Inversive void-shells

The void-shell is one of the trickiest structural classes to generate using sphere inversion, being one of only five that are missing in the 7x7 classification of self-similar geometry here.

A void-shell is the boundary surface of a space-filling tree, so is a fractal with no volume. There are two simple ways to make this using linear transformations, the Menger-shell:

and the Viscek-shell:

(both my own naming convention as I don't know of any better names).

My first successful attempt at this class with sphere inversions is based around the fairly well-known square-symmetric Apollonian gasket:

We use this pattern on each each principle axis. The large white circles are mapped to the full disk such that the two inner contact points become vertical or horizontal opposites. This creates a sort of cocoon with the triangular hole seen in the centre of the below images. 

The four white circles remain as 3-plane shapes after being transformed to the full disk. The remaining asymmetrical concave quadrilateral black areas are Mobius transformed into the inner concave quadrilateral, and this is then treated as a single-plane structure like the one above.

A tree-solid is simply the set inversion of the above. However, inkeeping with having the structure within a sphere, we can do this by:

1. orthographically transform the unit sphere to an infinite plane

2. repeatedly scale down any points larger than the above 'tri-disk's radius.

3. using the single-plane structure, but one-sided at the start, so features are only on the inside of the sphere

The result is an inversive tree-solid: