Rules for Inversive Classes

This classification of recursive shapes can be applied to inversive geometry in 3D, which is geometry that is symmetric to inversions around various 'generating spheres'. It is a 7x7 table that looks like this:

How we go from generating spheres to the class above is what I describe in my new paper preprint. It is a connection through about 4 layers of abstraction. 

Firstly, we have the generating spheres. The inversive limit sets are generated by repeatedly inverting points within these spheres such that the sphere surface remains unchanged. 

Secondly we have the connectivity graph, which represents many sets of generating spheres that all have the same connectivity. The nodes are the spheres and the edge numbers represent the 'branch order' n of the two intersecting surfaces. The dihedral angle of intersection is pi/n.

Thirdly we have the hypergraph, which represents many connectivity graphs with the same broad limit-set connectivity. In this diagram the branch orders are not important, just the presence of not of pairwise sphere intersections (an edge), triplet-wise sphere intersections (a semi-translucent face) and quad-wise sphere intersections (a solid volume). 

Fourthly and lastly we have an abstract hypergraph, which represents many types of hypergraph that all map to the same structural class in the 7x7 table. This structure is a schema for generating hypergraphs. The key component is the polyhedral sphere, which represents any polyhedron in the hypergraph that is intersected by a single orthogonal sphere. 

In the above abstract hypergraph, the red polyhedral sphere is the parent and the single curved edge represents one or many child edges in the hypergraph. In the hypergraph above it, there are four such edges and the spherical polyhedron is an irregular tetrahedron.

This particular abstract hypergraphs generates cluster-sponges. These are sponges that would be clusters if boundary points were removed. 

You can see this limit set in the table at the top of the page. These can be viewed in realtime here.

So this new paper is an extension of the previous paper but rather than creating a single family of limit sets for the five main diagonal classes, it is a method for generating a broad family of limit sets for all 49 classes.

Well... nearly, you can see in the top table that five limit sets are missing. These are easy to generate with linear limit-sets (see them rendered here) but hard to find for inversive limit sets. It is not certain whether they even can be generated, at least with simple sphere inversions. Working out whether these last 5 classes is possible with sphere inversions is the last major open question for this structural classification.