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Here is the new folding code

 // Comment out to use the original (Tglad) Mandelbox formula.

  // if (x > 1) x = 2.0 - x; else
  if (x < -1) x = -2.0 - x;
            
  // if (y > 1)    y = 2.0 - y;  else
  if (y < -1) y = -2.0 - y;
            
  // if (z > 1)   z = 2.0 - z; else
  if (z < -1) z = -2.0 - z;

// End Code
 

scale=1.04

minScale=0

300 iterations.

public override void Init() {base.Init();additionalPointInfo=new AdditionalPointInfo();}


// Simplified Mandelbox
public override long InSet(double ar, double ai, double aj,  double br, double bi, double bj, double bk, long zkl, bool invers) {

  double gr=50000; // bailoutValue
  double xx, yy, zz; // temp var
  long tw; // boring
  int n;
  double x=0,y=0,z=0;
  double r_n = 0;

  xx = x * x; yy = y * y; zz = z * z;
  tw = 0L;
 double r = Math.Sqrt(xx + yy + zz);

// This fixid to a special non Julia-variant
  double scale = 1.05;

  // Surface Coloring
  additionalPointInfo.red=0;
  additionalPointInfo.green=0;
  additionalPointInfo.blue=0;

  for (n = 1; n < zkl; n++) {

  // Original box fold is replaced by x=abs(x); y=abs(y); z=abs(z)
  // Inspired by the work of Kali.
   
  // Comment out to use the original (Tglad) Mandelbox formula.

  // if (x > 1) x = 2.0 - x; else
  if (x < -1) x = -2.0 - x;
             
  // if (y > 1)    y = 2.0 - y;  else
  if (y < -1) y = -2.0 - y;
             
  // if (z > 1)   z = 2.0 - z; else
  if (z < -1) z = -2.0 - z;
             
  //  ; fold sphere onto itself
  double fixedRadius = 1;
  double minRadius = 0.00001;

  xx=x*x;yy=y*y;zz=z*z;
  double length=   r = xx + yy + zz;

// Colors
  if(r!=0) {
  additionalPointInfo.red+=1.0*xx/r;
  additionalPointInfo.green+=1.0*yy/r;
  additionalPointInfo.blue+=zz/r;
  }
           
  // Bailout Condition
  if (length > gr) {  tw = n; break; }

 double mult=1;

 if (length < minRadius) {
 //     mult=Math.Sqrt(fixedRadius)/(minRadius);
 }   else
 if (length < fixedRadius) {
   mult=Math.Sqrt(fixedRadius)/(length);
 }
 mult*=scale;
 x*=mult;
 y*=mult;
 z*=mult;

// In Julia Mode: the fixed Julia values
// Iin Mandelbrot set mode: the room coordinates of the pixel to test
   x+=br;
   y+=bi;
   z+=bj;
 }

// For inside rendering
if (invers) {if (tw == 0) tw = 1; else tw = 0; }

  // Warning: this valaue can be very dangereus, if it is used for distanve esitmation:
  //
  return (tw);
}

The geometry is similar to the regular mandelbox, but the "last budda steps"-coloring method, described in: http://www.fractalforums.com/theory/coloring-the-mandelbulb/
define a nice surface coloring.

 

Typical Mandelbox structures: The spiral.

Infinity

The only real 3D part in this picture are the green holes. Use default surface painting.

This is not a sine wave.

The monster and the plant.

Two sides of the same thing.

My first journey into this nice box (which is really not a box - its a spherical segment).

http://www.youtube.com/watch?v=8c8Yfi25Za0

Amazing video trafassel!

Thanks.

Even the 4d variant looks very symmetric.

But the most symmetric structures can be found in some julias.

Default Julia with Seed = (0, 0, 0)

and

Julia with Seed = (0.045, 0, 0)

More Julia Sets

Wow, I love those last julia sets - very beautiful and interesting indeed!

If you like it, I will present some more AbsBox Julia sets.

 :joy: MARVELLOUS images

This video visualize the change of the julia seed.

http://www.youtube.com/watch?v=B_t_8SvXGE0

The AbsBox with Scale=1.

Trafassel, I am noticing that Mandelbulb renders of your great fractal are very different from yours. Probably, some things needs fixing...

1st; Mandelbulb starts from x=cx, y=cy, z=cz, w=cw=0 when you seem to start at x=y=z=w=0 (very important)
2nd; this part seems to be missing in Jesse's formula

double mult=1;

if (length < 1e-5) {
 //     mult=Math.Sqrt(fixedRadius)/(minRadius);
 }   else
 if (length < fixedRadius) {
   mult=Math.Sqrt(fixedRadius)/(length);
 }
 mult*=scale;
 x*=mult;
 y*=mult;
 z*=mult;

That may be cause of errors or something. :)

Really nice and interesting renders, trafassel, very good stuff!

What about using abs(a+f)-f (for each axis a) and varying f values ?

also abs(a+f1)-abs(a-f2)-a with different f1 and f2 values.

Hope somebody finds the "error", it would be great to see such images in Mandelbulb :)

DarkBeam, you are right. To work with the formula as julia set you have to replace

x=y=z=w=0; with  x=ar;y=ai;z=aj; (The position of the point to test).

The sphere inversion:
double mult=1;
if (length < 1e-5) {
 //     mult=Math.Sqrt(fixedRadius)/(minRadius);
 }   else
 if (length < fixedRadius) {
   mult=Math.Sqrt(fixedRadius)/(length);
 }

is the same as:
mult=1;
  if (r < fixedRadius && r>0) {
    mult=1.0/r;
  }
if minRadius is very small. In my tests both variants generates nearly the same pictures.

The pictures of the scale=1 AbsBox are generated with a very small bailout value. So in Reply #18  I have replaced:
double gr=50000; // bailoutValue
with
double gr=32; // bailoutValue

Here ist formula, I used for generation the julia movie. The get it to work in Gestaltlupe you have to press render two times. First to set the parameters. Second to render the picture. Hope, this will help you with the formula.


public override void Init() {base.Init();
if(GetString("intern.Formula.TempUpdateVal")!="33b6843e3102477181c51c97eaf7b578"){
SetParameterBulk(@"<Entry Key='Border.Max.x' Value='0.832922740362181' /><Entry Key='Border.Max.y' Value='1.590864959092' />
<Entry Key='Border.Max.z' Value='-0.01610155004377' /><Entry Key='Border.Min.x' Value='-3.20862914697999' />
<Entry Key='Border.Min.y' Value='-2.45068692825015' /><Entry Key='Border.Min.z' Value='-2.28947448667369' />
<Entry Key='Border.Min.zz' Value='0' />
<Entry Key='Formula.Static.Cycles' Value='120' /><Entry Key='Formula.Static.Formula' Value='-2' />
<Entry Key='Formula.Static.jx' Value='0.01' /><Entry Key='Formula.Static.jy' Value='0.08' /><Entry Key='Formula.Static.jz' Value='-0.01' />
<Entry Key='Formula.Static.jzz' Value='0' /><Entry Key='Formula.Static.MinCycle' Value='51' />
<Entry Key='Transformation.Camera.AngleZ' Value='12' /><Entry Key='Transformation.Perspective.Cameraposition' Value='0.3' />
<Entry Key='Transformation.Stereo.Angle' Value='-9' /><Entry Key='Transformation.Stereo.EyeDistance' Value='0.5' />
<Entry Key='View.Height' Value='720' /><Entry Key='View.Perspective' Value='1' /><Entry Key='View.Width' Value='1280' />
<Entry Key='intern.Formula.TempUpdateVal' Value='33b6843e3102477181c51c97eaf7b578' />");}
additionalPointInfo=new AdditionalPointInfo();}


// Simplified Mandelbox
public override long InSet(double ar, double ai, double aj,  double br, double bi, double bj, double bk, long zkl, bool invers) {

  double gr=50000; // bailoutValue
  double xx, yy, zz; // temp var
  long tw; // boring
  int n;
  double x=ar,y=ai,z=aj;
  double r_n = 0;
  double mult=1;

  xx = x * x; yy = y * y; zz = z * z;
  tw = 0;
  double r = xx + yy + zz;

  double scale = 1.05;

  // Surface Coloring
  additionalPointInfo.red=0;
  additionalPointInfo.green=0;
  additionalPointInfo.blue=0;

  for (n = 1; n < zkl; n++) {

  // Original box fold is replaced by x=abs(x); y=abs(y); z=abs(z)
  // Inspired by the work of Kali.
   
  // Comment out to use the original (Tglad) Mandelbox formula.

  // if (x > 1) x = 2.0 - x; else
  if (x < -1) x = -2.0 - x;
             
  // if (y > 1)    y = 2.0 - y;  else
  if (y < -1) y = -2.0 - y;
             
  // if (z > 1)   z = 2.0 - z; else
  if (z < -1) z = -2.0 - z;
             
  //  ; fold sphere onto itself

  xx=x*x;yy=y*y;zz=z*z;
  r = xx + yy + zz;

  if(r!=0) {
    additionalPointInfo.red+=xx/r;
    additionalPointInfo.green+=yy/r;
    additionalPointInfo.blue+=zz/r;
  }
           
  // Bailout Condition
  if (r > gr) {  tw = n; break; }

  mult=1;
  if (r < fixedRadius && r>0) {
    mult=1.0/r;
  }
  mult*=scale;
  x*=mult;
  y*=mult;
  z*=mult;

  // In Julia Mode: the fixed Julia values
  // In Mandelbrot set mode: the room coordinates of the pixel to test
  x+=br;
  y+=bi;
  z+=bj;
 }

  // For inside rendering
  if (invers) {if (tw == 0) tw = 1; else tw = 0; }

  return (tw);
}

:embarrass:

Very sorry to disturb for nothing... Now I tried again Jesse's formula with different parameters and I am getting almost perfect pictures too. :)

Scale=1 is rendered (but with awful de) and I managed to rendered your Apollonian julia ;)

The problem is that many iters are needed (slow convergence on scales almost 1) :)

Mandelbulb3Dv17{
P.....S....O/...kR/........dcze4Pd10.f2jgjIOPw.El.kbcJntYzvPSTETfd3yzKr12oIzLkzj
................................y.YKxj687.2.......EW./..................y.2.....
................/ME//....6.o0...lY....E2.....cBbscqueQpD/..........c./...w1....U
z.EnAnQD32../2.............................................2.....y1...sD...../..
.w1...sDlEeuPJMmGxX68stvHi.2zsbyiuux7goDiD8RvQ3w0xH3r0FQE.mHzkX6c6NTUmojAK67nbjN
1xfE9XgQ96GIzUCpt/0cfVoD......2O/.............kD.2....sD..kz0...................
.............................UTRU4.Yq/O.cQ5c/.wRU4.Ks/O.kX5c/UMSU4..............
.....................wzzz1.U..6.P....U0...EB....p....kB....F....1/...I1....UJp32
XM.U.0.MzzDg./kz....z1....6.1c..zzzz............MZCpCckYcz1..........2k.8.kXWF1.
QvM93P58iz9GMmnWK2zwz0........../6U0.wzzz1................................E.0c..
zzzz.................................2U.8.kzzzD.................................
/6U0.wzzz1....................................C....wk1D.zz/..wD.k1Dw.szD.U9...Dw
k1EzTpxe...wk1D.Qs5s....k1Dw..bTU1....Dwk1.ly/C....wk1D.Mw5s....k1Dw.kqTU1....Dw
k1.kz/C....wk1D....s....zz/..wD.yz1.s0..xzJrf0..................................
E....2....E.....I....A....EE0x4SBx4NoE4.rJaQ....................................
..........EnAnAnAnAwz.........zD................................................
................................................................................
........................}

picture :D

Another zoom into the AbsBox:

http://www.youtube.com/watch?v=gOS6vUIihM8

Very interesting movie Traf! You get a nice feel of the fractal...
Nice to see that they have Disco-strobe lights in the fractal world too! :)

Nice, that you like it.

The Disco-strobe lights comes from randomly generated shadows.