The basic idea is much like the complex methods which perform an angular remapping of the unit sphere with a radial power.
Z <- S(Z/|Z|) * |Z|^p +C.
The spherical remapping however is specifid geometrically. It starts with a identity mapped unit sphere. Then circlular patches are added to the sphere. Each patch remaps the sphere and may or may not map its self. The center of the patch maps the opposite side of the sphere.
The inverse mapping points are analogous to the negative real axis in the power 2 Mandelbrot and tends to promote the creation of orbits. It also results in a kind of fractal tower because it pulls higher values of C into the convergent set. (At this point I regret chosing fractower as my user name because I cannot talk about the algorithm without feeling self referential.
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The identity mapped regions suppresses the formation of orbits and is analogous to the positive real axix of the power 2 Mandelbrot. These regions are 1D or 2D which differes from complex bulbs which are only identiy mapped at discrete points.
The identity mapped and near identity mapped regions end up suppressing fractals over much of the surface. Fractals are found only where orbits form by jumping from tower to tower. This ends up recursively copying and distorting images of the towers. Magnifications of these regions show the same pattern of smooth areas with an occasional tower recursion.
I suspect that some areas that appear to be fractal may actually have only finite orbits. I am not sure how to evaluate this yet.
This construction has lots of knobs to play with. I hope at some point to create a 3D fractal version of the old biomorph game.
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