Structure

This morning I’ve been pondering two senses of the word structure. As a chemist, my first instinct is to think about molecular structure – how atoms are connected to each other within a molecule. A dictum of chemists is ‘structure determines function’, i.e., when you know the structure of a molecule you can start to predict how it will behave or react with different molecules. In this sense, structure refers to the physicality of the molecules.

However, there is another sense in which structure is used. Structure can refer to how things are ordered. At first glance this sounds similar to the “how atoms are arranged” definition above, but now I’m thinking about organizational structure. If you’re joining a new company, you should care about the ‘org chart’. It helps you figure out who you need to talk to when you need to get certain things done. Structure in this sense is not the physicality of any object, but the functional relationship between these objects.

I’m interested in how complex systems arise from seemingly ‘simpler’ systems. However, it isn’t clear how one measures complexity. What makes something simple versus complex? If I say “that’s a complex molecule”, I might be thinking of structure in the first sense. If I say, “that’s a complex mixture”, I might be thinking of structure in the second sense. Is there a third sense? Can these be quantified on a similar or related scale? If there is a scale, it’s unlikely to be linear. Different order parameters will come into play at different length and time scales.

If complexity resides somewhere between order and chaos, why is it observed? Or is it observed only because we the observers are coarse-graining through an anthropocentric lens? The world we live in, possibly the pocket of the universe we live in, is strange. It seems to have started in a low entropy state, but then we get to see all sorts of interesting things happen as the entropy increases. For some reason, en route to overall higher entropy, mini-pockets of low entropy structures (in both senses) show up. How do these structures persist? They could either change very slowly (first sense of structure). Or they could replicate quickly in a cycle of destruction and creation (second sense of structure).

I recently read a physicist’s dictum that “order can only arise from more order”. When we see mini-pockets of seemingly low entropy order arise, it is because fundamentally there is higher order behind it. Symmetry breaking is the rule as events progress through time. A perfect sphere seems simple because it possesses an infinite symmetry – our minds somehow grasp this abstraction. Lower symmetry polyhedral on the other hand have finite symmetry operations. Why do we instinctively think the lower symmetry cube is simpler than an octahedron or a dodecahedron? It could just as well be the other way around. Methane (CH₄) is more symmetric than Water (H₂O) and therefore more sphere-like. And what happens when thousands of such molecules interact with each other? Structure is more complicated than it seems.