A different 3d mandelbrot idea

[Tglad]

Hi, I'm new to this forum. I've been looking at the marvellous 3d mandelbulbs rendered in the last few days. Awesome (a shot from inside a cave looking out would be cool).
I wonder whether the remaining whipped cream effects are because the coordinate system has singularities... if many values (e.g. all thetas when omega is 180 degrees) have the same physical position then they will get the same c in the formula, which will dilute out the fractal variation in places... perhaps.

Anyway, there is no mapping (or direction to move) on a sphere that is singularity free... (the hairy ball problem). So it would be really interesting to try it on a donut which is singularity free, theta is around the small radius and omega angle is around the big radius.

So each iteration:
  theta *= n
  omega *= n
  vec.x = cos(omega)*(2-cos(theta))
  vec.y = sin(theta)
  vec.z = sin(omega)*(2-cos(theta))
  vec *= pow(vec.magnitude, n-1) // raise length from origin to power n
  vec += c

If would be interesting to see what this would look like. I have no way to try it out, let me know if anyone can help.
 Thanks,
   Tom.

[cbuchner1]

This idea has merits. Someone should investigate.

[kram1032]

several atempts on tori where made sofar and the results where that it just looked like the spherical mandelbrot but on a torus with more or less similar amount of whipped cream...

Actually I don't really care about the whipped cream all that much. I doubt there would be too much to do about it.
under circumstances, whipped cream is unavoidible for higher dimensions...

there was an other try with tetrahedron-foldings, resulting in a tetrahedralish Mandelbulb. It has less whipped cream overally but it's more or less concentrated on the bottom side of the tetrahedron...

I guess, though, in the end it's relatively unimportant, on which geometric object to map a 3D-Madelbrot.
In the end, it will just look the same basically but wrapped around that object.

I think what wasn't tried sofar was an alternating geometric system like...

spherical z²

+c

toric z²

+c

spherical....

or

spherical z²

+c

toric z²

-c

spherical....

or something alike...

[mrrgu]

Hi

Great idea;D

Started to implement this..
But realized the way my code wotks is that it starts with cartesian coordinates...
Converts to polat modifies angles and length vector and then back to cartesian.

So is there an inverse for the toroid equations ?

So I can get from (x,y,z) to (vec.magnitude,omega,theta)

Can not find this on the web.. maybe not possible? 

Hi, I'm new to this forum. I've been looking at the marvellous 3d mandelbulbs rendered in the last few days. Awesome (a shot from inside a cave looking out would be cool).
I wonder whether the remaining whipped cream effects are because the coordinate system has singularities... if many values (e.g. all thetas when omega is 180 degrees) have the same physical position then they will get the same c in the formula, which will dilute out the fractal variation in places... perhaps.

Anyway, there is no mapping (or direction to move) on a sphere that is singularity free... (the hairy ball problem). So it would be really interesting to try it on a donut which is singularity free, theta is around the small radius and omega angle is around the big radius.

So each iteration:
  theta *= n
  omega *= n
  vec.x = cos(omega)*(2-cos(theta))
  vec.y = sin(theta)
  vec.z = sin(omega)*(2-cos(theta))
  vec *= pow(vec.magnitude, n-1) // raise length from origin to power n
  vec += c

If would be interesting to see what this would look like. I have no way to try it out, let me know if anyone can help.
 Thanks,
   Tom.

[BradC]

There's a version of toroidal coordinates described in this thread: http://www.fractalforums.com/theory/toroidal-coordinates/

See the original message and also reply #4.

[Nahee_Enterprises]

    Hi, I'm new to this forum.  I've been looking at the marvellous
    3d mandelbulbs rendered in the last few days.  Awesome....

Greetings, and a belated Welcome to this particular Forum !!!     

Always interesting to have new ideas put forth for people to discuss and try out.  That is what this forum is all about!!!