Fractal classification continued

[Tglad]

A while ago on this forum I was developing a sort of periodic table of fractals to classify them, because such a thing doesn't seem to exist.
You can group them by how they are made (IFS, escape-time, diamond-square etc) but that's just how they are made. The table I built was based on how they self connect, in 2d it boils down to combinations of fundamentals- cluster, tree, sponge (plus void and solid for completeness), shown here: https://sites.google.com/site/simplextable/what-is-the-simplex-table

It is a triangular table, but I think it can possibly be extended to a full table (see below)
I'm very unsure about the void row and the solid column but I think I can add 3 new fractal types for the 2d case. A tree cluster (a cluster of tree fractals), a sponge tree (a tree fractal with holes in it like a sponge), and a sponge cluster which is just a cluster of sponges (nesting of sponges is allowed).

This top triangle of the table is different from the bottom triangle, the bottom triangle fractals based on contact points unlike the top triangle.

Anyway, if anyone has some ideas or alternatives to this table let me know. I'm still trying to work out the top and right row and column.
The images in the table are just examples, e.g. a tree-tree (just tree for short) could be a Koch snowflake, a cloud, a real tree or a river, it doesn't matter, a tree is just defined as containing branches (however short) but not loops.

[lkmitch]

Since you're using the same terms along both axes, it's not clear to me what is being represented.  Also, it seems that the table should be symmetric across the diagonal, so that (void, solid) has the same picture as (solid, void).  If your classification is not symmetric, how does (x, y) differ from (y, x)?

[Tglad]

Yes it's a little confusing, it reads across then down (I capitalised the bottom labels to indicate that they go first) so a Cluster tree is 2 across and 3 down. The meaning is much the same as how a dragon fly is a type of fly with some characteristics of a dragon rather than a type of dragon. So a Cluster tree is a tree, its structure splits into branches, containing no loops, no internal air pockets etc, but it is like a cluster, in fact if you removed any width from the border it would be a cluster. This rule is the same for all the classes in the table.


Conversely a Tree cluster would be a type of cluster (separate objects with a power law distribution of sizes), with tree characteristics, so each object is a tree.
I think I have worked out what the full table should be (attached), the Solid column and the void row were the tricky ones, I'm thinking that they are basically the same as the diagonal, i.e. the 5 main fractals (void, cluster, tree, sponge, solid), since for example a Cluster cluster (cluster of clusters) is the same as a Solid cluster (cluster of solid shapes) since a cluster is itself a recursive definition so cluster = cluster cluster = cluster cluster cluster ...
the Void blah row can be extracted since the whole table is anti symmetric in the sense that if you flip it around the north-east axis you get the colour inverse of the same table (solid and void parts exchange).
This all leads to a problem, what is a Solid void? The example is a half-plane of solid, but this is just one example, every fractal in this table can be transformed by any Mobius transform and remain in the same table cell. So the Solid void could be just a solid blob, or an empty blob gap in an infinite solid, or an infinite half plane.