On Monday 16 November 2009 19:23:17 fred Lunnon wrote:

I'm less taken by the geometry underpinning this project --- the use of humdrum spherical coordinates results in simple non-isotropic polygonal symmetry of the resulting surface, rather than genuinely polyhedral.

A more sophisticated approach --- already suggested elsewhere --- is to employ quaternions instead of complex numbers, though I doubt the possibilities have been well explored; one major difficulty is analysis in the non-commutative algebra \H, a topic mentioned briefly before in these posts.

It's alleged (on a page linked to from the one we're talking about) that the obvious quaternionic generalization of the Mandelbrot set is basically the ordinary Mandelbrot set plus some rotational symmetry and has no "extra" fractal structure to it. This seems plausible enough on the face of it, but I've not made any attempt to check. I share your feeling that it's all a bit ad hoc and humdrum, and that it should be possible to get as much or more aesthetic beauty with more mathematical elegance. -- g