Kaliset + boxfold = Nice new 2D / 3D fractal

[Kali]

I just tried applying the positive part of Tglad's boxfold (from Mandelbox) to my formula Kaliset (2D), and the results are very interesting.

Not only Julias provides nices images with this, you can find nice shots inside the Mandelbrot version.

This is the code for the iteration:

Code:

x=abs(x)
y=abs(y)
if x>1
   x=2-x
endif
if y>1
   y=2-y
endif
m=x*x+y*y
x=x/m*scale+cx
y=y/m*scale+cy

Exponential smoothing for coloring, as always (but I'm trying new methods and variations for coloring, I'll make a post on this soon)


Some images...

This is the whole set for scale 1:




This is the same with cx and cy values exchanged to reveal new patterns in flat zones, just like original Kaliset.




This is scale 2 (also with cx and cy flip)




And scale <1 works too. This is 0.5:





Some scenes from scale 1 Mandelbrot mode:






Scale 1 - Julia mode:






Scale 2 - Mandelbrot mode:






Scale 2 - Julia mode:






The last one looks like 3D  


I'll post some more variations later. I'll try rotations and other stuff I have in mind.

[Tglad]

Superb images 
One to try would be sphere inverse, translate by some constant complex number, then flip around an arbitrary axis, then +c.
Because the only mandelbrots that have no stretch are those that don't swap handedness. (doesn't extend to 3d tho)

[KRAFTWERK]

I like the blue "X-ray" images Kali! 

[Kali]

Superb images  grin
One to try would be sphere inverse, translate by some constant complex number, then flip around an arbitrary axis, then +c.

Thanks Tom, and I'll try that later 

I like the blue "X-ray" images Kali!

Haha, thanks... That image maps really look like x-ray
Btw, I just made a huge image of flipped scale 3 whole mbrot map - I will upload and post the link later.

---

I've done two more images with smaller and higher scales.

This is 0.05 scale mbrot (used some crazy colors ):




And a 10 scale julia:




I don't know about a 3D version, but what trafassel did with "Mandelbox variant 21" it's close to this... the differences are that I'm using abs function for extended patterns areas, and unrestricted sphere inverse instead of ballfold.

[Kali]

Well, I didn't get a non-streched mandelbrot mode, but adding a translation is nice for "tweaking" the Julias.

I did it this way: abs - boxfold - translation - inversion - addC (or second translation)

A couple of images I made with this:

[yv3]

woha this formula is really nice, very much intresting places in it. Thank you for providing the iteration method, i will integrate it into my tool as "Kaliverse". But its without boxfold because my bad math knowledge. I added 4 "modifiers", here's the code :

Code:

	
	static double	x, y, m;
	static int		n;

	x=m_CustomRange2;
	y=m_CustomRange2;

	for(n=0; n<m_Iterations; n++)
	{
		x=abs(x);
		y=abs(y);

		if(x>m_CustomRange4) x=m_CustomRange3-x;
		if(y>m_CustomRange4) y=m_CustomRange3-y;

		m=x*x + y*y;

		x=x/m * m_CustomRange1 + m_StartX; 
		y=y/m * m_CustomRange1 + m_StartY; 

		if((x*x)+(y*y) > m_Magnitude )
		{
			m_C=int( float(n) * m_ColorPerIteration ); // Return color index from range 0 to (m_NumColors-1)
			return;
		}
	}

	m_C=0;

[trafassel]

Kali,
each time, when I think, that all possible types of fractals are examined, you publish some pictures like, in Reply#4. Thank you for sharing the pictures and formulas.

 

[Kali]

@yv3: Thanks for your comment... I'm don't have too much math knowledge either, but I'm always borrowing it from the guys here who really have some

"Boxfold" is what you just did:

Code:

if(x>m_CustomRange4) x=m_CustomRange3-x;
if(y>m_CustomRange4) y=m_CustomRange3-y;

But other values instead of >1 and 2- produces discontinuities, at least with the values I tried.

For best results, you would want to use an inner coloring, instead of escapetime.
To implement exponential smoothing, do this:

Declare this variables (double): sum, mold
Then replace this code:

Code:

if((x*x)+(y*y) > m_Magnitude )
		{
			m_C=int( float(n) * m_ColorPerIteration ); // Return color index from range 0 to (m_NumColors-1)
			return;
		}

with this:

Code:

sum+=Math.Exp(-1/Math.Abs(mold-m))
mold=m

Then use the result of the variable sum after all the iterations as a color index. I suggest multiplying by a "color density" factor and then map the results to m_NumColors with "mod" function (I don't remember how it's implemented in C, but I think is with symbol %)

It should be like this, but I'm not sure:

Code:

m_C=int(sum*m_colordensity) % m_NumColors

---

@trafassel: You are welcome, thanks. I don't know if "Kaliset" and the derived formulas like this are such a novelty or really a great finding, but I'm glad you like it and I really appreciate your words...
Remember that "the possibilities are infinite", so I guess there is still a lot of things to discover 

[yv3]

@Kali:
I always thought that the exponential smoothing does not work on fixed palettes. Thank you very much! It looks beautiful!

[matsoljare]

Don't forget trying to vary the other parameters by the axis, as well....

[KRAFTWERK]

I love the "electric triangle" and the other image too in your last post Kali! 

[dwsel]

Recently I've been implementing various formulas of yours using good old Tierazon with its tiny formula parser. And then I came to the boxfold part:

Code:

if x>1
   x=2-x
endif

which literately says for me:

Code:

if (x>1)
   x=2-x;
else
   x=x;

It looks like boxfold can be calculated without if clause, just like:

Code:

-abs(x-1)+1;

Won't dropping 'if' in favour of 'abs' speed up calculations?

I'll see if it's a good path and post some images if I get kaliset + boxfold implemented in this way.
Have a nice day!

dwsel

[Kali]

Recently I've been implementing various formulas of yours using good old Tierazon with its tiny formula parser. And then I came to the boxfold part:

Code:

if x>1
   x=2-x
endif

which literately says for me:

Code:

if (x>1)
   x=2-x;
else
   x=x;

It looks like boxfold can be calculated without if clause, just like:

Code:

-abs(x-1)+1;

Won't dropping 'if' in favour of 'abs' speed up calculations?

I'll see if it's a good path and post some images if I get kaliset + boxfold implemented in this way.
Have a nice day!

dwsel

Well, I think it's a more elegant way to do it, but I'm not sure about performance improvement. If condition x>1 is false, no further operations are needed, x remains the same. but using your method, I count 4 math operations and a variable asignment performed on every iteration. Even when I don't know exactly how expensive is an "if" statement compared to abs, sum and change of signs, I think it should be at least the same. Anyway it's a good observation you made.

[DarkBeam]

I have coded a version of this fractal for mandelbulb but looks different. Tomorrow look again

[Kali]

I have coded a version of this fractal for mandelbulb but looks different. Tomorrow look again