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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  :D  It would look a little like L.P. Lee's mapping on this page: http://www.progonos.com/furuti/MapProj/Dither/ProjPoly/projPoly.html

It actually looks like a wild but beautiful mix of 2D-Mandelbrot and some kind of truncated sierpinski-sponge....

And it's interesting to see that the "flat" sides seem to feature a 5-rotational symmetry.
I didn't expect that from thetrahedrons but rather from dodecahedrons and their platonic solid duals....

Very nice :)
It actually does contain buds but 2D-ones....

Bah! It turns out I made a mistake in the +c code, and the actual result looks like below. Which personally looks much less interesting to me.
The previous shape was due to a rotation of 180 degrees after the very first iteration.

Anyway, after all that, I'm wondering whether there is a different approach to making the 3d mandelbrot equivalent, instead of just maximising conformality.
The amazing thing about the mandelbrot is that you can mutate it each iteration and still get an exact mandelbrot; you can add a random complex number each iteration, you can scale Z by any value each iteration, and you can rotate Z by any number of degrees each iteration and you still get a mandelbrot.
The standard mandelbulb formula sort of shares this property, you can add any vector, scale by any value and rotate around the j axis by any value each iteration and you get the same result. But it isn't the only mapping that does this in 3d.
Since there are many mappings that do this in 3d, a way to get the unique one might be to add an extra constraint, perhaps that you can rotate it around the i axis also and it gives the same shape, if that is possible then it would be more universal. So maybe the measure of the shape is how universal it is.

this even more looks like a sierpinski mandelbulb :)

And a VERY interesting property of the Mset in 3D is that the most interesting (artistic) results yet all where found by errors.

Maybe we could try to generalize the formula to include several variables which would usually be modified by accident if you have a typo (+ - and * / mixup, forget a constant....)

The original post tetrahedral mandelbulb is indeed still a fractal, my error means that it just has a slightly more complicated formula. Instead of V^2 + C where ^2 is the smooth tetrahedral fold, it is (V+C')^2 + C-C' where C' is the conjugate of C (negate i & j). It seems nicer to me, though any variation that uses this tetrahedral folding tends to look interesting because there is very little stretch.
Here's a closer up pic of where the sierpinski pyramid bit connects to the mandelbrot bit.

In fact, since ' is actually a conformal rotation around r, something that doesn't exist in Z you could think of ' as having no effect on Z, just as the conjugate operation has no effect in R. So this fractal could be said to reduce to the mandelbrot in 2d with the right interpretations.
Anyway, code posted here "http://www.fractalforums.com/3d-fractal-generation/a-new-3d-mandelbrot-like-fractal/15/?action=post;quote=11640;num_replies=16;sesc=aee003da38f141169bfe35f4f08fee04"
Here are a few close up shots, click below them for bigger image.

It's interesting, how this set looks like either mountains/cliffs or skin :)

Very nice :)

I improved the accuracy of the mapping so I think it is now pretty close to optimal for a spherical mapping. Unfortunately it did all the wrong things
- the mandelbrot part became less mandelbrot-like
- the sierpinski pyramid became less sierpinski pyramid-like
- the whispy floating cobwebs below became stronger rather than fading away

On the positive side, it still produces some very nice craggy cliffs, and unusual patterns. And it may be possible to optimise a little further by having a less spherical mapping.

Also, surprisingly, the optimised mapping turns the 2nd fractal in this post into the same as the alien one in my other post.