Even though fractals have been around for half a century, they haven't been really accepted yet as fundamental objects, even though they are generally a better description of natural objects than the euclidean geometry from the ancient greek times.
So, while there is an enormous body of work that categorises shapes such as polyhedrons, it seems fractals haven't really been categorised at all, especially 3d fractals, at least I can't see evidence of this in the public domain.
So today I decided to start categorising. The easiest way to start is to take a 3x3x3 cube (i.e. made of 27 unit cubes) and enumerate the different fractals that you can make with it. For example, if you remove the middle cube and all the face cubes, recursively, then you get the menger sponge.
We simplify the task by reducing the cubes to remove to just the symmetric categories of the middle cube (m), the face cubes (f), the edge cubes (e) and the corner cubes (c), this also has the generalising property that these four categories represent the volume, surface, borders and corners of any polyhedron that might be used in a recursive fractal.
For each category of cube, I choose whether it is outsidethe set (0), recursion applied (1) or inside the set (2). Hence each fractal that I enumerate can be given a trinary code cefm = 1201 (from 0000 to 2222).
Lastly, I want to categorise the *type* of fractals that you can get out of this, not the specific shape. Something more like topology than geometry, but recursive topology.
So here are the results
Most of the names of the types I made up, and I still don't know how to categorise most of these fractals. I think it demonstrates how small our vocabulary is when it comes to even simple fractals like this. Compare it to the euclidean truncated-stellated-hemi-hexahedrons, inflated ellipsoids and non-rational-B-spline-patches etc, where the vocabulary is rich.
I also think there are some fractals in this set that I have never seen before. The branching surfaces are probably the ideal conductors of heat, but I have never seen the shape before. The variation of this #12 which is ridden with holes, and #15 are novel, well to me at least.
People are slowly realising that not only are fractals good approximations of nature but they are also optimal configurations, e.g. a tree shape is the most efficient way to transport water, a fractal coastline is the most optimal way to dissipate the destructive energy of the waves etc. Categorising and creating a vocabulary for 3d fractals seems really useful in this regard.