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Author Topic: EM formulas for Mandelbulb3D  (Read 1960 times)
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Alef
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« on: May 23, 2017, 04:10:53 PM »

Hi I just made some JIT formulas for Mandelbulb3D.

The best one I think is NOTHING. Becouse if you want to take something off the hybrid. Throught 1 iteration of nothing interferes with fractal and DE.

I would upload few pics here:
http://www.fractalforums.com/gallery/em-formulas-for-mandelbulb3d-pictures/

There are some question about programming too. But later.

 *  *  *

EDIT
Updated formulas with another one package with 9 new formulas and one small update. This one should be better.

 *  *  *

EDIT:
+ 1 new formula, generating buildings.

UPDATE of all other formulas: improved math - faster generation, correction of found errors say of spherefold, expansion of FLD's variable Addition into Julia like multi value addition.
To handle them all is marked with "Formula version 1.1" in commentary.

Everything is in the single zip file including explanation of all the values of building generator.


==========================

EDIT:
UPDATE v1.2:
+ 3 new formulas. FLD mandelbrot set versions and one FLD.
6 formulas - updated and improved, including techna transform completely remade so that it now actualy works and not destroys DE, and  chudobox now is less boring.



--===+++++++++++++++++++===--
UPDATE: Version 1.3:
Added formulas:
+ JIT_EM_N_ElazoBulb.m3f
+ JIT_EM_N_KalezoBox.m3f
+ JIT_EM_U_Superseed_Lp.m3f
Chudobox, Domofractal, FLD_Boxbulb - added internal conditionals to speed up calculation with the default values

I alsou included in formula readmy article about mathematics of  most of the transformations used and the Lp spaces.

* EM_JIT_M3FormulasFULL_v13.zip (52.98 KB - downloaded 79 times.)
« Last Edit: August 22, 2017, 05:04:36 PM by Alef » Logged

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Alef
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« Reply #1 on: May 23, 2017, 04:34:56 PM »

There are some things about JIT through.
It seems to have more operators than listed. sqr, and then all of the round, ceil, floor, trunc. Of them just two is mentioned. sqr(x) probably should be faster than x*x, but that depends on compiler.

What is the TPIteration3D?  embarrass From bulb and Benesi formulas I figured out what is PIteration3D  

And how does one should write complex conditionals, like:
if x >1 && != 5 then x=325; y= 480; else x=520; else if endif. I used 2 conditionals and this could had slowed it down. Well, google will tell this, but not about TPIteration3D.

Alsou, folds seems to be integer based or else they create a lot of noise. None seems to had mentioned this. And DarkBeam had a lot talked about cut formulas destroying DE and generating a lot of noise. But this seems to be trait of float folding. Integers cuts more accurately.

Ha ha I uploaded it reprirocaly;)
« Last Edit: May 24, 2017, 04:59:07 PM by Alef » Logged

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Alef
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« Reply #2 on: May 23, 2017, 05:11:21 PM »

In searching I just found, it was a Newtonian Triples by jehovajah
 http://www.fractalforums.com/new-theories-and-research/newtonian-triples/
They would be a good start for a hybrid.
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Alef
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« Reply #3 on: May 24, 2017, 04:22:09 PM »

These drums are pretty unique. I hadn't seem them before, throught I don't know all of the formulas and combinations.



Its like this.

Fld Newtonian Triples:
Fold = 1, Scale = 2, Addition = 0 (With addition starts spontaneus simmetry brakeing, ha ha.)
Code:
Initialisation by software x,y,z = pixel coordinates

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) ;


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


///Addition
x := x + Addition ;
y := y + Addition ;
z := z + Addition ;


//Cube power
x1:= IntPower (x, 3) - IntPower (y, 3) + IntPower (z, 3) - 6*x*y*z;
y1:= 3*(sqr(x)*y- sqr(y)*z-x*sqr(z));
z1:= 3*(sqr(x)*z+ x*sqr(y) - y*sqr(z) );

 x := x1;
 y := y1;
 z := z1;


end;
[END]

First the space is folded and scaled. And then goes z=z^3 (or z=z^2) by Newtonian Triples until it converges/diverges. They clearly are  mathematicaly rigid construction and do some movement in 3D number space so that it creates this repeating shapes.

Algebraic 3D space fold. Fld is like they write in arabic and hebrew so it's mystical.

Code:
Mandelbulb3Dv18{
g.....9/...g2...w....II...kgJHzHnlp3.HK.K45Z2CAEFLWpYiCuGyn1Dsx8KDY2.50bpxB9.K/E
................................NUg9h1epNz1........A./..................y.2...wD
...Uz6.....M/.../MU0/......p/...U.....E3.....InUmgLr96rD/.......Cfed0dkpXm1....k
z.....cD12../..........wz.................................U0.....y1...sD...../..
.zXaNadD1BDSpNw2Cy1..........CHeSVydXWfjaBgH75kjjun3ZzguQ2kVzEyu6MBJUBsDFfjvzl3q
muHA4YupI/qUzmJsVXC5/QsDU.....2xc.............sD.6....sD..G.0...................
.............oAnAt1...sD....zIg7........................................6....k1.
.....4iSoz1.....tzDe.wzzz5UO.U2.K/...61...EG....m....c3...E4....Q....I14...UdF5Q
...U.GAqPzzhTWszxwJoznNSK/..0c..zrhe..EOf5nlcsxj6.Dbqxm4Iz1...........2/8.EBl35.
.o0FIhT67.2.hE2Jv5G0./..........3I13.wqN51../0biMaOyzsVsP8OPAXzD.czYfTLkRzH/pElz
5TtN...LVxTtT1yD.yXjDUIN0.2.WRvgcbtszKEA8oDU.06..YKaZ6Aeiz1858nQJ6Tnz.ERpBK.iAwj
/6V0.2FAZ.............klrlkCx9yj.6wTJHz2Vz1..wDyb1kljWrlsRGh8p6.LCbDL0AGLC5U.oIH
BpI1YlDi1.U6niekdtrzsTC.5z8S5HeTzXzt.QwfsRg9zxDyb1kljWrldxrzsTC.5z8S5HuTzXzt.Qwf
sRglzxDyb1kljWrl..kzsTCysxlzL0UdIKY6W6W6Ic4.zyTX................................
E....2..F2E.....I....A....UG7FpL3poL4l4NTtINrFrPiFZQdB1.........................
..............................zD..........2.....................................
................................................................................
........................}
{Titel: FldNewtonTriples3}
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Alef
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« Reply #4 on: May 30, 2017, 04:37:27 PM »

Here I did another bunch of formulas. This time with more of algebraic (and artistic) approach and understanding of process. So I think these formulas are better or at least more symmetric. The most intricate folding multiplications don't produces mandelbrot sets at all.
Realy this was more like an artistic inspiration.


I used this. throught I realy don't know what this does. But it's the main spin around for the Saturn:
x1:= ( x-w + y+z )*0.5;
y1:= ( x+w + y-z )*0.5;
z1:= ( x+w - y+z )*0.5;
w1:= ( y+z - x+w )*0.5;

I got it from hypercomplex 4D sinh when I delated all the trig.

In this compiler you can type sqr(x) or x*x. But the x*x is way faster. Probably becouse sqr(x) is turned into intPower(x,2) or even Power(x,2) what is not calculated by multiplication.

As a side I did tricorn conj(z*z)+c out of hypercomplex numbers. It looks just like mandelbulb tricorn. It seems that in 3D tricorn is somewhat universal like mandelbrot in 2D.


* A_fld_hybrid_standartBulb.jpg (63.89 KB, 460x330 - viewed 1039 times.)

* A1_Fld_Electronics.jpg (34.63 KB, 400x200 - viewed 886 times.)

* Fld_Goodplace_HyperCrystal.jpg (57.68 KB, 440x350 - viewed 1034 times.)
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Alef
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« Reply #5 on: June 06, 2017, 11:31:34 AM »

Nex update of these

+ One new formula, fractal house generator - domofractal. I made this after an artistic suggestion.
It is made of modified rectangle Menger Sponge, few folds and other additions. The only thing that can not be make as hybrid is Menger Sponge modification.
I think the best way to generate a house would be to make general shape by x, y, z folds and then load it into Mutagen, turn off formula change and watch. As there is no "one variable adds this and another that". They all interacts with each other.

_Folding1 placed before the formula will add 2 more storeys to the building.

Anyway, who needs some strange mathematical object if there are houses. Architects are free now ha ha. 



And I updated all other formulas: improved math - faster generation, correction of found errors,  expansion of FLD's variable Addition into Julia like multi value addition. Say I had mistake in spherefold of my box. Now it looks smooth without noise.
To handle them all of the updated formulas is marked with "Formula version 1.1" in commentary.

It is pure inspiration based so it is somewhat chaotic. But it is an artistic.


* Domorajon2.jpg (60.07 KB, 600x360 - viewed 376 times.)

* Domorajon3.jpg (53.11 KB, 480x360 - viewed 934 times.)

* Domorajon4.jpg (31.05 KB, 480x360 - viewed 934 times.)

* Fractal_Street.jpg (47.08 KB, 600x360 - viewed 108 times.)
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thargor6
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« Reply #6 on: June 16, 2017, 11:57:33 PM »

Thanks - will include those formulas in the next release and also push them to the repo at github

Cheers!
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Alef
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Posts: 1174



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« Reply #7 on: June 27, 2017, 01:01:49 PM »

Thanks A Beer Cup

Updated and improved few old and added couple of new formulas.
The most important addition is mandelbrot sets of FLDs. It don't sounds as notable found if these formulas is considered julia sets. But it stretched mandelbrot set until pixel was added by +c/Scale -> in next iteration pixel values is multiplied by scale.

With scale =1 cutouts shows scenery very mutch like in 2D. My contest entry is based on it. Scale =1.5 generated ring pattern.

Alsou I added BenesiTomag rotation to Chudobox, and freed it's folds, so that now it is usefull. Before that formula was just too boring.

This should be all. Now its summer again and formula research is mutch more computation hard and heat generating than rendering them. Hard disk monitor reports dangerous heating.





* Chudo_Cave.jpg (67.34 KB, 480x360 - viewed 869 times.)

* FLD_PilarsOfKhwarezmi.jpg (31.12 KB, 360x270 - viewed 781 times.)

* FLD_W_MandelbrotAndTricorn.jpg (32.74 KB, 400x430 - viewed 786 times.)

* FLD_MandelFLD_xari_scale15.jpg (61.04 KB, 480x310 - viewed 943 times.)
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Alef
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« Reply #8 on: August 22, 2017, 04:37:39 PM »

Update 1.3.
This could be a lucky number. I added two more formulas and then internal conditionals to some old to speed up calculation with default values, so not to calculate certain things witch then is not used.

All of them is based on Lp spaces and alsou on http://gannjondal.deviantart.com/ formula tutorials. In that tutorial there were just one formula I found very good and one abox version. It was good example on conditionals. The more later formulas is if not better then more elaborate and these are the most elaborate.

Elazobulb - Lp spaces distorts mandelbulb to Lp shapes. And two power in paralel creates very elaborate bulbs with lots of holes.
Kalezobox - boxfold fractal with Lp space sphere inversion between two radiuses. Spherefold was decunctructed in parts and then reconstructed. It is like Amazing Box but more different than most of it's versions.
They are kind of "smart formulas". Lp spaces is computationaly hard so I added conditionals to pick more lightweight equations when it is possible.

Superseed - I used to explore Lp spaces. It woun't do mutch more.

I alsou included in formula readmy mathematics of  most of transformations used. Becouse it's often the transformations witch constitute formula matters not a formulas itself. I often can decipher transformations present.

Compiler is weird. With conditionals it in certain formulas increases speed when I added conditional witch does nothing.
//=================================
// this useless conditional speed up the next one
// At least on my AMD PC.  
// Maybe becouse of weak compiler, some code optimisation or something like that, I have no idea why.
if Thickness = 4 then
   begin
   end;










Edit:
My mistake. It was https://en.wikipedia.org/wiki/Sigmoid_function  not sigma function I mentioned in readme about functions.

And the benessi Mag angle is
https://en.wikipedia.org/wiki/Magic_angle


* Elazobulb_hands.jpg (60.87 KB, 500x460 - viewed 436 times.)

* Elazobulb_hands_julia.jpg (57.77 KB, 500x500 - viewed 440 times.)

* Kalezobox_sC444vAdd03.jpg (96.4 KB, 560x610 - viewed 91 times.)
« Last Edit: October 10, 2017, 02:35:32 PM by Alef » Logged

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