Formula suggestion.

[Alef]

I think I found noteworthy formula.
Idea is like abox, but julia sets are very dissimilar, and mset alsou are different, throught my abox alsou was a bitt non ordinary. 4 variables IMHO is enought and here alsou negative min Radius values can be used. Only negative trait is strong dependancy on bailout value. I coined the name "baguabox" becouse sometimes it looks like bagua on korean flag and becouse then in my file it stands second after amazingbox;)

Mset:

Some side zoom:

Julia set

(My monitor have some problems with red colour.)

Code:

Baguabox (quaternion) {

// Mandelbox alternative.
// By Edgar Malinovsky 25.07.2012.
// don't forget credits.

parameter real bailout;
parameter quaternion julia;
quaternion C;
real zx, zy, zz, temp, modulus, radius;
parameter real add, Min_R, Scale, lenght;

parameter int settype;

void init(void)
 {
     if (settype=="Julia Set")
    {	
	z = pixel;
	C = julia;
    }
    else
    {
	z= 0;
	C= pixel;
    } 

}		

void loop(void)	
{
zx=real(z);
zy=imag(z);
zz=part_j(z);



 //Folding by pow 8 modulus having unit circle shaped as octagon pillow.

modulus =((zx)^8 + (zy)^8+ (zz)^8)^0.125;

if (modulus != 0)
{
	if (zx > lenght)
	{
	zx=-zx/modulus -add;
	}
	else if (zx < -lenght)
	{
	zx=-zx/modulus +add;
	}

	if (zy > lenght)
	{
	zy=-zy/modulus -add;
	}
	else if (zy < -lenght)
	{
	zy=-zy/modulus +add;
	}
	
	if (zz > lenght)
	{
	zz=-zz/modulus -add;
	}
	else if (zz < -lenght)
	{
	zz=-zz/modulus +add;
	}	
}


//spherefold. Similar to TGlads original, but allows to use a negative radius.

radius =sqr(zx) + sqr(zy) + sqr(zz);

if (radius < abs(Min_R) ) 
{
temp = Scale/Min_R;
}
else
{
temp = Scale;
}

//generating z value. 

	z = quaternion(zx, zy, zz, 0)* temp + C;

}
bool bailout(void)
	{
	return(  |z| < bailout );
	}
void description(void)
	{
		this.title = "Baguabox";

		bailout.caption = "Bailout Value";
		bailout.default = 15.0;
		bailout.min = 0.5;

    settype.caption = "Set type";
    settype.enum = "Mandelbrot Set\nJulia Set";
    settype.default = 0;
	
separator.label1.caption  = "Shape depends on folding parameters and bailout.";		
 	add.caption = "Add value";
	add.default = 0.1;
	
	lenght.caption = "Lenght";
	lenght.default = 1.8;
	
	Min_R.caption = "Radius";
	Min_R.default = 0.5;
	
	Scale.caption = "Scale";
	Scale.default = 1.5;	

    julia.caption = "Julia Parameter";
    julia.default = (0.35,-0.35,2.3,0);
    julia.hint = "4th value is not used";
    julia.visible = (settype=="Julia Set");
	}
}

p.s.
More about this in here:
http://www.fractalforums.com/new-theories-and-research/new-3d-pattern-box/
The same here:
https://sites.google.com/site/3dfractals/baguabox
And at the bottom of my formula file:
http://www.chaospro.de/formulas/display.php?fileid=222

Edited: changed spherefold, so result is wqithout messy regions.

[Alef]

Maybe aslou this, throught of corse it would have more niche use:
http://www.fractalforums.com/new-theories-and-research/imho-reason-behind-no-3d-mandelbrot/



https://sites.google.com/site/3dfractals/discussion

Don't know, but power 4 shape is funny:


And an inside rendering:

[Alef]

Probably most of folks had seen this.
http://www.fractalforums.com/theory/an-old-formula-revised/msg52114/#msg52114
But anywhay, it is pretty cool fractal.