Thursday, September 11, 2008

Hierarchy and Nested Structure

I've recently been thinking a lot about the organization of structures in 3-dimensional space, and I really would like to take a class on topology.

Anyway, with regard to hierarchy, it seems to me as if there is a qualitative distinction between two classes of hierarchical structures, but maybe the distinction isn't really there at all...I don't know.

Consider hierarchical structures like a tree or a ziggurat:

If you were going to write instructions about how to build such structures, how would you do it? They're constructed by using a root node, like the trunk or the base, and by adding smaller child elements of a similar type to the parent node. Then you just repeat that step a given number of times.

For example, you could say:

1) Start with a cylinder
2) Add two cylinders 50% the size of the original on one end, each facing 45 degrees in opposite directions
3) Repeat 10 times

That would give you a tree structure. You could do the same thing with boxes in order to build a pyramid/ziggurat. If you think about your body, many aspects of it are constructed this way. Your limbs, hands, and fingers are essentially tree structures.

Now consider the following hierarchical structures, such as nested Russian dolls or an onion:

These structures are hierarchical as well, aren't they? But aren't they different from the previous examples?

How would you write instructions for how to build a set of nested dolls? You could go from the top down or the bottom up, but either way, the difference with the previous examples is that each element completely contains its child element. Rather than the child elements being added externally, as in a tree, they're added internally. This is more intuitive if you think of cellular structure. An organ acts as a container for millions of cells, each of which acts as a container for numbers of sub-cellular components.

So some aspects of your morphology are like a tree structure, where elements are oriented relative to the external surface of their parent elements which act as substrates for child elements. Other aspects of your body are like nested dolls, where parents serve as containers to child elements.

I got this insight when thinking about the organization of elements in the neocortex. The neocortex is one big, flat sheet. But it contains many regions, which in turn contain subregions, which contain columns, which contain minicolumns, which contain neurons. So it is essentially a nested hierarchy. But neurons themselves act as substrates for their axons and dendrites, which are essentially tree structures. But neurons are also containers for sub-cellular structures, like the tiny sacs that contain neurotransmitters.

It seems to me that any system for specifying 3D structures that incorporates both kind of hierarchy can efficiently represent any arbitrary structure that has a high degree of hierarchy, such as biological organisms.

1 comment:

Kenny Wyland said...

I'm enjoying this whole hierarchy quest that you are on. It stimulates my own brain hierarchy.

When it comes to the onion and the nesting dolls, my gut doesn't think hierarchy, it thinks layers. I wouldn't call it a hierarchical structure, I would call it a layered structure. I still feel that a hierarchical structure has to provide the option to have elements at the same level that are completely unrelated except for a (potentially distant) ancestor.