
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« Reply #88 on: January 13, 2017, 08:15:28 PM »

The following images should demonstrate the difference in using the line of separation and using classic box folding. The only image without cut is the classical mandelbox julia without using the line of separation.

Nice, the boxfold version could well become quite popular 'general' fractal, if Luca were to implement a hybridisable version in mb3d. I would definitely explore it smiley 

(also added the brot frag to previous post)
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DarkBeam
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« Reply #89 on: January 13, 2017, 11:30:19 PM »

If you can use bitwise operators on the integer that derives from floor(), like I do in assembly, do a bw AND 1.
If the result is 1 change the sign of the corresponding variable after wrap() to obtain infinite fold. Else do a Tglad fold after wrap() using an appropriate wrap constant...
Sorry but I feel bad now, must sleep badly
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