Void Cluster

Fractal dust AKA a void-cluster is fairly easy to generate, for example the 3D Cantor dust. Here I continue working on developing less rectilinear examples by using sphere inversion.

The method I use is very similar to the previous Sphere Cluster in that it uses the dodecahedral or icosahedral symmetries together with the offset sphere inversion that typically generates the hyperbolic tesselation of the above polyhedrons.

The difference is with this void-cluster we do not apply the centralised sphere inversion when recursing into a sub-cluster, and we rescale the cluster from rk up to 1 each iteration unconditionally.

The result is an icosahedral/dodecahedral set of sub-clusters, together with the rk sized subcluster in the centre. Here for k=0.9:

Above left: icosahedral and right dodecahedral void-clusters

Above: same but alternating between the two types 

Here are the same variations for k=1 from an approximately symmetrical view angle:

The problem with these clusters is only that they don't seem to be a proper generalisation of the Sphere Cluster, becaue they aren't based on an inversion of a regular polyhedral tesselation.

Any hyperbolic tesselation when inverted will give a central sphere of finite size, leading to the Sphere Cluster when applied recursively. If you think of the reducing radius central sphere as equivalent to a growing radius hyperbolic disk, then the limit would be an infinite radius (and zero curvature) hyperbolic disk which is flat Euclidan space. The only regular polyhedral teselation of Euclidean space is cubic. 

So we invert the space, then pull all cubes in the tesselation into the central cube. This is the equivalent of the multiple dot products and offset sphere inversions in the Sphere Cluster. Then shrink the space by k and recurse.

The result for k from 0.4 up to 1.6 is:

 The significant points along the way are k=1 where it goes from a void-cluster to a void-sponge:

and k=2 where (despite rendering artifacts) it becomes a void-foam:

(slice-through visible).

This is a very interesting family of shapes as it covers three fractal classes, and it seems to be a generalisation of the Sphere Cluster (which itself seems to cover cluster, cluster-sponge and possible cluster-foam classes). But more interesting is that even the finest k=0.4 case seems like its fractal dimension would be 3, due to being the inversion of a 3D (cubic) lattice.

Here's a higher detail of k=0.5: