Hi everyone. This mandelbulb is new and very interesting looking, I'll post the source as soon as it is cleaned up.

I've used an optimised mapping to fold the sphere around itself with a smooth tetrahedral folding. The results are a little approximate as I don't know the optimal mapping that minimises stretch, it would be great for everyone and anyone to help with finding this mapping. The better the mapping the more beautiful the result I'm sure, and the deeper we will be able to zoom.

The nature of the tetrahedral fold is such that it doesn't produce buds. Here's the reasoning that lead to this fractal, based on many discussions on these forums:

The formula that produces the 2d mandelbrot can be applied in different numbers of dimensions. R^2+C in 1d, Z^2+C in 2d, Q^2+C in 4d and a 3d equivalent of Q^2+C in 3d by keeping the w component at 0. However, you can't think of these as mandelbrot equivalents because the nature of the fractal (which is the border) changes. In 1d it is a disconnected dust of bifurcation points, in 2d it is connected and unsmooth, in 3d it is the lathed mandelbrot (circles map to their two finite-size 3d equivalents, spheres and toruses). In 4d the border is smoother still.

So it seems that in higher dimensions the squaring function just isn't enough to cover the space fully, and in 1 dimension the space isn't big enough for a connected fractal.

It follows that to create a 3d equivalent of the mandelbrot, it needs to use an operation that also covers that extra dimension. As the mandelbulb has shown, the way to do this is to quadruple cover the space rather than double-cover it as the mandelbrot does. So how do you quadruple-cover the 3d space? How do you fold it back onto itself?

The mandelbulb does this by doubling the longitude, latitude and squaring the altitude of the 3d space. This works but produces stretched areas that look like taffy.

The stretched areas exist because any small sphere in 3d space maps to an ellipsoid when the triplex squaring is applied, some areas creating more stretched ellipsoids than others. On the 2d mandelbrot small circles always map to circles.

So, for a true 3d Mandelbrot we would like no stretching. Unfortunately this theorem from 1850 tells us we cannot do that "

http://en.wikipedia.org/wiki/Liouville's_theorem_(conformal_mappings)". None of the few mappings that are conformal (without stretch) ever fold the 3d space into itself. So we could conclude that there is no 3d equivalent, but if you consider 2d mandelbrot as a fractal that is as conformal as possible in 2d, then the equivalent in 3d would exist as a folding that is as conformal as possible given the number of dimensions it is in.

The fractal here uses a tetrahedral folding of the sphere using a smooth mapping. It is quasi-conformal (there is an upper bound on the amount of stretch) and the average stretch is low. It is non-smooth at only 3 corners. All in all, it looks to be the most conformal form of quadruple cover that is possible.

It just needs the discovery of this optimally conformal mapping. That's the difficult part

It would look a little like L.P. Lee's mapping on this page:

http://www.progonos.com/furuti/MapProj/Dither/ProjPoly/projPoly.html