David wrote: << This is a question relating to the elusive "holy grail" of a perfect Mandelbrot Set in 3D. As I understand it the problems with respect to getting "whipped cream" in non-linear pure-3D fractals is related to the facts that there are no complete algebraic fields in R3 and that (by Louiville's (conformality) theorem) conformal mappings are restricted to mobius transforms only.
I don't know why fractals in 3-space would need to be based on something so very analogous to the Mandelbrot or Julia sets in the plane. There are plenty of other mappings one might experiment with, like the real polynomial mappings (x,y,z) -> (P(x,y,z), Q(x,y,z)). I don't think a great deal is known about these even if P and Q are only quadratic polynomials. << With respect to Louiville's theorem: http://en.wikipedia.org/wiki/Liouville%27s_theorem_%28conformal_mappings%29 The proof shows that any Rn (n>2) mapping/transform that is not a mobius transform will not be conformal.
Just to be clear, conformal mappings R^n -> R^n already must satisfy the conclusions of this theorem even for n = 2. What makes this theorem of Liouville striking is that it's even true when the domain of the conformal mapping is any open *subset* of R^n. (Since, there is a huge profusion of conformal mappings U -> R^2 where U is a proper open subsets of R^2, but not for n > 2.) << My question (due to my own ignorance/lack of understanding) is: Does Louiville's theorem also discount the possibility of an Rn non-mobius mapping/transform being conformal in Rx space (x<n) if only Rx of the Rn space is considered ?
Yes, such possibilities are already excluded by the theorem, since [compositions of] translations, similarities, rotations on any affine subspace of R^x of R^n can be extended to the same type of composition on all of R^n. (Inversions would be excluded since they're not continuous on all of R^x.) << For example is it not possible that if we have Rn (n>3) then a non-Mobius mapping/transform in Rn could be such that all non-conformality is restricted to a particular Rx of Rn where x>=1 and x<n-2 ?
This is possible in many ways. E.g., (x,y,z) -> (x^2 - y^2, 2xy, z) is conformal except along the z-axis. (But isn't this the opposite of an example of your previous question?) --Dan Those who sleep faster get more rest.