Experimenting with formulas

[Alef]

Some 5 or more years agou i had vritten the formula for the Fractal Explorer. There are just two variants to implement hyperbolic sine into something mandelbrot like and it not so depends on bailout values. So I recoded the same for the Chaos pro. It have classical mandelbrot features, but have tons small julia satelite sets flying around main set and each other and produce leafy pattern (mandelbrot produces the same with abs(z) bailout test. ).

// formula z=z*sinh(z)+C*C  09.11.2001 by Edgars Malinovskis, first in Fractal Explorer 5 years agou.
MalinovskyDecoFractal {
// formula z=z*sinh(z)+C*C  09.11.2001 by Edgars Malinovskis, first in turbo pascal 5 years agou.
parameter int settype;
parameter complex julia;
parameter real bailout;
complex C;
parameter bool testBroken;

void init(void)
 {
   if (settype=="Julia Set")
   {   
   z = pixel;
   C = julia;
   }
   else if (settype=="Inverted Mandelbrot Set")
    {
    z= 0;
    C= 1/pixel;
    }
    else if (testBroken==true) //recreates z=pixel features
    {
    z = (1,0.5);    //other values are useless
    C= pixel;
    }
    else
   {
   z= 0;
   C= pixel;
   }
}

void loop(void)
{
      z= sinh(z)*z -sqr(C);
}

bool bailout(void)
   {
      return(  abs(real(z))+ abs(imag(z)) <= bailout  );
   }

void description(void)
   {
      this.title = "Malinovsky Deco Fractal";

      bailout.caption = "Bailout Value";
      bailout.default = 7.0;
      bailout.min = 0.5;
      bailout.hint = "Defines the bailout radius: If pixel falls out of it, the iteration stops. Larger number increases details";

    settype.caption = "Set type";
    settype.enum = "Mandelbrot Set\nJulia Set\nInverted Mandelbrot Set";
    settype.default = 0;

    testBroken.caption="Broken Fractal";
    testBroken.default=false;
    testBroken.visible = (settype=="Mandelbrot Set");

    julia.caption = "Julia Parameter";
    julia.default = (0,0.5);
    julia.hint = "Different values create different Julia Sets.";
    julia.visible = (settype=="Julia Set");
   }
}

Then using spider as base written formula giving menorah and angel like figures. Nothing superb, but fractal have infinite size and allways should be turned by 90 degrees. But after changeing the complex variables to quaternion so formula could work in 3D everyrhing, fractal appeared to be much more sophisticeted. Simple curves in 3D turned into complicated saturnian rings. Probably coz of inverted nature fractal is very chaotic and allways produce pretty counterintuitive results.

// Generates menorah like figures. Hyperbolised spider + variables change. 28.10.2001 by Edgars Malinovskis.
Menorah_Fractal{
//Quaternion Friendly!  23.11.2001 by Edgars Malinovskis. Optimised 27.11.2011.
parameter int settype;
parameter real bailout, multi;
quaternion c, p;
complex w;

   void init(void)
   {
   
      if (settype=="Real Coefficient")
    {   
        c = 1/pixel;
    z = c;
    p = c;      //allways 1/pixel.
    w=multi*0.01+0.5;      //+-1/100 gives results.
    }
      else if (settype=="Imaginary Coefficient")
    {
    c = 1/pixel;
    z = c;
    p = c;
    w=multi*0.01i+0.5;
   }
      else
    {
    c = recip(pixel);
    z = c;
    p = c;
    w=tan(multi)*0.01+0.5;
   }
}
   void loop(void)
   {
      z=sqr(z)+c+p;      //spider formula + 1/pixel
      c=w*c+z;
   }
   bool bailout(void)
   {   
      return(  abs(real(z))+ abs(imag(z))<= bailout  );
   }
   void description(void)
   {
      this.title = "Menorah Fractal (Quaternion friendly)";
      this.magn = 0.25;

      bailout.caption = "Bailout";
      bailout.default = 7.0;
      bailout.min = 1;
      bailout.hint = "Defines the bailout radius: If pixel falls out of it, the iteration stops. Larger number increases details";


      multi.caption = "-9<Menorah Coef<9";
      multi.default = 0;
      multi.min = -9;
      multi.max = 9;
      multi.hint = "Increases multiplicator by 1/100. Changes fractal geometry.";

      settype.caption = "type(if MC is not 0)";
      settype.enum = "Real Coefficient\nImaginary Coefficient\nTan Coeff + Recip inverse";
      settype.default = 0;
      settype.hint = "Add real, imaginary or tangential value to the multiplicator of the Fractal. Only if the Menorah Coefficient is nonZero.";
   }
}

Basic shape, lots of julias around, connected ones are found inside of the valleys.


Julia set with maximal bailout value. Just like the same of z^2+c but with sets around and leafy pattern.


Some more artsy deep zoom into one of the small satelites. At the core lies his majestry classical mandelbrot. Must be some julia set who look like this.


Fractal Menorah. Infinite, but prety simple.


The same Menorah in 3D. Countless rings in and out.


Core of the menorah quaternion. Hard to zoom to it.

[DarkBeam]

Mandelbrot set with selectable formulas?
New idea?
Just download Fractint, you will get identical results ... And stop pretending to invent and copyright formulas, 'flexible' formulas (some with hundreds of params) were already invented ages ago

[fractower]

Fractint is so cool. I missed a whole season of Survivor while diving into the the various formulas.

Lighten up DarkBeam. It is a long walk to the cutting edge and it is often hard to tell when you have gotten there. Anyone with the math and coding chops Asdam has displayed is one innovation away from leaving the rest of us behind. Or better yet leading us to a new playground.