basic inversive tree-shell

The tree-shell is probably the hardest structure to make with inversive geometry because it requires the structure to 'conspire' to meet along a line or curve. Other structures such as cluster-trees and tree-sponges only require single points to meet.

Ideally I'd like to find one that is nowhere differentiable, in particular that every surface patch contains approximations of the whole structure under conformal transformations. 

However due to the difficulty in this post I'll make one that is mainly smooth spherical surfaces. Nevertheless, it makes for a nice looking shape.

It is based on the 2D tree-solid from a previous post, but extended to 3D, and looks like this:

The 8 legs is arbitrary, as long as it is even. You can go higher like 10 here, but start to get a little self-intersection:

The method is a replacement fractal under Mobius transformations. There are three types of structure being used. Each type is constructed from a combination of the  types as shown:

 

Type 0:

Type 1:

Type 2:

It is a tree-shell because it would normally be a tree of recursive hemispheres, but the hemispheres meet along edges as seen between the two types on the right side of the above pictures. This forms a water-tight basin, at multiple scales, which makes it a shell.

Apart from the making a nowhere differentiable tree-shell (which would have a very different construction) I think there is probably a way to improve the above structure...

Notice the type 2 shape in the bottom image, the rightmost limb is significantly bigger than the limb next to it. That's because we use only three types, one meets the parent sphere at an angle of zero (the big one) and the others meet at an angle of 45 degrees.

I think it may be possible to interpolate between the small limb (left) and the big limb (right) so the meeting angle gradually drops to zero. 

Shader: https://www.shadertoy.com/view/tcycWD

Another approach is based on bridging the gap between two kissing spheres rather than the gap between two spheres that intersect at right angles. We can do it in a slightly more traditional way, placing unit inversion spheres on the edge centres of the unit cube, and radius 2 inversion spheres at the face centres of the unit cube.

It is still too lumpy to properly count as a shell, and as you can see, it still isn't making all of the surface recursive. Lastly, there is a slight problem at the circled areas, which should probably just be removed:

Removing four of the edges is smoother but nicer in terms of no real protrustions, and I removed the broken bit:

I think it still counts as a shell because the indentations do split into separate ones.