New 3D pattern box

[Alef]

_{After understanding how tglads formula works, there could ne found other boxes. Could be that square and circle nonlineary interferating creates all the variance, at least in positive scale.}

At first I wanted implement rotated  fold suggested by Kali in some kaliset threads. It didn't worked however, two circle - folds generated nothing, but then I found different fold. Modulus function deffined with power 8 have unit circle turned into octagonal pillow, going nicelly with spherefold, much better than spherefold in another thread.
http://www.fractalforums.com/new-theories-and-research/higer-dimension-modulus-or-something-odd/

This fold allowed add to additional number inside of fold, with some values aproaching mandelbox like pattern. As this works slower than tglads formula, remowed squares of min radius in spherefold, IMHO pretty unnesesary function executing two sqr operators in one iteration step. It less depends on renderer settings (maybe less discontiniuities with add=0), but equaly depends on maxiter and bailout.

Thus algorithm is following:
3D vector z=VECTORoctafold(z)*REALscale*REALcirclefold(octafold(z))+ VECTORc
Throught formula can work without circlefold, as when Min_R = 0.

Code:

//iterate:

zx=real(z);
zy=imag(z);
zz=part_j(z);

 //Folding by pow 8 modulus having octagon pillow shape as unit circle.
modulus =((zx)^8 + (zy)^8+ (zz)^8)^0.125;

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

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

//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;

p.s.
Now are rendering picttures.

EDITED: changed spherefold
EDITED: once more changed spherefold.

[Alef]

I was thinking about the name for the formula. MandelBoxVarEdgar would sound pleasing for the ears of programmers but quind of lacks poetics;) Patterns somewhat reminds Bagua (thing on South Korean flag), so formula could be called Baguabox. And in the world there are lots of asians, especialy around deviantart.com;) Third, hanging out with russians means that north asian mimics, like those of south koreans, looks quind of fammiliar, much more so than massive gesticulation of french. There are even russian rock legend, who is ethnic korean.


Not tested much, just picked best of 12 pictures. Julia set looks the most interesting, but were the fastest render, so of a bitt of a least quality.

Add=0.4 Lenght=1.8 minRadius=0.2 Scale=1.5


Add=0.15 Lenght=1.8 minRadius=0.5 Scale=1.5



Add=0.3 Lenght=1.7 minRadius=0.5 Scale=1.365




Add=-0.2 Lenght=1.9 minRadius=0.5 Scale=2




Add=0.25 Lenght=2 minRadius=0.5 Scale=1.5




julia baguabox Add=0.1 Lenght=1.8 minRadius=0.5 Scale=1.5  julia seed = 0.25, 0.15, 0.2







p.s.
The same here
https://sites.google.com/site/3dfractals/baguabox
And the full code here:
http://www.chaospro.de/formulas/display.php?fileid=222

[Alef]

Alsou switched to a bitt different spherefold using idea from Trafassel simplified code. Code is similar to TGlads spherefold, and visual result alsou is very similar, largest difference is more pronounced circle, but this spherefold allows to use negative min Radius. And it is slightly faster, thus balancing slower first fold. Spherefolding with negative radius gives nice squares instead of normal circles, so in formula I used this fold.

Code:

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

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

if (radius < 1 ) 
{
temp = Scale/Min_R;
}
else if (radius < Min_R && radius !=0) 
{
temp = Scale/radius;
}
else
{
temp = Scale;
}

// generating z value. 
z = quaternion(zx, zy, zz, 1)* temp + C;

Negative radius box:
a11  Add= 0.1  Lenght= 1.8  Radius= -0.65  Scale= 1.5


Here are the same fractal spot using my new spherefold. Differences are pretty small:




Alsou an previous julia set box, a bitt better render.

[Alef]

In my in formula file "Baguabox" goes nicely second after "Amazingbox" (throught that is a bitt different amazingbox) as it were intended. Probably one of differences is lack of any DE. Result of negative min radius considerably differs from negative scale and radius parameter is less influencial than scale.

Here is  Add=0.4  Lenght=1.8  Radius=0.2  Scale=1.5



The same fractal, zooming out and increasing bailout value I got this more chaotic shape:



More zooming out and widescreen camera resolution.



I think, julia set like this would be pretty ineteresting to animate.
 Add=0.1  L=1.8  Radius=0.5  Scale=1.3  Julia seed=0.35; -0.35; 2.3



Zoomed far out julia set. Add=0.4  Lenght=1.8  Radius=0.25  Scale=1.5  Julia seed=0.5; 0.5; 2.5


p.s.
Julia sets probably are the most interesting, much different from any other sets encountered. Only negative is formula dependance on bailout value.

[Alef]

Baguabox_sec_2        AddValue=0     Lenght=1   Radius=0.5  Scale=2


Baguabox_sec_3        AddValue=0  Lenght=0.5   Radius=0.5  Scale=3

[Alef]

I changed to spherefold to different one, so now it is much less chaotic in areas, where it was messy. With abs function I kept possibility of negative scale Min radius.

Bagua like spot:
 


Using standar mandelbox values lenght =1, scale =2.





Far outside videscreen view:

[Alef]

Very colourfull side zoom. Levels of detail depends on maximal iteration. So this is smoother.




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

[Alef]

Some far out side structure of fractal:


Julia Set of this fractal. Some far outside structure. Julia sets are much more orderly than msets:


And Julia set more in the middle of coordinate system: