Summary: I didn't find any holy grails.

Details: A few people have mentioned trying toroidal coordinates, so I did some experiments and here's what I got. The coordinate systems used in the images below are obtained by rotating polar coordinates about the z-axis. (The usual mathematical toroidal coordinate system is obtained by rotating *bipolar* coordinates about the z-axis, but I haven't tried that.) The coordinates I used are (r, theta, phi), where r is the distance from the center of the tube, theta is the angle measured (from a reference angle gamma) around the tube, and phi is the angle measured along the tube. I used 3 parameters (alpha, beta, gamma) to parametrize the coordinate system. Alpha is the distance from the origin to the center of the tube, beta is the radius of the tube at which "squaring" r leaves points unchanged, and gamma specifies the reference origin used when doubling theta. Squaring a point consists of doubling theta and phi, and squaring and scaling r such that points beta units from the center of the tube stay at distance beta. Points inside the tube move farther inside, and points outside move farther outside. Here's a picture:


I experimented with parameters ranging from the degenerate case of a sphere (alpha = 0, beta = 1), a spindle torus (alpha = 1/4, beta = 3/4), a horn torus (alpha = 1/2, beta = 1/2), and a ring torus (alpha = 3/4, beta = 1/4). See http://en.wikipedia.org/wiki/Torus for a description of these types of tori. In the degenerate case of a sphere, these coordinates are equivalent to spherical coordinates, and we get the regular Mandelbulb fractal. As beta decreases (and our torus gets "thinner"), the fractal tends to thin out and get wispier, but I don't think I see anything new and exciting happening.

Here are degree 2 examples:


Here are degree 7 examples:

Actually, this is quite nice

I only wrote the equations for transforming (x, y, z) <-> (r, theta, phi) above. I left out the "Mandelbrot" part of the formula... In between the two equations above should go this step, which raises to the nth power:

r := r^n / beta^(n-1)
theta := n * theta
phi := n * phi

And then as the very last step, add the Mandelbrot constant:

x := x + a
y := y + b
z := z + c

Great idea! I like these a little better than the ones up above:


A zoom into the last one looks like this:

The others should be zoomed, too!