FLD algebraic space fold

[Alef]

Mandelbrots meets mandelbox.
Hadn't introduced some of formulas to public. Consider this something like forum post and the wikipedia article;)


At first I wanted to implement something like Folding Int Power becouse it's often used. It and the lack of knowledge of JIT acted as catalisator for creation of new.


Algebraic Space Fold:

Code:

Initialisation:
x,y,z= pixel coordinates.

Code:

///Folding
///This step is necessary, as floats interfere with formula and generates something else.
	intFold:=round(Fold);
x := x + abs(x-intFold) - abs(x+intFold); 
y := y + abs(y-intFold) - abs(y+intFold);
z := z + abs(z-intFold) - abs(z+intFold);
w := w + abs(w-intFold) - abs(w+intFold);

x:=x * Scale;
y:=y * Scale;
z:=z * Scale;
w:=w * Scale;

It starts simmilar to Tglad's AmazingBox. I don't know where I got this, this fold is slightly different, and at least one operation faster. Sign could be more important for mandelbrot sets.

Here this folding must use integer numbers. Or else it generates something else. Not shure, but  maybe small remains of float can sway symmetry brakeing to the another direction.   

Code:

///Addition, new version - julia set like
x1 := x + vAdditionX ; 
y1 := y + vAdditionY ;
z1 := z + vAdditionZ ;
w1 := w + AdditionW ;

I added this later just to have more variety.


Large structure of all of these formulas is the same. If in one of the wievpoint one of FLD formula generated something, other will do the same. Zooms here increase level of detail as in 2D mandelbrot set.

Code:

// quaternion pow2
x := x1*x1 - y1*y1 - z1*z1 - w1*w1;
y := 2*x1*y1;
z := 2*x1*z1;
w := 2*w1*x1;

Any of the 3D or 4D multiplications would generate the fractal including all of the mandelbulbs.
However most of 3D mandelbrot used here do not generate nice symmetric fractals.
Instead, raising in power by mathematical correct multiplication formulas whitch don't produces mandelbrot fractals at all generates the most beautiful regular shapes.

Maybe this shows that generation of all of the 3D mandelbrots involves some sort of symmetry brakeing. Trigonometry based bulbs are both regular and fractal but some of symetry brakeing could be trigonometic, realy I don't know.


There were a nice thread by Jos Hendriks that multiplication of complex numbers is movement in complex plane:
http://www.fractalforums.com/index.php?topic=11008.0

I used that a lot. Here the most interesting are:

Code:

///hypercomplex multiplication, changed to "crystal" fractal
// what in 2D complex numbers is when i*i=1.
x:= x1*x1 + y1*y1 + z1*z1 + w1*w1;
y:= 2*x1*y1 + 2* z1*w1;
z:= 2*x1*z1 + 2* y1*w1;
w:= 2*x1*w1 + 2* y1*z1;

in 2D this coresponds to is split complex when i*i=1. Mandelbrot set of it is just square, but it have interesting cubic julia sets and buddhabrot renders.

Code:

[END]
Formula version 1.1

4D
As main fold -
crystal version of hypercomplex pow2 mandelbrot -
where y*y=1 is like x*x=1
Generates diamond like structures.

 * * *
By Edgar Malinovsky

These formulas is alsou quite a fast as it requires just few iterations, average = 3.6 and max = 6.

These formulas
http://www.fractalforums.com/mandelbulb-3d/em-formulas-for-mandelbulb3d/


Here is the pictures:

Standart mandelbulb


Hypercrystal


Pillars of Khwarezmi. A poetic name for bicomplex power of split complex power - the simplest ever:


This was generated by quaternion pow2:



Quat pow 2 hybridased with some simmilar formula:




It is not something very new. Hovewer I hadn't seen these sort of multiplication used like that.

[Alef]

Later I tried to make some mandelbrot set out of this. Just putting + C (pixel) at the end created scenery of stretching. I tried different things for +C until I found that +C/scale generates clean picture - scale in next iteration multiplies pixel values.



With Scale =1 there is mandelbrots and tricorns in cutouts. I had seen something like that here in fractalforums in 2D.


Here is 4D fold with different variables. It don't needs integers here, but sign of x should be important:

Code:

///folds of different lenght
x := x + ( abs(x- FoldX) - abs(x+ FoldX) ) ; 
y := y + ( abs(y- FoldY) - abs(y+ FoldY) ) ;
z := z + ( abs(z- FoldZ) - abs(z+ FoldZ) ) ;
w := w + ( abs(w- FoldW) - abs(w+ FoldW) ) ;

Then

Code:

x1:=x* Scale;
y1:=y* Scale;
z1:=z* Scale;
w1:=w* Scale;

The most basic here is the most beautifull

Code:

x := x1*x1 - y1*y1 - z1*z1 - w1*w1;
y := 2*x1*y1;
z := 2*x1*z1;
w := 2*w1*x1;

+c descaled

Code:

contrscale:= 1/scale;
x := x +  contrscale* PIteration3D^.J1; 
y := y +  contrscale* PIteration3D^.J2; 
z := z +  contrscale* PIteration3D^.J3; 

Well

Code:

4D
Formula version 1.2
(^_^) 
Mandelbrot version of 
FLD - algebraic space fold

With pow2 quaternion mandelbrot inside.

 * * *
By Edgar Malinovsky

Why there is no 3D mandelbrot:
1. No symmetry brakeing in space for canonic multidimensional multiplications.
2. 3D requires different visual qualities than 2D.


Scale = 1

Scale = 1.5


In attachment Scale=1. Kind of like 2D fractalforums pictures about abs and mirrors.

[Alef]

The best versions of multiplication is:

Code:

//simple quaternion pow2
x := x1*x1 - y1*y1 - z1*z1 - w1*w1;
y := 2*x1*y1;
z := 2*x1*z1;
w := 2*w1*x1;

and

Code:

//hyperCrystal - split hypercomplex numbers?
x:= x1*x1 + y1*y1 + z1*z1 + w1*w1;
y:= 2*x1*y1 + 2 *z1*w1;
z:= 2*x1*z1 + 2* y1*w1;
w:= 2*x1*w1 + 2* y1*z1;

With some additions it generates crystals.

Code:

//newtonian triples - Cube power
x:= x1*x1*x1 - y1*y1*y1 + z1*z1*z1 - 6*x1*y1*z1;
y:= ( x1*x1*y1 - y1*y1*z1 - x1*z1*z1 )*3;
z:= ( x1*x1*z1 + x1*y1*y1 - y1*z1*z1 )*3;

Plain mandelbrot is turned by 45 degree extruded set. But alsou generates drum like shapes.
https://jehovajah.wordpress.com/2012/09/13/the-newtonian-triple/

And of corse all of the mandelbulbs. What is less surpising, as they are beautifull alone.

Fold=1, Scale =2
Newtonian triple (not mandelbrot set)

[wes]

Looks like one of my experiments with folding from http://www.fractalforums.com/new-theories-and-research/extending-the-definition-of-a-burning-ship/
 

[Alef]

Yes. When I first saw it, it reminded me those pictures;)

Folding is mirror and should do same function like in Burning Ship. Somewhere here were mandelbrot set with boxfold in 2D. It was just repeating mandelbrots sets and burning ships or tricorns.

[wes]

Somewhere here were mandelbrot set with boxfold in 2D. It was just repeating mandelbrots sets and burning ships or tricorns.

I'd like to see those To me, all the most visually interesting fractals involve folds/mirrors. I'd like to experiment with them some more.

[kjknohw]

Some of these images seem better at preserving 3D in all directions than the mandelbulb at z^2.