Formulas of Aleksandrov

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Offline Alef

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« on: December 28, 2017, 01:33:01 PM »
I registered just to post this.  This place just sounds more smart, and maybe it could be good for 3D mandelbrot.

Complex numbers
Formulas of Alexandrov
Alexandrov Georgiy Minkovich, 12.04.1950. russian mathematician In 1982 defended dissertation of doctoral candidate blablabla.

In 2016 with the Monte Carlo method he found general trigonometric representation of complex numbers:

Formula of Muavr looks like this:

There were found interesting properties of complex numbers:

a,b,n - any numbers.

This sounds interesting. Well, trigonometric representation of complex numbers is just what we were doing at But this formula looks longer so could be more accurate representation of angles. And maybe Monte Carlo simulation could help to find 3D analog of complex numbers. And the 3D Mandelbrot.

Hence this must be tested on mandelbulb. I think in more conservative academic enviroment they could not know about mandelbulb.

But I don't think real 3D Mandelbrot would be visualy mutch better than Mandelbulb. It would be just slightly more true mandelbrot. So far IQ-bulb I think looks the best.

Offline Alef

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« Reply #1 on: January 03, 2018, 01:51:00 PM »
Simplest mandelbulb Z->Z*Z+C:
Code: [Select]
r  := sqrt(x*x + y*y + z*z );
th := ArcTan2(y, x) * n;
ph := ArcSin(z/r) * n;
r  := Power(r , n);

x  := r * cos(ph) * cos(th);
y  := r * cos(ph) * sin(th);
z  := r * sin(ph);

x := x + Сx;
y := y + Сy;
z := z + Сz;

From aleksandrovs formulas:
Code: [Select]
r := sqrt(x*x + y*y + z*z );
r := power(r, n);
mcangle:=ArcTan(abs(y) / x;);
mcangle:= (pi*0.5 - pi*abs(x)*0.5/x + mcangle )*n;


   x := x1 + Сx;
   y := y1 + Сy;

When I united these it rendered mandelbulb but angles or signs were messed up. So I did:

I just mirrored y formula to z. In more precise terms z is symmetrical to y.
Code: [Select]
r  := sqrt(x*x + y*y + z*z );
r:= power(r, F_Power);

mcangle:= abs(y) / x;
zangle :=abs(z) / x;
zangle :=ArcTan(zangle);

mcangle:= (pi*0.5 - pi*abs(x)*0.5/x + mcangle +zangle)* F_Power;

x:= cos(mcangle)*r;
y:= sin(mcangle)*r*abs(y)/y;
z:= sin(mcangle) *r*abs(z)/z;

   x := x+ Cx;
   y := y+ Cy;
   z := z+ Cz;

One of synthesis of mandelbulb and this thing:

Code: [Select]
r  := sqrt(x*x + y*y + z*z );

mcangle:= abs(y) / x;
mcangle:=  (pi*0.5 - pi*abs(x)*0.5/x + mcangle ) * F_Power;

theta := ArcCOs(z/r) * F_Power;

r  := Power (r, F_Power);

x  := r * cos(mcangle) * sin(theta);
y  := r * sin(mcangle) * sin(theta) *abs(y)/y;
z  := r * cos(theta) ;

   x := x+ Cx;
   y := y+ Cy;
   z := z+ Cz;

All of these must start by this:
Code: [Select]
/// division by zero!
 if x = 0 then
    x := 1E-15;

 if y = 0 then
    y := 1E-15;

 if z = 0 then
    z := 1E-15;

abs(y)/y is same as sign() but 0/0 creates mess in cutouts. Without cutouts like z=0 or y=0 pixels rarely get value 0.

Rendered these with Mandelbulb3D version 1.99 Formula is in JIT and is at the bottom.
« Last Edit: January 03, 2018, 03:33:00 PM by Alef »

Offline Alef

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« Reply #2 on: January 03, 2018, 01:52:56 PM »
The formulas of Aleksandrov - Muavr formula looks like those for mandelbulb. |y|/y is sign(y) function. And ArcTan(pi/2-sign(y)*p/2+angle) functionaly is folding or mirror.

1st - very unusual formula. It could be all around mandelbrot set in all 2D cutouts made of swirling ribbons. -> Mandelbrot (broken) in all 3D space when mandelbulb it is only along axis (good). But it don't renders so nice is DE sensitive and it requires low bailout like bailot = 4.

2st - More beautifull and complex version of mandelbulb, allways conected but with holes but without no 2D mandelbrot in any cutout. It have very nice julia sets with swirling threads but z value of julia do not affect it. So it is not correct formula.
I got some other sythesis of mandelbulb and Aleksandrovs formula but the result was less nice.

1st Maybe simmilar Monte Carlo simmulation could create expansion of complex numbers in 3D space. Then out of this a 3D mandelbrot set could be made. It could be just like imagined 3D Mandelbrot, swirling braids going out from the object. Or like fluffy connected balls.

2nd I do not understand these angles. Trigonometry is not my favored discipline. So maybe someone with better understanding of triginometry can made better mix of Daniel White and Paul Nylander triplex with Aleksandrovs formula.
 This is my try and error. The most mandelbrotish I could got was fractal formula with full mandelbrot set in one cutout and half of mandelbrot set in other.

Offline Alef

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« Reply #3 on: January 03, 2018, 03:37:21 PM »
Not very artistic but larger pictures of that

I just searched to contact Aleksandrov. I found that he is pretty remarkable in internet with both math, poetry  and aphorisms. 
And his formula for e:
\[ e = \prod \limits _{n=1}^{\infty } \left [ \left (\frac {2n}{2n-1}\right)^{2}{\left ({\frac { 2n^2+n-1}{2n^2+n }}\right)}^{2\,n} \right ] \]
So I invited him to fractalforums.

Offline Kalter Rauch

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« Reply #4 on: April 21, 2018, 09:51:17 PM »
Alef...your Power 3 Juliabulb in the grass
artistically says something about Nature's Powers.

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