• May 18, 2021, 04:34:33 PM

### Author Topic:  Mandelbrot 3D: Mandelnest  (Read 3854 times)

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#### Jeannot

• Fractal Phenom
• Posts: 56
• Mathematical philosophy
##### Mandelbrot 3D: Mandelnest
« on: February 10, 2021, 02:09:21 PM »
Hello
This family of 3D fractals has complete cubic symmetry (unlike Mandelbulb). Here images power 11.
Ever heard of it ? Already actually known ?
Otherwise I am available to provide the code to those who would like to improve the rendering.

There is à convex dual form.

#### Jeannot

• Fractal Phenom
• Posts: 56
• Mathematical philosophy
##### Re: Mandelbrot 3D: Mandelnest
« Reply #1 on: February 10, 2021, 04:59:42 PM »
one convexe more

#### 3DickUlus

• Posts: 2217
##### Re: Mandelbrot 3D: Mandelnest
« Reply #2 on: February 11, 2021, 01:40:27 AM »
yess please code would be nice

#### Jeannot

• Fractal Phenom
• Posts: 56
• Mathematical philosophy
##### Re: Mandelbrot 3D: Mandelnest
« Reply #3 on: February 11, 2021, 11:56:43 AM »
Thanks. Here is the code.
The method is inspired by projective geometry (homogeneous coordinates), but it is very simple.
I use the usual Mandelbrot formula: Zn+1= Zn^p + c in space R3.
The multiplication in R3 is defined on the angles: at each iteration Zn is normalized and so we have for the 3 components -1<= x < =+1.
We can therefore naturally associate the 3 components with sines or cosines: x= cos(a0) and a0=acos(x).
Then classically Z^p is done by an+1= p x an and the module is increased to Mn+1= Mn^p.

Z=normalize(Z);
a0=asin(Z[0]);  // convex form, or a0=acos(Z[0]) for concave dual form.
a1=asin(Z[1]);
a2=asin(Z[2]);
a0=P*a0;
a1=P*a1;
a2=P*a2;
M=pow(M,P);

Z[0]=sin(a0);
Z[1]=sin(a1);
Z[2]=sin(a2);
Z=normalize(Z);
for(int a=0; a<3;a++) Z[a]*=M;
Z=Z+C;
M=norm(Z);
N++;

Of course, the rendering can be significantly improved. I use my own graphics program openGL not very sophisticated.
Notice to candidates.
I take this opportunity to introduce myself: I am French (city of Poitiers), I am 61 years old and I have spent a lot of time trying to define a complex multiplication for R3, without of course achieving it

#### 3DickUlus

• Posts: 2217
##### Re: Mandelbrot 3D: Mandelnest
« Reply #4 on: February 12, 2021, 12:37:54 AM »
Thanks for sharing, this looks really interesting, I haven't seen this variation before.

edit: oh, and welcome to the forum

#### Jeannot

• Fractal Phenom
• Posts: 56
• Mathematical philosophy
##### Re: Mandelbrot 3D: Mandelnest
« Reply #5 on: February 12, 2021, 10:35:17 AM »
Thanks
Yes, cubic symmetry seems to be new and bulbs are different, so I believe it is not a simple variation of Mandelbulb.

#### Pupukuusikko

• Fractal Freshman
• Posts: 6
##### Re: Mandelbrot 3D: Mandelnest
« Reply #6 on: February 12, 2021, 06:32:34 PM »
Hello, surely a nice looking bulb with a simple formula. I tried to implement it in Fragmentarium, and did achieve similar looking bulbs.
In Fragmentarium I think the formula can be rewritten as:
Code: [Select]
  r = length(z);  z = normalize(sin(Power*asin(z/r))) * pow(r,Power);
Complete but crude frag with Buddhi's DE is attached, maybe the Fragmentarium masters around here will refine / correct it

#### claude

• 3f
• Posts: 1870
##### Re: Mandelbrot 3D: Mandelnest
« Reply #7 on: February 12, 2021, 10:21:27 PM »
this seems different from the Mandelbulb approach because it defines a 3D power without trying to do a 3D multiplication first.

I tried to do it in FragM with vectors of dual numbers for automatic differentiation for DE, but the 20000 lines of GLSL (after preprocessing) take 5mins to compile which makes debugging the blank screen impossible

#### lycium

• Posts: 77
##### Re: Mandelbrot 3D: Mandelnest
« Reply #8 on: February 12, 2021, 11:26:12 PM »
I tried to do it in FragM with vectors of dual numbers for automatic differentiation for DE, but the 20000 lines of GLSL (after preprocessing) take 5mins to compile which makes debugging the blank screen impossible

If only you had some other way / project to debug it...

#### Sabine62

• 3f
• Posts: 1205
##### Re: Mandelbrot 3D: Mandelnest
« Reply #9 on: February 13, 2021, 12:51:21 PM »
Hello and welcome, Jeannot:)

Mangled Pupukuusikko's code to run with simple DE, not sure if it's an improvement, though
Rendered concave and convex pow 3-16 for anyone to check if the code is correct,c.q. does it doe what it should

Update: at pow10 no difference between convex and concave...

To thine own self be true

#### Jeannot

• Fractal Phenom
• Posts: 56
• Mathematical philosophy
##### Re: Mandelbrot 3D: Mandelnest
« Reply #10 on: February 13, 2021, 06:37:24 PM »
Hello Sabine62,

Sure it’s an improvement ! Thanks a lot, beautiful shaded and woody colors. I like much P12 and P16.
I don’t know about you, but I find them really beautiful, and perhaps are they original (I hope so) .
The orientation (1,1,1) in the corner, give also a very nice look if you have time.

Your editions generally correspond to mine, but we have one difference: on my side I observe a dissymmetry on the cosine at even powers, and you on the cosine odd power if I interstand well, on those you call convexe and I concave (I call convexe the sinus because an egg appears in the middle of the nests )  Strange ! For example, the 11 ones sine and cosine are completely symetric in my edition above, and not in yours. The power 2 cosine give something dissymetric also for me (but not really interesting). For me power 10 cosine is dissymetric and so different from sine.

#### 3DickUlus

• Posts: 2217
##### Re: Mandelbrot 3D: Mandelnest
« Reply #11 on: February 13, 2021, 08:00:49 PM »
@Pupukuusikko I saw that optimization too, down to one line using vec3
@Pup & @Sabine62 ha, you beat me to it, nice work
@Claude were you using "Raymond" ? I'm seeing about 800 lines of glsl generated by standard includes.
@Jeannot     a new toy to play with tnx!

#### Jeannot

• Fractal Phenom
• Posts: 56
• Mathematical philosophy
##### Re: Mandelbrot 3D: Mandelnest
« Reply #12 on: February 13, 2021, 09:08:36 PM »
->Sabine62: I understood the problem of symmetry: logically if we use acos to get the angles, we must conversely use cos to calculate x,y,z and not sin... (I forgot to mention it in the code) but if we mix acos and sin (as you seem to have done) it is interesting anyway!

->3DickUlus: tnx, I’m not familiar... but I can get to know

#### Sabine62

• 3f
• Posts: 1205
##### Re: Mandelbrot 3D: Mandelnest
« Reply #13 on: February 13, 2021, 10:57:46 PM »
Sin/cos: Indeed, the behaviour is just the other way round

Thank you very much for sharing your code, lovely results with Julia too!

Below a pow3 Julia as example plus complete .frag (unpack into one separate folder to run)

#### hgjf2

• Fractal Feline
• Posts: 169
##### Re: Mandelbrot 3D: Mandelnest
« Reply #14 on: February 14, 2021, 08:52:31 AM »

Nice image of fractal almost Julia 3D set (with some stretches) , but is one of the fractals (Julia like) as in the former fractal site WWW.FRACTAL3D.COM by Ashara from Nagoya University. The webpage with (Julia like sets) wasn't ready.
Least you was revealed that couldn't given this former site.
COOL!
Fractal researcher

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