### Ideas on 3d Apfelmännchen

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#### jukzi

• Fractal Freshman
• Posts: 9

#### Ideas on 3d Apfelmännchen

« on: May 22, 2020, 10:12:29 PM »
Hi,
As far as i know its proven to be impossible to calculate a Mandelbrot in 3D as in 2D.
However it should be possible construct a solid that looks like the inner set by replacing circles with spheres. The circles in the 2d mandelbrot look like they are placed like Ford circles projected on a circle (or cardioid for the root). So i tried it using Ford spheres projected on a sphere (/extruded cardioid). And of cause recursivly subsituting any sphere with the same.
I did not find similar work like that. What do you think about the idea?
The main task to get a more beautiful picture would be to get an idea how to draw the spheres:
1. The spheres are not dense enough to look like a solid surface.
2. The main beauty of a Mandelbrot are the colors outside the actual set. In 3d one could calculate a color value for each voxel based on the distance to the next sphere. But the outside of a 3d solid is solid so one could not see it anyway.
3. One can also simply add satellite cardioids above every sphere however i have no idea how to place them so that beautiful spiral forms appear
Any Ideas how to improve?

#### jukzi

• Fractal Freshman
• Posts: 9

#### Colored spheres

« Reply #1 on: May 22, 2020, 10:27:50 PM »
another picture with spheres colored by coordinate on the parent surface

#### hgjf2

• Fractal Friar
• Posts: 111

#### Re: Ideas on 3d Apfelmännchen

« Reply #2 on: May 23, 2020, 08:12:37 AM »
Fine 3D Mandelbrot!! Great!

Can you show whick is the formula for this fractal?
I yet searching formulas for true 3D Mandelbrot set

#### mclarekin

• Fractal Freak
• Posts: 710

#### Re: Ideas on 3d Apfelmännchen

« Reply #3 on: May 23, 2020, 09:40:28 AM »
super cool

it looks a bit like a juliabulb, but it is clearly different

#### jukzi

• Fractal Freshman
• Posts: 9

#### Re: Ideas on 3d Apfelmännchen

« Reply #4 on: May 23, 2020, 10:06:31 AM »
Can you show whick is the formula for this fractal?

I wrote a hacky java program for it. See attachment.
You can run it with "java -jar apple3dSource.zip" or compile the source yourself

#### mclarekin

• Fractal Freak
• Posts: 710

#### Re: Ideas on 3d Apfelmännchen

« Reply #5 on: May 23, 2020, 10:45:01 AM »
it should be possible to iterate and render this in Fragmentarium using raymarching and distance estimation. As it is just constructed from primitive spheres following a rule.

#### jukzi

• Fractal Freshman
• Posts: 9

#### Re: Ideas on 3d Apfelmännchen

« Reply #6 on: May 23, 2020, 11:04:44 AM »
Im not familiar to Fragmentarium. But as it sounds like a good idea - can you do it please? I dont even know how to draw a simple sphere there.

#### mclarekin

• Fractal Freak
• Posts: 710

#### Re: Ideas on 3d Apfelmännchen

« Reply #7 on: May 23, 2020, 11:30:15 AM »
i can make a simple sphere but i would have to learn about cardinoids and the mandelbrot set. I will see if I can interest someone who is more clever than me

• Posts: 159

#### Re: Ideas on 3d Apfelmännchen

« Reply #8 on: May 23, 2020, 10:49:10 PM »
i can make a simple sphere but i would have to learn about cardinoids and the mandelbrot set. I will see if I can interest someone who is more clever than me

I was curious, so I tried out a few formulas for 3d cardioids.
If you're using ray-marching, I think this as an inside/outside test would work quite well. I looked up a polar formula for cardioids and approached that as a surface of revolution.

It can be easily modified to give distance estimates too, if that's more useful.

Code: [Select]
function isInside(x,y,z){   var r = (x^2+z^2+y^2)^.5   var xzR = (x^2+z^2)^.5   var theta = atan(xzR/p.y);   if(p.y < 0){theta = theta+3.1415} //(this part can often be replaced by atan2(x,y), when available)   if (r <  1 - cos(theta)) return true   else return false}
x, y, and z may have to be swapped around to get proper orientation.

#### jukzi

• Fractal Freshman
• Posts: 9

#### Re: Ideas on 3d Apfelmännchen

« Reply #9 on: May 24, 2020, 02:47:42 AM »
great. In Fragmentarium (glsl) that would be like
{code}
float cardioid(vec3 p){
float r = length(p);
float xzR = length(vec2(p.x,p.z));
float theta = atan(xzR,p.y);
return (r -  (1.0 - cos(theta)));
}
{code}
But the distance estimate looks wrong inside the hole. So if you can please provide a working distance formula for the hole...

• Posts: 159

#### Re: Ideas on 3d Apfelmännchen

« Reply #10 on: May 24, 2020, 02:59:47 AM »
Hm, are you sure? Other than what looks to me like rendering errors, it seems right to me... What seems wrong about it? Maybe I just can't see it from the one image.

#### mclarekin

• Fractal Freak
• Posts: 710

#### Re: Ideas on 3d Apfelmännchen

« Reply #11 on: May 24, 2020, 06:15:05 AM »
return 0.5 * (r -  (1.0 - cos(theta))); // where 0.5 was a guess

or

uniform float DEtweak; slider[0,0.5,1.0]

return DEtweak  * (r -  (1.0 - cos(theta)));

or

reduce fudgefactor on Raytracer tab

#### hgjf2

• Fractal Friar
• Posts: 111

#### Re: Ideas on 3d Apfelmännchen

« Reply #12 on: May 24, 2020, 07:23:21 AM »
Thanks!

#### jukzi

• Fractal Freshman
• Posts: 9

#### Re: Ideas on 3d Apfelmännchen

« Reply #13 on: May 24, 2020, 07:53:57 AM »
return 0.5 * (r -  (1.0 - cos(theta))); // where 0.5 was a guess
Doesnt work when looking from all angles: Near the pole the error getting too big :-(. Furthermore for a recursive definition we need exact values anyway.

• Posts: 159

#### Re: Ideas on 3d Apfelmännchen

« Reply #14 on: May 24, 2020, 08:09:24 AM »
Well, I looked up info on the main cardioid of the Mandelbrot Set specifically, it looks like it shifts to the right by exactly 0.25, leaving the cusp on the right side at positive 0.25, and its size is multiplied by exactly 0.25 so that it's left side is at -0.75. These values should extend into 3d pretty easily to get an exact position and size for the surface-of-revolution equivalent.

I wouldn't really know what to do to make it render more accurately after this. The math should be solid from this point, so maybe it's just a matter of turning up the settings and choosing an angle that hides the parts that still don't render correctly.

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