* Tom Karzes <karzes@sonic.net> [Aug 11. 2011 07:51]:
Me neither. There is, however, a very natural 4-dimensional set that combines both the standard Mandelbrot set and all of its corresponding Julia sets into a single entity.
Let one plane determine c0, and an orthogonal plane determine c1. The product is 4-dimensional, and the iteration is:
z(0) = c0 z(i) = z(i-1)^2 + c1
The planar slice obtained by setting c0 to 0 and varying c1 is the Mandelbrot set. The planar slice obtained by setting c1 to some constant and varying c0 is the Julia set corresponding to c1.
[...]
Neat! Back in 1993 (or so) I used quaternions, only to become slightly diappointed because all cut planes containing the real axis produce the usual image 8-) The other cuts looked "random" to me, and that is what I observe with most "new" attempts. It might be more interesting to visualize the Julia sets. Maybe some live in a proper 3D subspace and no projection or cut would be required. Another thing is to look at finite fields. Of course, nothing escapes anymore, so one may rather look at the decomposition into orbits of different size. Take GF(q^2) to get 2D right from the start, q being a prime, or maybe a prime power (dunno whether GF(2^(2*k)) could possibly give something nice). My attempts of this back then, again, produced rather "random" images. However I did know too little about the (well known) theorems of elementary field theory.