Hi
I talked to Twinbee about this on deviantart and he told me to come here, so here I am.
Hi lycium.
I had one idea for why the standard Mandelbrotset behaves that assymmetric in this 3D-Variant.
Most likely, that is, because it already does so in the 2D-Version.
Well, nearly.
In the 2D-Version, z²+c produces only mirror-symmetries and relative to the zero-point, inside the caridode, one could say, it features "single-axial radial symmetry"
z³+c features double mirror symmetries or "2-axial radial symmetry"
z
^{4}+c features 3-axial radial symmetry and so on
z
^{n}+c in general features an (n-1)-axial radial symmetry.
As things get twisted into higher dimensions, things with only one axis of symmetry start to look odd. they twist in a rather strange way and that happened to the z²+c-variant.
Though, I have a couple of ideas:
Right now, you basically use two totally unrelated angles and transform them in one and the same way to get your results.
What if you actually search for a relation between them?
One way to do this would be to interpret the two angles as complex unit vectors.
your formula for the angles then would be:
(r,theta,phi)
^{n} =
(r
^{n},2*pi*cos(arg(z))*n,2*pi*sin(arg(z))*n)
Other relations could be thinkable... though I only can give an example which probably does not work directly.
http://www.wolframalpha.com/input/?i=sqrt%28%28x%29^2%2B1%2F%28x%29^2%29%3D2
the idea here is that one angle always has to be the inverse of the other angle. But as the angles are real-valued, so that complex solutions wouldn't work, you only get four different values to work with.
A different way to get nice angular relations might be via the riemann-sphere.
Taking the radius as is but looking at where the iteration would have been found on the riemann-sphere, express it as the two angles (the radius of 1, as said is overwritten by the resulting radius) and plot the new position accordingly.
Though, as on the bottom of that sphere, you have zero and on the top you have infinity, this might lead to a different kind of distortion. - 3D but not as spherical as the original intention^^
A very exotic variant would be a weird clifford algebra without any real part.
Something like
i*i=-j
j*j=-e
e*e=0
i*e=j
j*e=i
i*j=e
i*j*e=e*e=0
no idea if that would actually describe something usable but it might be worth a try
Also, for that, you would probably need to figure out, if it at all works, how it works:
is j*i=i*j or not?
Ok, those where some random quick ideas. I hope, you like them