Menger Sponge 45 Rotation

[trafassel]

Formula.Parameters: angleX=0 angleY=0 angleZ=45 bailout=6000000000 offsetX=0.88 offsetY=1.272 offsetZ=1.4 scale=1.5 zUpperBound=12 Formula.Static: Cycles=75 Julia=1 jx=0.05 jy=0 jz=0 Scene: CenterX=0.641905372680213 CenterY=0.339207258021939 CenterZ=-0.262591944640448 Radius=0.249229000878406 Transformation.Camera: AngleX=175.668590010273 AngleY=-27.0356750479037 AngleZ=-171.080170533749 IsometricProjection=0 Position=1

[trafassel]

Formula.Parameters: angleX=0 angleY=0 angleZ=45 bailout=3 offsetX=2 offsetY=2.5 offsetZ=2.5 scale=1.56 Formula.Static: Cycles=75 Julia=1 jx=0.05 jy=0 jz=0
Scene: CenterX=-1.33456370239741 CenterY=0.263163682905943 CenterZ=0.216002920489848 Radius=0.672939928707691 Transformation.Camera: AngleX=-44.373124365217 AngleY=63.4414563477275 AngleZ=-96.3279863410774 IsometricProjection=0 Position=1

[trafassel]

Here I remove the coordinate swap and add some additional conditions. The result fractal ist the same Menger Sponge, but the rotation only applies to one plane.


double _scale=3;
double _offsetX=2;
double _offsetY=2;
double _offsetZ=2;
double _bailout=3;
double _angleX=0;
double _angleY=45;
double _angleZ=0;
// _cycles=8

public override bool GetBool(double x,double y,double z)
{
  for (int n=1;n < _cycles;  n++)
  {
    x=Math.Abs(x);
    y=Math.Abs(y);
    z=Math.Abs(z);
    if(x>_bailout)return false;
    if(y>_bailout)return false;
    if(z>_bailout)return false;
    x=_scale*x-_offsetX*(_scale-1);
    y=_scale*y-_offsetY*(_scale-1);
    z=_scale*z-_offsetZ*(_scale-1);
    if(x<y && x<z && x<-0.5*_offsetX*(_scale-1.0))x+=_offsetX*(_scale-1.0);
    else if(y<x && y<z && y<-0.5*_offsetY*(_scale-1.0))y+=_offsetY*(_scale-1.0);
    else if(z<x && z<y && z<-0.5*_offsetZ*(_scale-1.0))z+=_offsetZ*(_scale-1.0);
    if(_angleX!=0)Rotate(_angleX,ref y,ref z);
    if(_angleY!=0)Rotate(_angleY,ref x,ref z);
    if(_angleZ!=0)Rotate(_angleZ,ref x,ref y);
    x+=_jx;
    y+=_jy;
    z+=_jz;
  }
  return true;
}


void Rotate(double angle,ref double x,ref double y)
{
  double re=Math.Cos(angle );
  double im=Math.Sin(angle );
  double a=re * x-im * y;
  y=re * y+im * x;
  x=a;
}

[mclarekin]

I must try out the conditional rotate. With some fractals I can only  choose to rotate one of the three axis (otherwise too distorted), this might make those formula faster 

     if(_angleX!=0)Rotate(_angleX,ref y,ref z);
    if(_angleY!=0)Rotate(_angleY,ref x,ref z);
    if(_angleZ!=0)Rotate(_angleZ,ref x,ref y);


I have been exploring  mixing conditional and non-conditional Menger iterations in 4D but have not tried it in 3D, but it would look like this at the end of a standard 3D Menger..

   z = fabs( z  + preAdd );
   
   if (z.x - z.y < 0.0f){temp = z.y; z.y = z.x; z.x = temp;}
   if (z.x - z.z < 0.0f){temp = z.z; z.z = z.x; z.x = temp;}   
   if (z.y - z.z < 0.0f){temp = z.z; z.z = z.y; z.y = temp;}
   z *= 3.0f;
   z.x -= 2.0f;
   z.y -= 2.0f;

   if (i >= startIter && i < stopIter)
   {
      if (z.z > 1.0f) z.z -= 2.0f;
   }
   else
   {
      z.z-=2.0f;
   }


Those additional conditions you have added look interesting

[trafassel]

The additional conditions I have only added, because I skipped the coordinate swap. The coordinates are now not sorted to x<y<z.

Also equivalent to the original formula ist my following naive approach. This is not optimized code, but its simple and is perhaps more stable to additional transformations.

Code:

  double boxSize=3;
  double holeSize=1;
  int iterations=6;
  x*=boxSize;
  y*=boxSize;
  z*=boxSize;
  if(x> boxSize)return false;
  if(y> boxSize)return false;
  if(z> boxSize)return false;
  if(x<-boxSize)return false;
  if(y<-boxSize)return false;
  if(z<-boxSize)return false;
  for (int n = 1;n < iterations;  n++)
  {
    double xa=Math.Abs(x);
    double ya=Math.Abs(y);
    double za=Math.Abs(z);
    int holeCount=0;
    if(xa<holeSize) holeCount++;
    if(ya<holeSize) holeCount++;
    if(za<holeSize) holeCount++;
    if(holeCount>=2) return false;
    if(x> holeSize) x-=2*holeSize;
    if(x<-holeSize) x+=2*holeSize;
    if(y> holeSize) y-=2*holeSize;
    if(y<-holeSize) y+=2*holeSize;
    if(z> holeSize) z-=2*holeSize;
    if(z<-holeSize) z+=2*holeSize;
    x*=boxSize;
    y*=boxSize;
    z*=boxSize;
  }
  return true;

[DarkBeam]

 

[trafassel]

The 45 degree variant.

[trafassel]

Coloring the surface according to the direction of the translation gives a nice surface coloring.

(Scene: CenterX=-0.932873944699854 CenterY=0.899375847775146 CenterZ=1.16693107216321 Radius=0.18692775797436 Transformation.Camera: AngleX=-171.967389672199 AngleY=-47.8088737441608 AngleZ=73.6703503820553 IsometricProjection=0 Position=1 )
 

Code:

public override bool GetBool(double x,double y,double z)
{

double boxSize=3;
double holeSize=1.2;
double scalex=1.1;
double scaley=1.1;
double scalez=0.9;

  x*=boxSize;
  y*=boxSize;
  z*=boxSize;
  if(x> boxSize*scalex)return false;
  if(y> boxSize*scalex)return false;
  if(z> boxSize*scaley)return false;
  if(x<-boxSize*scaley)return false;
  if(y<-boxSize*scalez)return false;
  if(z<-boxSize*scalez)return false;
  for (int n=1;n < _cycles;  n++)
  {
    double xa=Math.Abs(x);
    double ya=Math.Abs(y);
    double za=Math.Abs(z);
    int holeCount=0;
    if(xa<holeSize)holeCount++;
    if(ya<holeSize)holeCount++;
    if(za<holeSize)holeCount++;
    if(holeCount>=2)return false;
    if(x> holeSize)
    {
      Red++;
      x-=2*holeSize;
    }
    if(x<-holeSize)
    {
      Red--;
      x+=2*holeSize;
    }
    if(y> holeSize)
    {
      Blue++;
      y-=2*holeSize;
    }
    if(y<-holeSize)
    {
      Blue--;
      y+=2*holeSize;
    }
    if(z> holeSize)
    {
      Green++;
      z-=2*holeSize;
    }
    if(z<-holeSize)
    {
      Green--;
      z+=2*holeSize;
    }
    x*=boxSize*scalex;
    y*=boxSize*scaley;
    z*=boxSize*scalez;
  }
  return true;
}

[mclarekin]

The surface coloring looks great. I am going to try it out

[trafassel]

The surface coloring looks great. I am going to try it out.

This would be cool. Some areas are grey because I implemented in my renderer a "color threshold". Without this threshold the corresponding surface would be drawn in pastel colors.

[trafassel]

I like simple formulas. Here is another one. I removed the box boundary restriction and change the "hole indicator".

The scene is located at:

CenterX=0.0537731690329398 CenterY=0.986837787113877 CenterZ=0.993711945778967 Radius=0.0175530009711978
Transformation.Camera: AngleX=-104.327115785263 AngleY=-20.6946703978129 AngleZ=-132.683623253497 IsometricProjection=0 Position=1

Code:

public override bool GetBool(double x,double y,double z)
{
  double boxSize=1.8;
  double holeSize=0.62;
  for (int n=1;n < 17;  n++)
  {
    if(Math.Abs(x)>=holeSize && 
       Math.Abs(y)>=holeSize && 
       Math.Abs(z)>=holeSize) return false;
    if(x> holeSize) x-=2*holeSize;
    if(x<-holeSize) x+=2*holeSize;
    if(y> holeSize) y-=2*holeSize;
    if(y<-holeSize) y+=2*holeSize;
    if(z> holeSize) z-=2*holeSize;
    if(z<-holeSize) z+=2*holeSize;
    x*=boxSize;
    y*=boxSize;
    z*=boxSize;
  }
  return true;
}