That’s when I understood that spatial structure is a Deep Fundamental Theory.
And it doesn’t stop there. The same thing explains the structure of our roadways, blood vessels, telecomm networks, and even why the first order differential equations for electric currents, masses on springs, and water in pipes are the same.
(The exact deep structure of physical space which explains all of these is differential topology, which I think is what Vaniver was gesturing towards with “geometry except for the parallel postulate”.)
Can you go into more detail here? I have done a decent amount of maths but always had trouble in physics due to my lack of physical intuition, so it might be completely obvious but I’m not clear about what is “that same thing” or how it explains all your examples? Is it about shortest path? What aspect of differential topology (a really large field) captures it?
(Maybe you literally can’t explain it to me without me seeing the deep theory, which would be frustrating, but I’d want to know if that was the case. )
There’s more than just differential topology going on, but it’s the thing that unifies it all. You can think of differential topology as being about spaces you can divide into cells, and the boundaries of those cells. Conservation laws are naturally expressed here as constraints that the net flow across the boundary must be zero. This makes conserved quantities into resources, for which the use of is convergently minimized. Minimal structures with certain constraints are thus led to forming the same network-like shapes, obeying the same sorts of laws. (See chapter 3 of Grady’s Discrete Calculus for details of how this works in the electric circuit case.)
Can you go into more detail here? I have done a decent amount of maths but always had trouble in physics due to my lack of physical intuition, so it might be completely obvious but I’m not clear about what is “that same thing” or how it explains all your examples? Is it about shortest path? What aspect of differential topology (a really large field) captures it?
(Maybe you literally can’t explain it to me without me seeing the deep theory, which would be frustrating, but I’d want to know if that was the case. )
There’s more than just differential topology going on, but it’s the thing that unifies it all. You can think of differential topology as being about spaces you can divide into cells, and the boundaries of those cells. Conservation laws are naturally expressed here as constraints that the net flow across the boundary must be zero. This makes conserved quantities into resources, for which the use of is convergently minimized. Minimal structures with certain constraints are thus led to forming the same network-like shapes, obeying the same sorts of laws. (See chapter 3 of Grady’s Discrete Calculus for details of how this works in the electric circuit case.)