Inversive cluster-shells

Here is a cluster-shell:

to tell that is is made from a cluster (of balls) we can take a cross section through it:

here it is from a more top-down angle:

This one is a bit of a complicated shape to engineer, it has five steps. Step 1, make an airtight strip of spheres:

Then add a strip at half that scale below it, and half again below that etc, making the resulting strip twice as deep:

Now do the same stacking but on a 2D grid of these strips instead, with the big spheres at the grid vertices:

It is now already a cluster-shell, but it has smooth areas and it is unbounded, so we can do better.

Next stereographically project this onto a sphere:

Lastly replace each sphere in the above with this shape, as shown in the first image. Here's a last pic from a more side on angle:

The inversive tree-foam is just the set complement of this. Here I have applied a sphere inversion around the largest inner ball to make it bounded in a sphere, and I'm rendering it backwards (with transparent and solid swapped) so you can see the inner structure:

------------ OLD --------------------

The below almost works but doesn't quite work, but I'm leaving in here for reference:

Cluster-shells are another structural class that are hard to generate with inversive geometry. They therefore are one of the five unspecified classes in this description of all 49 classes of the 7x7 structural classification of self-similar shapes in 3D.

One way that I *think* counts as a cluster-shell is shown here:

It is built in a similar way to the cluster-tree. It starts with a Ford sphere packing, using the triangular lattice version, here the spheres are shown at half size:

At full size they all are packed in so no sphere can move:

Between each touching pair of spheres we can make a watertight concave triangle bridging them. This is true for the underneath spheres too:

This is a cluster-foam as the lower 'triangular mesh' of spheres butt up against the big spheres. We can now stereographically project this infinite mesh of spheres onto the (Riemann) sphere:

Much like the sphere tree (cluster-tree) structure, we can then recursively replace each sphere with this above shape (ignore the thin gaps):

The problem with this is that it remains a cluster-foam as the recursed sphere is aligned with the small underneath sphere mesh, so it has millions of air pockets.

Instead we can turn the sphere shape upside down every time we recurse each sphere. Giving the resulting shape, the (probable) cluster-shell:

A slice-through shows the solid spheres from which is it made:

Unforunately this isn't a true cluster shell as some of the intermediate spheres don't match together properly.