self-similar cross sections by class

What are the classes of cross sections through self-similar objects, according to my self-similar shape classification? Let's look at the 2D self-similar structures first:

2D

These are approximate, I will try to update these tables when I get more time and find mistakes. They are the 'almost surely' cross section structures. For instance a cross section through a 2D void-cluster almost surely will not hit any points in 2D so will be a completely empty void-void structure. Likewise, a cross-section through a cluster-cluster will almost surely cut through infinitely many of the blobs (over all cluster-clusters, not just a geometrically aligned one), making it also a cluster-cluster in 1D.

3D

In 3D things get more difficult so expect errors in this table. The main difficulty is that we get shapes that are combinations of more than two structural types. For instance a cross section through a sponge-sponge will definitely have separated objects (a cluster) but the objects will also have loops (sponge) when the cross section aligns with the loop, and hierarchical protrusions (tree) when we capture just the start of these loops in the cross-section. So it has the characteristics of all three. 

Typically this is written from the denser to the sparser type, in this case a sponge-tree-cluster. This is either interpreted as sponge-tree cluster: a cluster of sponge-trees (trees with holes in them in 2D); or as a sponge tree-cluster: a cluster of trees, but made of sponge (holes in it). These are the same thing, suggesting that A-B-C classes are associative. i.e. (A-B)-C = A-(B-C).

There is less obvious a pattern than in the 2D case, but it does have some consistency. The first word (the qualifier) is void, cluster, tree, sponge, sponge, sponge, solid from left to right and the last word (the noun) is void, cluster, cluster, cluster, tree, sponge, solid from top to bottom. 

These have both dealt with the two missing classes (shell & foam) at different ends. The first by repeating the second-last type (sponge) and the second by repeating the second type (cluster).

In order to accomodate the middle terms, there is a simpler way to work this out. Firstly, we look at the A-A diagonals. This has a pattern to it. 

Then all of the other cells of the form A-B are just the cross section result for A-A followed by the corss section result for B-B. There is some reduction going on. A tree-cluster-cluster will reduce to a tree-cluster and a tree-cluster-tree-cluster will reduce to a tree-cluster. Otherwise the cross section of the non-diagonal cells are just the piecewise combination of the cross section of the corresponding diagonal cells.

As for the A-A diagonals, they follow this pattern:

Numerically (with void-solid being 1-5) it is: 1-1, 2-2, 3-2, 4-3-2, 4-3, 4-4, 5-5. In 2D the pattern is 1-1, 2-2, 2-2, 2-2, 3-3. In general the n-dimensional cross section of the n+1 dimensional structure is:

with repeated terms removed in the triplet. Then X(A,B) is just X(A)-X(B) with appropriate reduction of repeated terms. '-' being a hyphen here.

The use of these tables is that you can guess the class of a higher dimensional structure just from its cross section. The most common case of a cross section is when the structure is n+1D so is a moving structure (a behaviour), and you have access to still images of it. So the above table can suggest the 2+1D structure from its 2D structure at any one time.

To guess a 3+1D structure we would need a 4D table.

My guess as to the A-A pattern would be: 1-1, 2-2, 3-2, 4-3-2, 5-4-3, 6-5-4, 6-5, 6-6, 7-7.

So the cross section of a shell-shell (number 5-5) would be: 5-4-3 which is a shell-sponge-tree. 

tables